## Highlights - Anyhow, we acquire the right to use without explanations terms like “Hausdorff space,” or “compact space,” or “countable base space,” and so on, and also use \(explicitly or implicitly\) facts like “a bijective continuous map of a compact space onto a Hausdorff space is a homeomorphism,” or “a compact subset of a Hausdorff space is closed\.” (Anatoly Fomenko 14) - The symbol R1 always means the union \(inductive limit\) of the chain R1 R2 R3 : : : ; thus, R1 is the set of sequences \.x1 ; x2 ; x3 ; : : : / (Anatoly Fomenko 14) - Show that none of the spaces R1 ; S1 ; D1 is metrizable\. (Anatoly Fomenko 15) - There are other definitions of R1 in the literature\. For example, \(1\) the “Hilbert space” e” P ` for which the series 2 2 is the set of all real sequences \.x1 ; x2 ; x3 ; : : : / xi converges;Pthe topology is defined by the metric d2 \.\.x1 ; x2 ; x3 ; : : : /; \.y1 ; y2 ; y3 ; : : : // D \.yi xi /2 ; \(2\) the Tychonoff space T is the set of all real sequences \.x1 ; x2 ; x3 ; : : : / with the base of topology formed by the sets f\.x1 ; x2 ; x3 ; : : : / 2 T j \.x1 ; : : : ; xn / 2 Ug for all n and all open U Rn [a sequence Xi D \.xi1 ; xi2 ; xi3 ; : : : / in T converges to X D \.x1 ; x2 ; x3 ; : : : / 2 T if and only if limi!1 xin D xn for every n]\. (Anatoly Fomenko 15) - Which of the inclusion maps R1 ! `2 ; R1 ! T; `2 ! T \(if any\) are continuous? Which of them \(if any\) are homeomorphisms (Anatoly Fomenko 15) - EXERCISE 6\. Prove that the real projective line is homeomorphic to the circle S1 \. (Anatoly Fomenko 15) - The points with xi ¤ 0 form the ith principal affine chart\. The correspondence \.x0 W x1 W W x n / $ \.x0 =xi ; : : : ; xi 1 =xi :xiC1 =xi ; : : : ; xn =xi / yields a homeomorphism of the affine chart onto Rn and equips the former with coordinates\. (Anatoly Fomenko 15) - If we assign to a point of Sn RnC1 a line passing through this point and the origin, we get a continuous map Sn ! RPn (Anatoly Fomenko 16) - Thus, RPn may also be obtained from Dn by identifying all pairs of opposite points on the boundary sphere (Anatoly Fomenko 16) - Like RPn , the space CPn is covered by n C 1 affine charts (Anatoly Fomenko 16) - We call a topology weaker if it has more open sets, that is, fewer limit points \(for us, the weakest topology is the discrete topology\)\. Informally speaking, we call a topology weak if the attraction forces between the points are weak\. The opposite terminology considers points as repelling each other; from this point of view, the discrete topology is the strongest\. (Anatoly Fomenko 16) - EXERCISE 7\. Prove that projective lines CP1 and HP1 are homeomorphic, respec2 and S4 tively, to S2 and S4 \. (Anatoly Fomenko 17) - We get a definition of a quaternionic projective space HPn \. One should notice, however, that , because of noncommutativity of the algebra H, one has to distinguish between left and right lines\. (Anatoly Fomenko 17) - but even such an exotic object as the Cayley projective plane are very important for topology\. (Anatoly Fomenko 17) - Suppose that in some space Rn two operations are defined: multiplication, a; b 7! ab, and conjugation, a 7! a\. Then we define similar operations in R2n D Rn Rn by the formulas \.a; b/ \.c; d/ D \.ac bd; bc C ad/; \.a; b/ D \.a; b/: Starting from the usual multiplication and identical conjugation \.a D a/ in R 1 D R, we get \(bilinear\) multiplications and conjugations in R2 ; R4 ; R8 ; R16 ; : : : (Anatoly Fomenko 17) - The multiplication in R4 is the multiplication of quaternions\. It is bilinear, associative, and admits a unique division \(that is, the equation ax D b has a unique solution if a ¤ 0\) but is not commutative\. The multiplication in R8 is still worse: Not only it is not commutative, but also it is not associative [although the associativity relations involving only two letters, such as \.ab/a D a\.ba/; \.ab/b D ab2 ; \.ab/a 1 D a\.ba 1 /, etc\., hold]\. Still this multiplication possesses a unique division\. The algebra R 8 with this multiplication is called the Cayley algebra or octonion algebra and is denoted as Ca\. (Anatoly Fomenko 17) - Much later in this book we will consider \(and prove\) the famous Frobenius conjecture: If the space Rn possesses a bilinear multiplication with a unique division, then n D 1; 2; 4, or 8\. (Anatoly Fomenko 17) - there are, for example, nonassociative bilinear multiplications with a unique division in R4 (Anatoly Fomenko 17) - The nonassociativity of the Cayley multiplication impedes defining any lines in the space Can with n 3\. (Anatoly Fomenko 17) - defined as the space of all k-dimensional subspaces of the space Rn \.2 The topology in G\.n; k/ may be described as induced by the embedding G\.n; k/ ! End\.Rn / which assigns to a P 2 G\.n; k/ the orthogonal projection Rn ! P combined with the inclusion map P ! Rn ; a more convenient description of the same topology arises from a realization of G\.n; k/ as a subspace of a projective space (Anatoly Fomenko 18) - A real Grassmann manifold G\.n; k/ is defined as the space of all k-dimensional subspaces of the space Rn \.2 The topology (Anatoly Fomenko 18) - The numbers j1 ;:::;jk \.P/ are called Plücker coordinates of P (Anatoly Fomenko 19) - There are obvious complex and quaternionic versions of the equalities G\.n; k/ D G\.n; n k/; G\.n; 1/ D RPn 1 ; also, GC \.n; k/ D GC \.n; n k/ and GC \.n; 1/ D Sn 1 \. The embeddings G\.n; k/ ! G\.n C 1; k/ and G\.n; k/ ! G\.n C 1; k C 1/ also have complex, quaternionic, and oriented analogs\. (Anatoly Fomenko 19) - A flag of type \.k1 ; : : : ; ks / in Rn is a chain V1 Vs of subspaces of the space Rn such that dim Vi D ki \. (Anatoly Fomenko 20) - V1 Vs of subspaces of the space Rn such that dim Vi D ki \. The set of flags has a natural topology [for example, as a subset of G\.n; k1 / G\.n; ks /] and becomes a “flag manifold” F\.nI k1 ; : : : ; ks /\. (Anatoly Fomenko 20) - The spaces F\.nI 1; 2; : : : ; n 1/; CF\.nI 1; 2; : : : ; n 1/; HF\.nI 1; 2; : : : ; n 1/; and FC \.nI 1; 2; : : : ; n 1/ are called \(understandably\) manifolds of full flags\. (Anatoly Fomenko 20) - The compact classical groups include the group O\.n/ of orthogonal n n matrices, the group U\.n/ of unitary n n matrices, the groups SO\.n/ and SU\.n/ of matrices from O\.n/ and U\.n/ with determinant 1, and the group Sp\.n/ of quaternionic matrices of unitary transformations of Hn \. (Anatoly Fomenko 20) - tice that the group SO\.2/ of rotations of the plane around the origin is homeomorphic to a circle\. The group SO\.3/ is homeomorphic to RP3 ; (Anatoly Fomenko 20) - ; the homeomorphism assigns to counterclockwise rotation by an angle ˛ of R3 around an oriented axis ` a point of ` at the distance ˛= from the origin \(in the positive direction\)\. Since the rotation by the angle around an oriented axis is not different from the rotation by the angle around the same axis with the opposite orientation, the image of this map is the unit ball in R3 with the opposite points on the boundary identifie (Anatoly Fomenko 20) - ely imaginary quaternions where 0 ¤ q The group SU\.2/ is homeomorphic to S3 (Anatoly Fomenko 20) - Finally, the groups U\.1/ and Sp\.1/, which are isomorphic, respectively, to the groups SO\.2/ and SU\.2/, are homeomorphic to S 1 and S 3 \. (Anatoly Fomenko 20) - The spaces of orthonormal k-frames in Rn \(topologized as a subset of Rn Rn \) is called the Stiefel manifold and is denoted as V\.n; k/\. (Anatoly Fomenko 21) - Stiefel manifolds generalize classical groups: V\.n; n/ D O\.n/; CV\.n; n/ D U\.n/; HV\.n; n/ D Sp\.n/; V\.n; n 1/ D SO\.n/; CV\.n; n 1/ D SU\.n (Anatoly Fomenko 21) - Notice also that V\.n; 1/ D Sn 1 ; CV\.n; 1/ D S 2n 1 ; HV\.n; 1/ D S4n 1 \. (Anatoly Fomenko 21) - The action of the group O\.n/ in Rn gives rise to its actions in Sn 1 ; Dn ; G\.n; k/; and V\.n; k/\. The subgroup SO\.n/ of O\.n/ acts also in GC \.n; k/\. There is also an action of O\.k/ in V\.n; k/: The matrices from O\.k/ are applied to the vectors of the frame\. (Anatoly Fomenko 21) - Thus, almost all classical spaces described above are homogeneous spaces of compact classical groups; that is, they can be described as quotient spaces of these groups over some subgroups\. Here are these descriptions: (Anatoly Fomenko 21) - S n 1 D O\.n/=O\.n 1/ D SO\.n/=SO\.n 1/I S 2n 1 D U\.n/=U\.n 1/ D SU\.n/=SU\.n 1/I S 4n 1 D Sp\.n/=Sp\.n 1/I G\.n; k/ D O\.n/=O\.k/ O\.n k/I CG\.n; k/ D U\.n/=U\.k/ U\.n k/I HG\.n; k/ D Sp\.n/=Sp\.k/ Sp\.n k/I GC \.n; k/ D SO\.n/=SO\.k/ SO\.n k/I V\.n; k/ D O\.n/=O\.n k/ D \.if n > k/ SO\.n/=SO\.n k/I CV\.n; k/ D U\.n/=U\.n k/ D \.if n > k/ SU\.n/=SU\.n k/I HV\.n; k/ D Sp\.n/=Sp\.n k/: (Anatoly Fomenko 21) - In particular, the manifold of full flags, CF\.nI 1; 2; : : : ; n 1/, is the quotient space of the group U\.n/ over its “maximal torus” U\.1/ U\.1/\. (Anatoly Fomenko 21) - The action of the group O\.k/ in V\.n; k/, as well as its complex and quaternionic analogs, are free\. With respect to these actions, V\.n; k/=O\.k/ D G\.n; k/; V\.n; k/=SO\.k/ D GC \.n; k/; CV\.n; k/=U\.k/ D CG\.n; k/; HV\.n; k/=Sp\.k/ D HG\.n; k/: (Anatoly Fomenko 23) - EXERCISE 9\. Prove that the projective plane with one hole is homeomorphic to the Möbius band \(thus, the Möbius band is a “classical surface”\)\. (Anatoly Fomenko 23) - can (Anatoly Fomenko 24) - EXERCISE 21\. Prove that the subset of the complex projective plane CP2 consisting of points whose homogeneous coordinates satisfy the equation x n0 C xn1 C xn2 D 0 is homeomorphic to the sphere with \.n 1/\.n 2/ 2 handles\. (Anatoly Fomenko 26) - The Grassmann manifold GC \.4; 2/ is homeomorphic to S2 S2 \. (Anatoly Fomenko 29) - EXERCISE 2\. Show that the group SO\.4/ is homeomorphic to S3 SO\.3/, that is, to S 3 RP3 \. (Anatoly Fomenko 30) - Smashing the upper base of the cylinder into one point gives the cone CX over X; thus, CX D \.X I/=\.X 1/ (Anatoly Fomenko 30) - Fig\. 6 Cylinder, cone, and suspension (Anatoly Fomenko 30) - This term can be justified by the fact that the suspension may be thought of as the union of two cones attached to each other by the bases; (Anatoly Fomenko 31) - (Anatoly Fomenko 31) - EXERCISE 4\. Prove that no closed \(that is, without holes\) classical surface except S 2 is homeomorphic to a suspension over any other space\. (Anatoly Fomenko 31) - Let f W X ! Y be an arbitrary continuous map\. The space obtained by attaching the cylinder X I to Y by means of the map X 0 D X f ! Y is called the cylinder of the map f a (Anatoly Fomenko 31) - The space obtained by attaching the cone CX to Y by means of the same map is called the cone of the map f a (Anatoly Fomenko 31) - The cylinder of f contains both X and Y; the cone of f contains Y\. (Anatoly Fomenko 31) - Fig\. 7 Cylinder and cone of a map (Anatoly Fomenko 31) - The set C\.X; Y/ of all continuous maps of a space X into a space Y is furnished by compact–open topology \(which can be thought of as the topology of uniform convergency on compact sets (Anatoly Fomenko 33) - The base of open sets of this topology consists of sets of the form U\.K; O/, where K is a compact subset of X and O is an open subset of Y; the set U\.K; O/ consists of continuous maps f W X ! Y such that f \.K/ O\. (Anatoly Fomenko 33) - A space is connected if it does not contain proper subsets which are both open and closed\. (Anatoly Fomenko 34) - For example, the cone over a space X with a base point x0 is obtained from the usual cone CX by collapsing the segment x0 I to a point which is chosen for the base point of the modified cone; (Anatoly Fomenko 35) - Suspensions and joins are modified in a similar way \(in the join the segment joining the base points is collapsed to a point\), and the images of segments collapsed are taken for the base points; (Anatoly Fomenko 35) - For a base point space X D \.X; x0 / the path space EX is defined as the space E\.X; x0 / of paths beginning at x0 , and the loop space X is defined as the space \.X; x0 / of loops beginning \(and ending\) at x0 ; (Anatoly Fomenko 35) - Alternatively, one can define the bouquet X \_ Y as the subspace of the product X Y composed of points \.x; y/ for which x D x0 or y D y0 \. The quotient space \(see Sect\. 2\.2\) X#Y D \.X Y/=\.X \_ Y/ is called the smash product or tensor product 5 of X and Y\. The base points in X \_ Y and X#Y are obvious\. (Anatoly Fomenko 36) - The set of homotopy classes in C\.X; Y/ is denoted as \.X; Y/\. Example 1\. The set \.X; I/ consists \(for every X\) of one element\. Example 2\. The set \.; Y/ \(where denotes a one-point space\) is the set of path components \(maximal path connected components\) of Y\. (Anatoly Fomenko 37) - EXERCISE 2\. Prove that a space that is homotopy equivalent to a path connected space is path connected (Anatoly Fomenko 40) - A space X is called contractible if the identity map idX W X ! X is homotopic to a constant map taking the whole space X to one point (Anatoly Fomenko 40) - EXERCISE 5\. Prove that the space E\.X; x0 / is contractible for any space X and any point x0 2 X\. (Anatoly Fomenko 40) - EXERCISE 8\. The previous statement is called the homotopy invariance of the operation of suspension\. Prove that the operations of product, join, mapping spaces, path and loop spaces are homotopy invariant in a similar sense\. (Anatoly Fomenko 41) - A subspace A of a space X is called a retract of X if there is a continuous map rW X ! X \(“retraction”\) such that r\.X/ D A and r\.a/ D a for every a 2 A (Anatoly Fomenko 41) - If a retraction r W X ! X of X onto A is homotopic to the identity idX W X ! X, then A is called a deformation retract of X\. (Anatoly Fomenko 41) - If a homotopy joining r with idX may be made fixed on A [that is, Ft \.a/ D a for all t 2 I; a 2 A], then A is called a strong deformation retract of X\. (Anatoly Fomenko 41) - Moreover, A is a deformation retract of X if and only if the inclusion map A ! X is a homotopy equivalence (Anatoly Fomenko 41) - the difference between deformation retracts and strong deformation retracts arises only in really pathological cases\. (Anatoly Fomenko 41) - Fig\. 11 Homotopy equivalence with no deformation retraction (Anatoly Fomenko 42) - We say that a subspace A of a topological space X is contractible in X if the inclusion map A ! X is homotopic to a constant map A ! X\. It is clear that if A is contractible \(in our usual sense; see Sect\. 3\.3\), then it is contractible in X, but the converse is not necessarily true (Anatoly Fomenko 42) - We consider the invariants of two different kinds: X 7! \.X; Y/ and X 7! \.Y; X/ \(for a fixed Y\)\. Each of these kinds gives rise to a theory, and, for a long time, the two theories remain parallel or, better to say, dual\. This duality is important for homotopy topology; it is called the Eckmann–Hilton duality\. (Anatoly Fomenko 45) - An important example of an Hspace: the loop space Z (Anatoly Fomenko 46) - An important example of an H 0 space: the suspension †Z (Anatoly Fomenko 46) - turns the suspension upside down\. (Anatoly Fomenko 47) - assigns to a loop the same loop passed in the opposite direction\. (Anatoly Fomenko 47) - Another important example of an Hspace is a topological group\. (Anatoly Fomenko 47) - Theorem\. The set b \.X; Y/ possesses a natural \(with respect to X\) group structure if and only if Y is an H-space\. Theorem\. The set b \.Y; X/ possesses a natural \(with respect to X\) group structure if and only if Y is an H 0 -space\. (Anatoly Fomenko 47) - Since Kn KnC1 , the set b \.X; Kn / is a group for any X (Anatoly Fomenko 49) - Since for n 1, Sn D †Sn 1 , the set b \.Sn ; X/ is a group for any X (Anatoly Fomenko 49) - This group is called the nth \(integral\) cohomology group of X and is denoted as H n \.X/ [or H n \.XI Z/] (Anatoly Fomenko 49) - This group is called the nth homotopy group of X and is denoted as n \.X/ (Anatoly Fomenko 49) - Decades ago, the computation of the homotopy groups of spheres seemed to the topologists a very important problem (Anatoly Fomenko 50) - The computation of the cohomology groups of the spaces Kn turned out to be a very important problem (Anatoly Fomenko 50) - A CW complex is a Hausdorff space X with a fixed partition X D 1 qD0 i2Iq e q i of X into pairwise disjoint set \(cells \) e q i such that for every cell e q i there exists a continuous map f q i q W Dq ! X \(a characteristic map of the cell ei i \) whose restriction to Int D is a homeomorphism Int D q q ei i whose restriction to S q 1 D Dq Int Dq maps S q 1 q into the union of cells of dimensions < q \(the dimension of the cell ei , dim e q i is, by definition, q\)\. The following two axioms are assumed satisfied\. q \(C\) The boundary eP i i q D ei i q ei i q D fi i \.Sq 1 / is contained in a finite union of cells\. (Anatoly Fomenko 50) - q \(W\) A set F X is closed if and only if for any cell ei i q the intersection F \ ei i is q closed \(in other words, \.fi / 1 \.F/ is closed in Dq \)\. (Anatoly Fomenko 51) - They abbreviate the expressions “closure finite” and “weak topology”\. (Anatoly Fomenko 51) - EXERCISE 1\. Prove that the topology described in axiom \(W\) is the weakest of all topologies with respect to which all characteristic maps are continuous\. (Anatoly Fomenko 51) - The dimension of a CW complex is the supremum of dimensions of all its cells, and the dimension of the nth skeleton may be less than n\. (Anatoly Fomenko 51) - A CW complex is called locally finite if every point has a neighborhood which is contained in some finite CW subcomplex\. (Anatoly Fomenko 51) - EXERCISE 3\. Prove that every compact subset of a CW complex is contained in some finite CW subcomplex\. (Anatoly Fomenko 51) - EXERCISE 4\. Prove that a CW complex is finite \(locally finite\) if and only if it is compact \(locally compact\)\. (Anatoly Fomenko 51) - EXERCISE 5\. Prove that a map of a CW complex into any topological space is continuous if and only if its restriction to every finite CW subcomplex is continuous\. (Anatoly Fomenko 51) - A continuous map f of a CW complex X into a CW complex Y is called cellular if f \.skn X/ skn Y for every n\. (Anatoly Fomenko 51) - A cell does not need to be mapped into a cell, but can be spread along several cells of the same or smaller \(but not bigger!\) dimensions\. (Anatoly Fomenko 51) - EXERCISE 7\. Let X 0 and X 00 be the segment I decomposed into cells as shown in Fig\. 13\. Are the identity maps f W X 0 ! X 00 and gW X 00 ! X 0 cellular? \(Answer: yes for f , no for g\.\) (Anatoly Fomenko 52) - Fig\. 13 For Exercise 7 (Anatoly Fomenko 52) - Remark 1\. The closure of a cell does not need to be a CW subcomplex\. Here is the example (Anatoly Fomenko 52) - EXERCISE 8\. Prove that a CW complex is metrizable if and only if it is locally finite\. (Anatoly Fomenko 53) - But we encounter an unexpected obstacle when we try to introduce a CW structure into a product and, the more so, smash product or join of two CW complexes\. Say, cells of the product of two CW complexes, X Y, are defined in the most natural way, as products of cells of X and Y, but there arises trouble with Axiom \(W\): It does not hold, in general (Anatoly Fomenko 53) - As to the mapping spaces, they are too big to have any hope of being decomposed into cells\. Still, there is the following theorem proven by Milnor\. Theorem \(Milnor [56]\)\. If X and Y are CW complexes, then the space Y X is homotopy equivalent to a CW comple (Anatoly Fomenko 54) - like the usual making a sphere from a ball by gluing all points of the boundary sphere into one point: \.x1 ; : : : ; xn / 7! cos ; x1 sin : : : ; xn sin ; where D x 21 C C x2n 2n and sin D for D 0\. (Anatoly Fomenko 56) - The other classical CW decomposition of S consists of 2n C 2 cells e0˙ ; : : : ; en˙ , where e q ˙ D f\.x1 ; : : : ; xnC1 / 2 Sn j xqC2 D D xnC1 D 0; ˙xqC1 > 0g\. (Anatoly Fomenko 56) - All these CW decompositions, except the first one, work for S 1 \. (Anatoly Fomenko 56) - Notice, however, that no one of these CW decompositions will work for D 1 \. (Anatoly Fomenko 56) - One more description is provided by the chain of inclusions ; D RP 1 RP RP RPn W We set e q D RPq Rq 1 \. A characteristic map for eq may be chosen as the composition of the canonical projection D q ! RPq \(see Sect\. 1\.2\) and the inclusion RPq ! RPn \. For n D 1, this construction provides a CW decomposition of RP1 with one cell in every dimension\. The construction also has complex, quaternionic, and Cayley analo (Anatoly Fomenko 57) - The CW decomposition of the Grassmann manifold G\.n; k/ described below is very important in topology \(in particular, for the theory of characteristic classes; see Lecture 19 ahead\) and also in algebra, algebraic geometry, and combinatorics\. The cells of this decomposition are called Schubert cells (Anatoly Fomenko 57) - There is a remarkable property of Schubert cells: Embeddings of G\.n; k/ to G\.n C 1; k/ and G\.n C 1; n C 1/ map every cell e\.m1 ; : : : ; ms / onto a cell with the same notation\. For this reason, the spaces G\.1; k/ and G\.1; 1/ are decomposed into cells corresponding to Young diagrams: In the second case they correspond to all Young diagrams, while in the first case they correspond to Young diagrams contained in the infinite horizontal half-strip of height k\. (Anatoly Fomenko 60) - EXERCISE 16\. The CW decompositions of RPn D G\.n C 1; 1/; CPn D CG\.n C 1; 1/; HPn D HG\.n C 1; 1/ constructed above are particular cases of the Schubert decomposition\. (Anatoly Fomenko 60) - The flag manifolds have natural CW decompositions which generalize the Schubert decomposition of the Grassmann manifolds\. (Anatoly Fomenko 60) - In particular, the manifold F\.nI 1; : : : ; n 1/ of full flags is decomposed into the union of cells corresponding to usual permutations of numbers 1; : : : ; n, and the dimension of a cell is equal to the number of inversions in a permutation\. (Anatoly Fomenko 61) - The most common CW decomposition of every classical surface has one twodimensional cell and one zero-dimensional cell\. Also, a sphere with g handles has 2g one-dimensional cells \(see Fig\. 18 for g D 2\), a projective plane with g handle (Anatoly Fomenko 61) - has 2g C 1 one-dimensional cells, and a Klein bottle with g handles has 2g C 2 one-dimensional cells\. (Anatoly Fomenko 62) - Corollary 1\. Let \.X; A/ be a CW pair\. If A is contractible, then X=A X\. More precisely: The projection X ! X=A is a homotopy equivalence\. (Anatoly Fomenko 63) - Corollary 2\. If \.X; A/ is a CW pair, then X=A X[CA, where CA is a cone over A\. (Anatoly Fomenko 64) - Theorem\. Every continuous map of one CW complex into another CW complex is homotopic to a cellular map\. (Anatoly Fomenko 64) - Definition\. A space X is called n-connected if for q n the set \.Sq ; X/ consists of one element \(that is, any two continuous maps S q ! X with q n are homotopic\)\. (Anatoly Fomenko 70) - Theorem\. If X is a CW complex with one vertex \(D zero-dimensional cell\) and without other cells of dimensions < q and Y is a CW complex of dimension < q, then every continuous map Y ! X is homotopic to a constant map (Anatoly Fomenko 70) - This follows directly from the cellular approximation theorem, since the qth skeleton of Y is the whole Y, and the qth skeleton of X is one point\. (Anatoly Fomenko 70) - In particular, if m < q, then \.Sm ; Sq / D b \.Sm ; Sq / D 0 \(that is, consists of one element\)\. (Anatoly Fomenko 70) - XERCISE 18\. Prove that each of the following two conditions is equivalent to n-connectedness\. \(1\) For q n, the set b \.Sq ; X/ consists of one element\. \(2\) For q n, every continuous map Sq ! X can be extended to a continuous map D qC1 ! X\.1 (Anatoly Fomenko 70) - EXERCISE 19\. Prove that 0-connectedness is the same as path connectedness\. (Anatoly Fomenko 70) - The term 1-connected\(ness\) is usually replaced by the term “simply connected\(ness\)\.” (Anatoly Fomenko 70) - Theorem\. Let n 0\. An n-connected CW complex is homotopy equivalent to a CW complex with only one vertex and without cells of dimensions 1; 2; : : : ; n\. \(In particular, every path connected CW complex is homotopy equivalent to a CW complex with only one vertex\.\) (Anatoly Fomenko 70) - Proof (Anatoly Fomenko 70) - Corollary\. If a CW complex X is n-connected, and a CW complex Y is n-dimensional, then the set \.Y; X/ consists of one element\. The same is true for b \.Y; X/ if X and Y have base points which are zero-dimensional cells\. (Anatoly Fomenko 71) - Remark\. The procedure of killing k-dimensional cells used in the last proof includes attaching cells of dimension k C 2\. (Anatoly Fomenko 71) - an n-connected n C 1-dimensional CW complex must be homotopy equivalent to a (Anatoly Fomenko 71) - bouquet of \.n C 1/-dimensional spheres (Anatoly Fomenko 72) - EXERCISE 20\. Prove that a connected CW complex X always contains a contractible one-dimensional CW subcomplex \(“a tree”\) Y, which contains all vertices of X (Anatoly Fomenko 72) - Theorem\. For any spaces with base points, \.X; x0 / and \.Y; y0 /, 1 \.X Y; \.x0 ; y0 // Š 1 \.X; x0 / \.Y; y0 /: (Anatoly Fomenko 74) - Theorem\. If the space X is path connected, then 1 \.X; x0 / 1 \.X; x1 / for any points x0 ; x1 2 X\. (Anatoly Fomenko 74) - Remark\. The statement 1 \.X/ is trivial means precisely that X is simply connected\. (Anatoly Fomenko 74) - Theorem\. The group 1 \.S1 / is isomorphic to the group Z of integers\. Proof (Anatoly Fomenko 75) - Just imagine, for a moment, that 1 \.S1 / be 0\. Then the fundamental groups of all spaces would be zeroes! Indeed, let 2 1 \.X; x0 / be represented by a loop sW I ! X\. Then there is a map f W S1 ! X such that s D f ı h1 , and Œs D f Œh1 D 0\. (Anatoly Fomenko 76) - e say that a path connected space T covers a path connected space X if there is a continuous map pW T ! X such that every point x 2 X has a neighborhood U whose inverse image p 1 \.U/ T falls into a disjoint union of open sets U˛ T such that for every ˛, p maps homeomorphically U˛ onto U\. in this situation, the map pW T ! X is called a covering\. (Anatoly Fomenko 76) - EXERCISE 2\. Prove that for any g 2 a sphere with g handles can cover a sphere with two handles\. \(Think about which classical surfaces can cover other classical surfaces (Anatoly Fomenko 77) - Sometimes we also require that the space is “semilocally simply connected,” which means that for every neighborhood U of x there exists a neighborhood V of x such that V U and every loop in V is homotopic to a constant loop in the whole space\. (Anatoly Fomenko 77) - Usually this means that the space is locally path connected ; that is, for every point x and every neighborhood U of x there exists a neighborhood V of x such that V U and any two points in V can be connected by a path in U\. (Anatoly Fomenko 77) - These properties will be needed to check the continuity of some maps (Anatoly Fomenko 77) - Fig\. 26 Proof of the lifting path lemma (Anatoly Fomenko 78) - Theorem\. If pW T ! X is a covering, then p W 1 \.T;e x0 / ! 1 \.X; x0 / is a monomorphism \(a one-to-one homomorphism\)\. (Anatoly Fomenko 80) - Theorem\. Let X be the “figure-eight space,” S 1 \_ S1 , and let i; jW S1 ! X be the two natural embeddings of S into S 1 \_ S1 \. Let ˛; ˇ 2 1 \.X/ be the images of the generator of 1 \.S1 / D Z with respect to i and j \. Then ˛ˇ ¤ ˇ˛\. Proof\. Consider a fivefold covering of the figure-eight space shown in Fig\. 29 (Anatoly Fomenko 81) - The commutativity of the group 1 \.S1 \_ S1 / would have implied the commutativity of the fundamental groups of all spaces\. Indeed, any two loops s; t with the same beginning of any topological space X form a map f W S1 \_ S1 ! X, and s D f ı ˛; t D f ı ˇ; thus, Œ˛ Œˇ D Œˇ Œ˛ would have implied Œs Œt D Œt Œs\. (Anatoly Fomenko 82) - Remark\. This result has an importance comparable with that of the computation of 1 \.S1 / \(see the remark in Sect\. 6\.2\)\. The c (Anatoly Fomenko 82) - EXERCISE 4\. Prove that if X is a topological group \(not necessarily commutative!\) or, at least, an H-space, then the fundamental group of X is commutative\. (Anatoly Fomenko 82) - Fig\. 29 A fivefold covering of the figure-eight space (Anatoly Fomenko 82) - The commutativity of the group 1 \.S1 \_ S1 / would have implied the commutativity of the fundamental groups of all spaces\. Indeed, any two loops s; t with the same beginning of any topological space X form a map f W S1 \_ S1 ! X, and s D f ı ˛; t D f ı ˇ; thus, Œ˛ Œˇ D Œˇ Œ˛ would have implied Œs Œt D Œt Œs\. (Anatoly Fomenko 82) - Remark\. This result has an importance comparable with that of the computation of 1 \.S1 / \(see the remark in Sect\. 6\.2\)\. The c (Anatoly Fomenko 82) - EXERCISE 4\. Prove that if X is a topological group \(not necessarily commutative!\) or, at least, an H-space, then the fundamental group of X is commutative\. (Anatoly Fomenko 82) - Definition\. Let pW T ! X be a covering\. A deck transformation of this covering is a homeomorphism f W T ! T such that p ı f D p\. [This condition means, in particular, that f \.p 1 \.x// D p 1 \.x/ for every x 2 X\.] (Anatoly Fomenko 83) - Proposition 1\. Let y 2 T\. A deck transformation f W T ! T is fully determined by the image f \.y/ of y\. In particular, if a deck transformation f has a fixed point, then f D id\. (Anatoly Fomenko 83) - Proposition 2\. Let y; y0 2 T and p\.y/ D p\.y0 /\. A deck transformation f W T ! T such that f \.y/ D y0 exists if and only if p 1 \.T; y/ D p 1 \.T; y0 /\. (Anatoly Fomenko 83) - x0 2 T; x0 D p\.e x0 /\. The group Theorem\. Let pW T ! X be a covering, and let e D of deck transformations of the covering p is isomorphic to the quotient of the x0 / 1 D p 1 \.T;e normalizer N D f 2 1 \.X; x0 / j p 1 \.T;e x0 /g of the group of covering p 1 \.T;e x0 / over this group\. (Anatoly Fomenko 83) - Definition\. Let pW T ! X be a covering\. A deck transformation of this covering is a homeomorphism f W T ! T such that p ı f D p\. [This condition means, in particular, that f \.p 1 \.x// D p 1 \.x/ for every x 2 X\.] (Anatoly Fomenko 83) - Proposition 1\. Let y 2 T\. A deck transformation f W T ! T is fully determined by the image f \.y/ of y\. In particular, if a deck transformation f has a fixed point, then f D id\. (Anatoly Fomenko 83) - Proposition 2\. Let y; y0 2 T and p\.y/ D p\.y0 /\. A deck transformation f W T ! T such that f \.y/ D y0 exists if and only if p 1 \.T; y/ D p 1 \.T; y0 /\. (Anatoly Fomenko 83) - x0 2 T; x0 D p\.e x0 /\. The group Theorem\. Let pW T ! X be a covering, and let e D of deck transformations of the covering p is isomorphic to the quotient of the x0 / 1 D p 1 \.T;e normalizer N D f 2 1 \.X; x0 / j p 1 \.T;e x0 /g of the group of covering p 1 \.T;e x0 / over this group\. (Anatoly Fomenko 83) - Definition\. A covering pW T ! X is called regular if the group of covering p 1 \.T;e x0 / is a normal subgroup of 1 \.X; x0 / \( (Anatoly Fomenko 84) - does not depend on the choice of e Equivalent Definition\. A covering pW T ! X is regular if the group of deck transformations acts transitively on p 1 \.x0 / \( (Anatoly Fomenko 84) - Thus, for a regular covering pW T ! X, the orbits of the group D of deck transformations coincide with inverse images p 1 \.x0 / of points of X in T (Anatoly Fomenko 84) - This means that X D T=D, (Anatoly Fomenko 84) - Let T be a connected topological space with a discrete action of a group D \(meaning that every point y 2 T has a neighborhood U such that sets d U for all d 2 D are mutually disjoint\)\. Then the projection T ! T=D is a regular covering, and all regular coverings can be constructed in this way\. (Anatoly Fomenko 84) - EXERCISE 5\. Prove that any twofold covering is regular \(this is equivalent to the well-known algebraic fact: For any group, any subgroup of index 2 is normal\)\. (Anatoly Fomenko 84) - EXERCISE 6\. Construct an example of an irregular three (Anatoly Fomenko 84) - Definition\. A covering pW T ! X is called universal if the space T is simply connected\. (Anatoly Fomenko 84) - Our observation above shows that for every point x0 2 X there is a one-toone correspondence between 1 \.X; x0 / and p 1 \.x0 /\. Moreover, it is possible to give this correspondence an appearance of a group isomorphism\. Namely, let there be a discrete action of a group D in a simply connected space T; then 1 \.T=D/ Š D\. (Anatoly Fomenko 84) - Example 1\. The covering pW R ! S1 \(see Sect\. 6\.4\) is a universal covering corresponding to the action of Z in R, (Anatoly Fomenko 84) - Definition\. A covering pW T ! X is called regular if the group of covering p 1 \.T;e x0 / is a normal subgroup of 1 \.X; x0 / \( (Anatoly Fomenko 84) - does not depend on the choice of e Equivalent Definition\. A covering pW T ! X is regular if the group of deck transformations acts transitively on p 1 \.x0 / \( (Anatoly Fomenko 84) - Thus, for a regular covering pW T ! X, the orbits of the group D of deck transformations coincide with inverse images p 1 \.x0 / of points of X in T (Anatoly Fomenko 84) - This means that X D T=D, (Anatoly Fomenko 84) - Let T be a connected topological space with a discrete action of a group D \(meaning that every point y 2 T has a neighborhood U such that sets d U for all d 2 D are mutually disjoint\)\. Then the projection T ! T=D is a regular covering, and all regular coverings can be constructed in this way\. (Anatoly Fomenko 84) - EXERCISE 5\. Prove that any twofold covering is regular \(this is equivalent to the well-known algebraic fact: For any group, any subgroup of index 2 is normal\)\. (Anatoly Fomenko 84) - EXERCISE 6\. Construct an example of an irregular three (Anatoly Fomenko 84) - Definition\. A covering pW T ! X is called universal if the space T is simply connected\. (Anatoly Fomenko 84) - Our observation above shows that for every point x0 2 X there is a one-toone correspondence between 1 \.X; x0 / and p 1 \.x0 /\. Moreover, it is possible to give this correspondence an appearance of a group isomorphism\. Namely, let there be a discrete action of a group D in a simply connected space T; then 1 \.T=D/ Š D\. (Anatoly Fomenko 84) - Example 1\. The covering pW R ! S1 \(see Sect\. 6\.4\) is a universal covering corresponding to the action of Z in R, (Anatoly Fomenko 84) - Example 2\. The covering pW Sn ! RPn \(again see Sect\. 6\.4\) is a universal covering with the group Z2 acting in Sn by the antipodal map\. Thus, the fundamental group of RPn is isomorphic to Z2 \. (Anatoly Fomenko 85) - For example, there are many known discrete subgroups in the group SO\.3/: the dihedral groups, the groups of symmetries of Platonic solids, and so forth\. For each of these groups there arises a regular covering SO\.3/ ! SO\.3/= ; since SO\.3/ is \(canonically homeomorphic to RP3 ; see Sect\. 1\.7\), we can combine this covering with the covering S 3 ! RP3 from the previous example and obtain a universal covering over SO\.3/= (Anatoly Fomenko 85) - Example 3\. If is a discrete subgroup of a topological group G, then there arises a regular covering G ! G= \. Fo (Anatoly Fomenko 85) - Example 4\. Let X be the union of all lines x D n; n 2 Z and y D n; n 2 Z \(an infinite sheet of graph paper\)\. The group Z Z acts in X in the obvious way \(and this action is discrete\)\. The quotient X=\.Z Z/ is, obviously, the figure-eight space \(the map pW X ! S1 \_ S1 maps every vertical segment Œ\.m; n/; \.m; n C 1/ onto the left S and every horizontal segment Œ\.m; n/; \.m C 1; n/ onto the right S1 \)\. Thus, we have a regular covering X ! S1 \_ S1 \. (Anatoly Fomenko 85) - Example 5\. Figure 30 presents a space of a universal covering over S 1 \_ S1 \(we leave details to the reader\)\. (Anatoly Fomenko 85) - EXERCISE 7\. Prove that every classical surface \(Sect\. 1\.10\) without holes, except S 2 and RP2 has a universal covering with the space homeomorphic to R2 (Anatoly Fomenko 85) - Example 2\. The covering pW Sn ! RPn \(again see Sect\. 6\.4\) is a universal covering with the group Z2 acting in Sn by the antipodal map\. Thus, the fundamental group of RPn is isomorphic to Z2 \. (Anatoly Fomenko 85) - For example, there are many known discrete subgroups in the group SO\.3/: the dihedral groups, the groups of symmetries of Platonic solids, and so forth\. For each of these groups there arises a regular covering SO\.3/ ! SO\.3/= ; since SO\.3/ is \(canonically homeomorphic to RP3 ; see Sect\. 1\.7\), we can combine this covering with the covering S 3 ! RP3 from the previous example and obtain a universal covering over SO\.3/= (Anatoly Fomenko 85) - Example 3\. If is a discrete subgroup of a topological group G, then there arises a regular covering G ! G= \. Fo (Anatoly Fomenko 85) - Example 4\. Let X be the union of all lines x D n; n 2 Z and y D n; n 2 Z \(an infinite sheet of graph paper\)\. The group Z Z acts in X in the obvious way \(and this action is discrete\)\. The quotient X=\.Z Z/ is, obviously, the figure-eight space \(the map pW X ! S1 \_ S1 maps every vertical segment Œ\.m; n/; \.m; n C 1/ onto the left S and every horizontal segment Œ\.m; n/; \.m C 1; n/ onto the right S1 \)\. Thus, we have a regular covering X ! S1 \_ S1 \. (Anatoly Fomenko 85) - Example 5\. Figure 30 presents a space of a universal covering over S 1 \_ S1 \(we leave details to the reader\)\. (Anatoly Fomenko 85) - EXERCISE 7\. Prove that every classical surface \(Sect\. 1\.10\) without holes, except S 2 and RP2 has a universal covering with the space homeomorphic to R2 (Anatoly Fomenko 85) - Fig\. 30 For Example 5: the space of a universal covering over the figure-eight space (Anatoly Fomenko 86) - Fig\. 30 For Example 5: the space of a universal covering over the figure-eight space (Anatoly Fomenko 86) - Theorem\. Let X be a sufficiently good path connected space with a base point x0 \. Then \(1\) For every subgroup H 1 \.X; x0 / there exists a unique, up to a base point equivalence, covering pW \.T;e x0 / ! \.X; x0 / such that p 1 \.T;e x0 / D H\. (Anatoly Fomenko 87) - Notice that this is the first \(and last\) case, when we need to use not only local connectedness, but also semilocal simply connectedness of the space X\. (Anatoly Fomenko 87) - Theorem\. Let X be a sufficiently good path connected space with a base point x0 \. Then \(1\) For every subgroup H 1 \.X; x0 / there exists a unique, up to a base point equivalence, covering pW \.T;e x0 / ! \.X; x0 / such that p 1 \.T;e x0 / D H\. (Anatoly Fomenko 87) - Notice that this is the first \(and last\) case, when we need to use not only local connectedness, but also semilocal simply connectedness of the space X\. (Anatoly Fomenko 87) - Theorem \(Van Kampen 3 \)\. Let Ai ; Ri be systems of generators and relations for for the group the groups 1 \.Ui ; x0 / \.i D 1; 2/\. Let B be a system of generators 1 \.U1 \ U2 ; x0 /\. Then the group 1 \.X; x 0 / is generated by the set A 1 fo A2 with the set of relations R1 x0 he R2 he B where the relation corresponding to b 2 B is j1 \.b/ D ` ` j2 \.b/, where, in turn, j1 \.b/ is regarded as a word in A1 and j2 \.b/ is regarded as a word in A2 \. (Anatoly Fomenko 91) - Theorem \(Van Kampen 3 \)\. Let Ai ; Ri be systems of generators and relations for for the group the groups 1 \.Ui ; x0 / \.i D 1; 2/\. Let B be a system of generators 1 \.U1 \ U2 ; x0 /\. Then the group 1 \.X; x 0 / is generated by the set A 1 fo A2 with the set of relations R1 x0 he R2 he B where the relation corresponding to b 2 B is j1 \.b/ D ` ` j2 \.b/, where, in turn, j1 \.b/ is regarded as a word in A1 and j2 \.b/ is regarded as a word in A2 \. (Anatoly Fomenko 91) - EXERCISE 4\. Prove that the group Z2 Z2 has a \(normal\) subgroup of index 2 isomorphic to Z\. (Anatoly Fomenko 94) - EXERCISE 5\. Prove that SL\.2; Z/ Š Z4 Z2 Z6 \. (Anatoly Fomenko 94) - EXERCISE 4\. Prove that the group Z2 Z2 has a \(normal\) subgroup of index 2 isomorphic to Z\. (Anatoly Fomenko 94) - EXERCISE 5\. Prove that SL\.2; Z/ Š Z4 Z2 Z6 \. (Anatoly Fomenko 94) - Theorem\. For sufficiently good spaces X; Y with base points, 1 \.X \_ Y/ D 1 \.X/ 1 \.Y/\. (Anatoly Fomenko 95) - Theorem 1\. If the knots K and K are isotopic, then 1 \.R3 K/ Š 1 \.R3 K 0 /\. (Anatoly Fomenko 95) - The following result is highly nontrivial\. Theorem 2\. If 1 \.R3 K/ Š Z, then K is an unknot\. (Anatoly Fomenko 95) - Theorem\. For sufficiently good spaces X; Y with base points, 1 \.X \_ Y/ D 1 \.X/ 1 \.Y/\. (Anatoly Fomenko 95) - Theorem 1\. If the knots K and K are isotopic, then 1 \.R3 K/ Š 1 \.R3 K 0 /\. (Anatoly Fomenko 95) - The following result is highly nontrivial\. Theorem 2\. If 1 \.R3 K/ Š Z, then K is an unknot\. (Anatoly Fomenko 95) - It could be expected that if 1 \.R3 K/ Š 1 \.R3 K 0 /, then the knots K and K 0 are isotopic\. However, it is wrong\. The simplest example: The trefoil knot \(see ahead\) is known to be not isotopic to its mirror image, but the two knots \(the trefoil and the mirror trefoil\), certainly, share the fundamental group of the complement\. There are more interesting examples, when 1 \.R3 K/ Š 1 \.R3 K 0 /, but K is not isotopic to either K 0 , or to the mirror image of K 0 (Anatoly Fomenko 96) - Prove that two knot diagrams represent the same knot \(the same isotopy class of knots\) if and only if they can be obtained from each other by a series of transformations called Reidemeister moves\. Move 1 (Anatoly Fomenko 96) - Fig\. 31 Knot diagrams: trefoil knot and figure-eight knot (Anatoly Fomenko 96) - It could be expected that if 1 \.R3 K/ Š 1 \.R3 K 0 /, then the knots K and K 0 are isotopic\. However, it is wrong\. The simplest example: The trefoil knot \(see ahead\) is known to be not isotopic to its mirror image, but the two knots \(the trefoil and the mirror trefoil\), certainly, share the fundamental group of the complement\. There are more interesting examples, when 1 \.R3 K/ Š 1 \.R3 K 0 /, but K is not isotopic to either K 0 , or to the mirror image of K 0 (Anatoly Fomenko 96) - Prove that two knot diagrams represent the same knot \(the same isotopy class of knots\) if and only if they can be obtained from each other by a series of transformations called Reidemeister moves\. Move 1 (Anatoly Fomenko 96) - Fig\. 31 Knot diagrams: trefoil knot and figure-eight knot (Anatoly Fomenko 96) - Our knot K will consist of the diagram \(several disjoint curves, that is, smooth curvilinear intervals in the plane\) and the arcs \(gates \) joining the ends of the curves below the plane \(our sheet of paper\)\. We apply Van Kampen’s theorem\. Let U be the intersection of R3 K with the half-space above the plane, and let V be the intersection of R3 K with the half-space below the plane (Anatoly Fomenko 97) - thus, 1 \.V/ is a free group “generated” by the gates\. Finally, U \ V is a perforated plane; its fundamental group is a free group “generated” by the curves\. Thus, 1 \.R3 K/ has a system of generators and relations where the generators correspond to the gates and the relations correspond to the curves\. (Anatoly Fomenko 97) - Obviously, there are equal numbers of gates and curves\. We mark the gates as a; b; : : : and the curves as A; B; : : : \. To specify the generators in 1 \.V/, we need to orient the gates; for this purpose, we simply fix an orientation of the knot and then take for the generators of 1 \.V/ loops which go through the gates in the direction of the knot\. For the trefoil diagram all this is done in Fig\. 32\. (Anatoly Fomenko 97) - Our knot K will consist of the diagram \(several disjoint curves, that is, smooth curvilinear intervals in the plane\) and the arcs \(gates \) joining the ends of the curves below the plane \(our sheet of paper\)\. We apply Van Kampen’s theorem\. Let U be the intersection of R3 K with the half-space above the plane, and let V be the intersection of R3 K with the half-space below the plane (Anatoly Fomenko 97) - thus, 1 \.V/ is a free group “generated” by the gates\. Finally, U \ V is a perforated plane; its fundamental group is a free group “generated” by the curves\. Thus, 1 \.R3 K/ has a system of generators and relations where the generators correspond to the gates and the relations correspond to the curves\. (Anatoly Fomenko 97) - Obviously, there are equal numbers of gates and curves\. We mark the gates as a; b; : : : and the curves as A; B; : : : \. To specify the generators in 1 \.V/, we need to orient the gates; for this purpose, we simply fix an orientation of the knot and then take for the generators of 1 \.V/ loops which go through the gates in the direction of the knot\. For the trefoil diagram all this is done in Fig\. 32\. (Anatoly Fomenko 97) - Theorem 3\. The fundamental group of the complement to the trefoil is a group with two generators, a and b, and one relation: aba D bab\. One can take for the generators u D ab and v D bab; then the relation takes the form u 3 D v 2 \. Theorem 4\. The fundamental group of the complement to the trefoil is not commutative\. (Anatoly Fomenko 98) - Corollary\. The trefoil knot is not isotopic to an unknot (Anatoly Fomenko 98) - Theorem 3\. The fundamental group of the complement to the trefoil is a group with two generators, a and b, and one relation: aba D bab\. One can take for the generators u D ab and v D bab; then the relation takes the form u 3 D v 2 \. Theorem 4\. The fundamental group of the complement to the trefoil is not commutative\. (Anatoly Fomenko 98) - Corollary\. The trefoil knot is not isotopic to an unknot (Anatoly Fomenko 98) - EXERCISE 11\. Find the fundamental group of the complement to the figure-eight knot \(it should be presented as a group with two generators and one relation\) (Anatoly Fomenko 100) - EXERCISE 13\. The Hopf link Hn is presented in Fig\. 34, right \(for n D 6; n is the number of components\)\. Prove that the group n \.R3 Hn / has n generators a1 ; a2 ; : : : ; an with n 1 relations: a1 a2 : : : an D an a1 a2 : : : an 1 D an 1 an a1 : : : an 2 D D a2 a3 : : : an a1 : Prove \(algebraically\) that the same group is isomorphic to a product of Z and a free group with n 1 generators\. \(Actually, S3 Hn is homeomorphic to the product of S 1 and S 2 minus n points; you can try to prove this\.\) (Anatoly Fomenko 100) - EXERCISE 14\. Let 2 R2 be a diagram of a knot K 2 R3 \. An admissible 3-coloring of is a coloring into the colors #1, #2, and #3 such that at every crossing, either only one color is used or all three colors are used\. (Anatoly Fomenko 100) - EXERCISE 11\. Find the fundamental group of the complement to the figure-eight knot \(it should be presented as a group with two generators and one relation\) (Anatoly Fomenko 100) - EXERCISE 13\. The Hopf link Hn is presented in Fig\. 34, right \(for n D 6; n is the number of components\)\. Prove that the group n \.R3 Hn / has n generators a1 ; a2 ; : : : ; an with n 1 relations: a1 a2 : : : an D an a1 a2 : : : an 1 D an 1 an a1 : : : an 2 D D a2 a3 : : : an a1 : Prove \(algebraically\) that the same group is isomorphic to a product of Z and a free group with n 1 generators\. \(Actually, S3 Hn is homeomorphic to the product of S 1 and S 2 minus n points; you can try to prove this\.\) (Anatoly Fomenko 100) - EXERCISE 14\. Let 2 R2 be a diagram of a knot K 2 R3 \. An admissible 3-coloring of is a coloring into the colors #1, #2, and #3 such that at every crossing, either only one color is used or all three colors are used\. (Anatoly Fomenko 100) - EXERCISE 15\. \(Sequel of Exercise 14\) Prove that the number of admissible colorings from Exercise 14 is precisely 3 less than the number of homomorphisms 1 \.X; x0 / ! S3 \. (Anatoly Fomenko 101) - EXERCISE 15\. \(Sequel of Exercise 14\) Prove that the number of admissible colorings from Exercise 14 is precisely 3 less than the number of homomorphisms 1 \.X; x0 / ! S3 \. (Anatoly Fomenko 101) - In Sect\. 9\.3, we derived a presentation of the fundamental group of a knot or a link, in which the generators correspond to gates and relations correspond to arcs\. (Anatoly Fomenko 102) - in which the generators correspond to gates and relations correspond to arcs\. There exists a presentation, also coming from Van Kampen’s theorem, in which generators correspond to arcs and relations correspond to gates\. Many people find it more convenient than the presentation of Sect\. 9\.3\. It is known by the name Wirtinger presentation\. To obtain his presentation (Anatoly Fomenko 102) - Thus, 1 \.U1 ; x0 / is trivial, and each of 1 \.U1 \U2 ; x0 / and 1 \.U2 ; x0 / is a free group with generators corresponding, respectively, to gates and curves of the diagram\. (Anatoly Fomenko 102) - In Sect\. 9\.3, we derived a presentation of the fundamental group of a knot or a link, in which the generators correspond to gates and relations correspond to arcs\. (Anatoly Fomenko 102) - in which the generators correspond to gates and relations correspond to arcs\. There exists a presentation, also coming from Van Kampen’s theorem, in which generators correspond to arcs and relations correspond to gates\. Many people find it more convenient than the presentation of Sect\. 9\.3\. It is known by the name Wirtinger presentation\. To obtain his presentation (Anatoly Fomenko 102) - Thus, 1 \.U1 ; x0 / is trivial, and each of 1 \.U1 \U2 ; x0 / and 1 \.U2 ; x0 / is a free group with generators corresponding, respectively, to gates and curves of the diagram\. (Anatoly Fomenko 102) - Fig\. 36 Proof of the attaching cell theorem (Anatoly Fomenko 103) - Fig\. 36 Proof of the attaching cell theorem (Anatoly Fomenko 103) - Moreover, if X has no zero-dimensional cells different from x0 , then 1 \.X; x0 / has a system of generators corresponding to one-dimensional cells \(classes of characteristic maps D 1 D I ! X\) with a system of relations corresponding to two-dimensional cells \(classes of attaching maps S 1 ! X\)\. (Anatoly Fomenko 104) - Moreover, if X has no zero-dimensional cells different from x0 , then 1 \.X; x0 / has a system of generators corresponding to one-dimensional cells \(classes of characteristic maps D 1 D I ! X\) with a system of relations corresponding to two-dimensional cells \(classes of attaching maps S 1 ! X\)\. (Anatoly Fomenko 104) - The standard CW decomposition of the Klein bottle K has two one-dimensional cells and one two-dimensional cell\. Thus, 1 \.K/ has two generators, denoted by c and d, and one relation, which can be read in Fig\. 4d in Lecture 1: cdc d D 1\. (Anatoly Fomenko 105) - Every handle results in two additional generators to the system of generators and in modifying the only relation by the multiplication of its left-hand side by the commutators of these additional generators\. With this in mind, we get the following description of the fundamental groups: 1 \.S2 with g handles/ D ha1 ; b1 ; : : : ; ag ; bg j Œa1 ; b1 : : : Œag ; bg D 1i; 1 \.RP2 with g handles/ D hc; a1 ; b1 ; : : : ; ag ; bg j c2 Œa1 ; b1 : : : Œag ; bg D 1i; 1 \.K with g handles/ D hc; d; a1 ; b1 ; : : : ; ag ; bg j cdc 1 dŒa1 ; b1 : : : Œag ; bg D 1i\. (Anatoly Fomenko 105) - Recall that the set n \.X; x0 / was defined as the set of base point homotopy classes of continuous maps of the sphere S n into X\. These maps are called spheroids\. In a different way, a spheroid can be defined as a continuous map of the cube I n into X taking the boundary @I n of the cube into x0 \. (Anatoly Fomenko 105) - The standard CW decomposition of the Klein bottle K has two one-dimensional cells and one two-dimensional cell\. Thus, 1 \.K/ has two generators, denoted by c and d, and one relation, which can be read in Fig\. 4d in Lecture 1: cdc d D 1\. (Anatoly Fomenko 105) - Every handle results in two additional generators to the system of generators and in modifying the only relation by the multiplication of its left-hand side by the commutators of these additional generators\. With this in mind, we get the following description of the fundamental groups: 1 \.S2 with g handles/ D ha1 ; b1 ; : : : ; ag ; bg j Œa1 ; b1 : : : Œag ; bg D 1i; 1 \.RP2 with g handles/ D hc; a1 ; b1 ; : : : ; ag ; bg j c2 Œa1 ; b1 : : : Œag ; bg D 1i; 1 \.K with g handles/ D hc; d; a1 ; b1 ; : : : ; ag ; bg j cdc 1 dŒa1 ; b1 : : : Œag ; bg D 1i\. (Anatoly Fomenko 105) - Recall that the set n \.X; x0 / was defined as the set of base point homotopy classes of continuous maps of the sphere S n into X\. These maps are called spheroids\. In a different way, a spheroid can be defined as a continuous map of the cube I n into X taking the boundary @I n of the cube into x0 \. (Anatoly Fomenko 105) - Fig\. 38 A homotopy between f C g and g C f (Anatoly Fomenko 106) - EXERCISE 1\. Prove that n \.X Y; \.x0 ; y0 // D n \.X; x0 / n \.Y; y0 / (Anatoly Fomenko 106) - For n D 1 the homotopy group is the fundamental group\. For n > 1 the homotopy group acquires a new feature: It is commutative\. Theorem\. If n > 1, then the group n \.X; x0 / is commutative for any \.X; x0 /\. (Anatoly Fomenko 106) - Notice in conclusion that a continuous map 'W \.X; x0 / ! \.Y; y0 / can be applied to spheroids, f 7! ' ı f , and, consequently, to a homomorphism ' W n \.X; x0 / ! n \.Y; y0 /\. The latter depends only on the homotopy class of '\. It is clear also that id D id and \.' ı / D ' ı \. Hence, homotopy equivalent spaces with base points have isomorphic homotopy groups\. (Anatoly Fomenko 106) - Fig\. 38 A homotopy between f C g and g C f (Anatoly Fomenko 106) - EXERCISE 1\. Prove that n \.X Y; \.x0 ; y0 // D n \.X; x0 / n \.Y; y0 / (Anatoly Fomenko 106) - For n D 1 the homotopy group is the fundamental group\. For n > 1 the homotopy group acquires a new feature: It is commutative\. Theorem\. If n > 1, then the group n \.X; x0 / is commutative for any \.X; x0 /\. (Anatoly Fomenko 106) - Notice in conclusion that a continuous map 'W \.X; x0 / ! \.Y; y0 / can be applied to spheroids, f 7! ' ı f , and, consequently, to a homomorphism ' W n \.X; x0 / ! n \.Y; y0 /\. The latter depends only on the homotopy class of '\. It is clear also that id D id and \.' ı / D ' ı \. Hence, homotopy equivalent spaces with base points have isomorphic homotopy groups\. (Anatoly Fomenko 106) - A path uW I ! X joining points x0 ; x1 2 X gives rise to an isomorphism u# W n \.X; x0 / ! n \.X; x1 /\. (Anatoly Fomenko 107) - The construction of u# is shown in Fig\. 40: First, we construct a map ! of the sphere Sn onto a bouquet Sn \_ I \(taking the base point of S into the endpoint of I distant from S \) and then assign to a spheroid f W Sn ! X taking the base point of S n into x0 the spheroid u# W Sn ! ! ! Sn \_ I f \_u ! X; taking the base point of S n into x1 \. (Anatoly Fomenko 107) - As seen in the example of the fundamental group, the isomorphism u# may be different for different paths u although it remains the same when the path u is replaced by a homotopic path\. In particular, loops representing an element ˛ 2 1 \.X; x0 / determine the same automorphism of n \.X; x0 /, which we can denote as ˛# \. In this way, we get a group action, or a representation, of 1 \.X; x0 / in n \.X; x0 /\. (Anatoly Fomenko 107) - A path uW I ! X joining points x0 ; x1 2 X gives rise to an isomorphism u# W n \.X; x0 / ! n \.X; x1 /\. (Anatoly Fomenko 107) - The construction of u# is shown in Fig\. 40: First, we construct a map ! of the sphere Sn onto a bouquet Sn \_ I \(taking the base point of S into the endpoint of I distant from S \) and then assign to a spheroid f W Sn ! X taking the base point of S n into x0 the spheroid u# W Sn ! ! ! Sn \_ I f \_u ! X; taking the base point of S n into x1 \. (Anatoly Fomenko 107) - As seen in the example of the fundamental group, the isomorphism u# may be different for different paths u although it remains the same when the path u is replaced by a homotopic path\. In particular, loops representing an element ˛ 2 1 \.X; x0 / determine the same automorphism of n \.X; x0 /, which we can denote as ˛# \. In this way, we get a group action, or a representation, of 1 \.X; x0 / in n \.X; x0 /\. (Anatoly Fomenko 107) - If for some n the isomorphism u# does not depend on the path u at all, the space X is called n-simple\. It follows from results of Lecture 6 that X is 1-simple if and only if the group 1 \.X/ is commutative (Anatoly Fomenko 108) - Spaces which are n-simple for all n are called simple\. For example, simply connected spaces are simple\. EXERCISE 2\. Prove that topological groups and H-spaces are simple \(compare with Exercise 4 in Sect\. 6\.7\)\. (Anatoly Fomenko 108) - x0 2 T, and let x0 D p\.e x0 / 2 X\. If Theorem 1\. Let pW T ! X be a covering, let e n 2, then p W n \.T;e x0 / ! n \.X; x0 / is an isomorphism (Anatoly Fomenko 108) - EXERCISE 3\. Prove that if X is a bouquet of circles, then n \.X/ D 0 for all n 2\. \(Prove that the universal covering of X is contractible; see Example 5 in Sect\. 6\.9\.\) (Anatoly Fomenko 108) - EXERCISE 4\. Prove that if X is a classical surface \(Sect\. 1\.10\) different from S 2 and RP2 , then n \.X/ D 0 for all n 2\. (Anatoly Fomenko 108) - X ! X, Let X be a surface S with a handle\. We consider the infinite covering pW where X is a cylinder with infinitely many copies of S attached \(see Fig\. 41\) (Anatoly Fomenko 108) - If for some n the isomorphism u# does not depend on the path u at all, the space X is called n-simple\. It follows from results of Lecture 6 that X is 1-simple if and only if the group 1 \.X/ is commutative (Anatoly Fomenko 108) - Spaces which are n-simple for all n are called simple\. For example, simply connected spaces are simple\. EXERCISE 2\. Prove that topological groups and H-spaces are simple \(compare with Exercise 4 in Sect\. 6\.7\)\. (Anatoly Fomenko 108) - x0 2 T, and let x0 D p\.e x0 / 2 X\. If Theorem 1\. Let pW T ! X be a covering, let e n 2, then p W n \.T;e x0 / ! n \.X; x0 / is an isomorphism (Anatoly Fomenko 108) - EXERCISE 3\. Prove that if X is a bouquet of circles, then n \.X/ D 0 for all n 2\. \(Prove that the universal covering of X is contractible; see Example 5 in Sect\. 6\.9\.\) (Anatoly Fomenko 108) - EXERCISE 4\. Prove that if X is a classical surface \(Sect\. 1\.10\) different from S 2 and RP2 , then n \.X/ D 0 for all n 2\. (Anatoly Fomenko 108) - X ! X, Let X be a surface S with a handle\. We consider the infinite covering pW where X is a cylinder with infinitely many copies of S attached \(see Fig\. 41\) (Anatoly Fomenko 108) - Fig\. 41 A covering over a surface S with a handle (Anatoly Fomenko 109) - Fig\. 41 A covering over a surface S with a handle (Anatoly Fomenko 109) - First, absolute homotopy groups may be regarded as a particular case of relative homotopy groups\. Namely, n \.X; x0 / D \.X; x0 ; x0 / for n 1\. Second, relative homotopy groups may be regarded as a particular case of absolute homotopy groups\. Namely, there exists a construction which assigns to a pair X; A with a base point x0 a space Y with a base point y0 such that n \.X; A; x0 / D n 1 \.Y; y0 / for n 1\. This explains why n \.X; A; x0 / is a group only for n 2 and a commutative group only for n 3\. (Anatoly Fomenko 111) - Fourth \(and the most important!\), there are connecting homomorphisms n \.X; A; x0 / ! n 1 \.A; x0 /: The homomorphism @ takes the class of a relative spheroid f W \.I n ; @I n ; @I n I n 1 / ! \.X; A; x0 / into the class of the absolute spheroid f jI n 1 W \.I n 1 ; @I n 1 / ! \.A; x0 / (Anatoly Fomenko 111) - First, absolute homotopy groups may be regarded as a particular case of relative homotopy groups\. Namely, n \.X; x0 / D \.X; x0 ; x0 / for n 1\. Second, relative homotopy groups may be regarded as a particular case of absolute homotopy groups\. Namely, there exists a construction which assigns to a pair X; A with a base point x0 a space Y with a base point y0 such that n \.X; A; x0 / D n 1 \.Y; y0 / for n 1\. This explains why n \.X; A; x0 / is a group only for n 2 and a commutative group only for n 3\. (Anatoly Fomenko 111) - Fourth \(and the most important!\), there are connecting homomorphisms n \.X; A; x0 / ! n 1 \.A; x0 /: The homomorphism @ takes the class of a relative spheroid f W \.I n ; @I n ; @I n I n 1 / ! \.X; A; x0 / into the class of the absolute spheroid f jI n 1 W \.I n 1 ; @I n 1 / ! \.A; x0 / (Anatoly Fomenko 111) - The “homotopy sequence of a pair” is the name given to the sequence @ @ n \.A; x0 / i i n \.X; x0 / j n \.X; A; x0 / @ @ n 1 \.A; x0 / i i ! 1 \.X; x0 / j ! 1 \.X; A; x0 / @ @ ! 0 \.A; x0 / i i ! 0 \.X; x0 /; where j and @ are homomorphisms described earlier in this chapter and i is induced by the inclusion map iW A ! X\. (Anatoly Fomenko 112) - The “homotopy sequence of a pair” is the name given to the sequence @ @ n \.A; x0 / i i n \.X; x0 / j n \.X; A; x0 / @ @ n 1 \.A; x0 / i i ! 1 \.X; x0 / j ! 1 \.X; A; x0 / @ @ ! 0 \.A; x0 / i i ! 0 \.X; x0 /; where j and @ are homomorphisms described earlier in this chapter and i is induced by the inclusion map iW A ! X\. (Anatoly Fomenko 112) - COROLLARIES\. If A is contractible, then n \.X/ Š n \.X; A/ \(more precisely, j is an isomorphism\); if X is contractible, then n \.X; A/ Š n 1 \.A/ \(more precisely, @ is an isomorphism\); if A is a deformation retract of X, then n \.X; A/ D 0 for n 1\. (Anatoly Fomenko 114) - EXERCISE 13\. If A is a retract \(not necessarily a deformation retract\) of X, then, for all n, (Anatoly Fomenko 114) - moreover, n \.X/ Š n \.X; A/ ˚ n \.A/\. (Anatoly Fomenko 114) - EXERCISE 14\. If A is contractible to a point within X, then (Anatoly Fomenko 114) - moreover, n \.X; A/ Š n \.X/ ˚ n 1 \.A/\. (Anatoly Fomenko 114) - EXERCISE 15\. If there exists a homotopy ft W X ! X driving X into A, that is, such that f0 D id and f1 \.X/ A, then (Anatoly Fomenko 114) - moreover, n \.A/ D n \.X/ ˚ nC1 \.X; A/\. (Anatoly Fomenko 114) - COROLLARIES\. If A is contractible, then n \.X/ Š n \.X; A/ \(more precisely, j is an isomorphism\); if X is contractible, then n \.X; A/ Š n 1 \.A/ \(more precisely, @ is an isomorphism\); if A is a deformation retract of X, then n \.X; A/ D 0 for n 1\. (Anatoly Fomenko 114) - EXERCISE 13\. If A is a retract \(not necessarily a deformation retract\) of X, then, for all n, (Anatoly Fomenko 114) - oreover, n \.X/ Š n \.X; A/ ˚ n \.A/\. (Anatoly Fomenko 114) - moreover, n \.X/ Š n \.X; A/ ˚ n \.A/\. (Anatoly Fomenko 114) - EXERCISE 14\. If A is contractible to a point within X, then (Anatoly Fomenko 114) - moreover, n \.X; A/ Š n \.X/ ˚ n 1 \.A/\. (Anatoly Fomenko 114) - EXERCISE 15\. If there exists a homotopy ft W X ! X driving X into A, that is, such that f0 D id and f1 \.X/ A, then (Anatoly Fomenko 114) - moreover, n \.A/ D n \.X/ ˚ nC1 \.X; A/\. (Anatoly Fomenko 114) - EXERCISE 17\. Let 1 ! A0 ! ! An ! 1 be an exact sequence\. \(1\) Prove that if all the groups Ai are finite and qi D jAi j, then be niD0 q \. 1/i i Q D Ai are finitely generated Abelian groups and ri D rank Ai , then 2\) niD0 Prove that if \.1/i ri D 0\. (Anatoly Fomenko 116) - we will consider a more general notion of fibrations whose main difference from coverings is that the second factor is not assumed to be discrete any more\. (Anatoly Fomenko 116) - EXERCISE 17\. Let 1 ! A0 ! ! An ! 1 be an exact sequence\. \(1\) Prove that if all the groups Ai are finite and qi D jAi j, then be niD0 q \. 1/i i Q D Ai are finitely generated Abelian groups and ri D rank Ai , then 2\) niD0 Prove that if \.1/i ri D 0\. (Anatoly Fomenko 116) - we will consider a more general notion of fibrations whose main difference from coverings is that the second factor is not assumed to be discrete any more\. (Anatoly Fomenko 116) - A fibration, or a locally trivial fibration, is a quadruple \.E; B; F; p/, where E; B, and F are topological spaces and p is a continuous map E ! B such that for every point x 2 B there exist a neighborhood U and a homeomorphism f W p 1 \.U/ ! U F such that the diagram f p \(U \) U ×F U p projection is commutative\. (Anatoly Fomenko 118) - Sometimes, the term fibration is attributed to the map pW E ! B; the term fibered space is also used: This is what the space E may be called\. (Anatoly Fomenko 118) - Example 2\. Let E D S3 D f\.z1 ; z2 / 2 C2 j jz1 j2 C jz2 j2 D 1g; B D S2 D CP1 ; F D S 1 D fz 2 C j jzj D 1g; p\.z1 ; z2 / D \.z1 W z2 /\. (Anatoly Fomenko 118) - A fibration, or a locally trivial fibration, is a quadruple \.E; B; F; p/, where E; B, and F are topological spaces and p is a continuous map E ! B such that for every point x 2 B there exist a neighborhood U and a homeomorphism f W p 1 \.U/ ! U F such that the diagram f p \(U \) U ×F U p projection is commutative\. (Anatoly Fomenko 118) - Sometimes, the term fibration is attributed to the map pW E ! B; the term fibered space is also used: This is what the space E may be called\. (Anatoly Fomenko 118) - Example 2\. Let E D S3 D f\.z1 ; z2 / 2 C2 j jz1 j2 C jz2 j2 D 1g; B D S2 D CP1 ; F D S 1 D fz 2 C j jzj D 1g; p\.z1 ; z2 / D \.z1 W z2 /\. (Anatoly Fomenko 118) - Example 5\. Let X; Y be compact smooth manifolds and f W X ! Y be a submersion, that is, a smooth map whose differential at every point is an epimorphism\. Let y0 be a point of Y\. 1 EXERCISE 5\. Prove that if the space Y is \(path\) connected, then \.X; Y; f \.y0 /; f / is a fibration\. (Anatoly Fomenko 119) - Fibrations, like coverings, possess a covering homotopy property \(CHP\)\. What is lost when we pass from coverings to fibrations is the uniqueness\. Here is the precise statement\. ' W X ! E be Theorem\. Let \.E; B; F; p/ be a fibration, let X be a CW complex, et et e '\. a continuous map, and let ˆW X X I ! B be a homot p py suc h that ê jX D 0 D p ı (Anatoly Fomenko 119) - Example 5\. Let X; Y be compact smooth manifolds and f W X ! Y be a submersion, that is, a smooth map whose differential at every point is an epimorphism\. Let y0 be a point of Y\. 1 EXERCISE 5\. Prove that if the space Y is \(path\) connected, then \.X; Y; f \.y0 /; f / is a fibration\. (Anatoly Fomenko 119) - Fibrations, like coverings, possess a covering homotopy property \(CHP\)\. What is lost when we pass from coverings to fibrations is the uniqueness\. Here is the precise statement\. ' W X ! E be Theorem\. Let \.E; B; F; p/ be a fibration, let X be a CW complex, et et e '\. a continuous map, and let ˆW X X I ! B be a homot p py suc h that ê jX D 0 D p ı (Anatoly Fomenko 119) - A Serre fibration is a triple \.E; B; p/ where E; B are topological spaces and p is a continuous map E ! B which satisfies the relative form of CHP \(as stated in Sect\. 9\.2\)\. A Serre fibration is not necessarily a locally trivial fibration \(see Fig\. 46\) although the theorem in Sect\. 9\.2 states that a locally trivial fibration is a Serre fibratio (Anatoly Fomenko 122) - A Serre fibration is a triple \.E; B; p/ where E; B are topological spaces and p is a continuous map E ! B which satisfies the relative form of CHP \(as stated in Sect\. 9\.2\)\. A Serre fibration is not necessarily a locally trivial fibration \(see Fig\. 46\) although the theorem in Sect\. 9\.2 states that a locally trivial fibration is a Serre fibratio (Anatoly Fomenko 122) - Example 2 \(Path fibration\)\. Let W be an arbitrary topological space with a base point w0 \. Put E D E\.W; w0 / \(the space of paths of W beginning at w0 \), B D W, and define pW E ! B by the formula p\.s/ D s\.1/\. Then \.E; B; p/ is a strong Serre (Anatoly Fomenko 123) - Example 2 \(Path fibration\)\. Let W be an arbitrary topological space with a base point w0 \. Put E D E\.W; w0 / \(the space of paths of W beginning at w0 \), B D W, and define pW E ! B by the formula p\.s/ D s\.1/\. Then \.E; B; p/ is a strong Serre (Anatoly Fomenko 123) - Fig\. 48 The path fibration (Anatoly Fomenko 124) - The example of a Serre fibration in Fig\. 46 shows that the fibers of a Serre fibration, that is, inverse images of points of the base, do not need to be homeomorphic to each other\. Still these fibers turn out to have some resemblance to each other\. (Anatoly Fomenko 124) - Definition\. We will say that a topological space S is weakly homotopy equivalent to a topological space T if, for CW complexes X, there exist bijections \.X; S/ $ \.X; T/, natural with respect to X (Anatoly Fomenko 124) - More precisely, for every CW complex X there is fixed a bijection 'X W \.X; S/ ! \.Y; T/ such that for every continuous \(or cellular; it makes no difference in view of the cellular approximation theorem\) map f W X ! Y, the diagram \.X; ' X 'X X; x x ? ? ? X x X x ? ? ? ? \.Y; S/ ' Y ? 'Y ! \.Y; T/ is commutative (Anatoly Fomenko 124) - Fig\. 48 The path fibration (Anatoly Fomenko 124) - The example of a Serre fibration in Fig\. 46 shows that the fibers of a Serre fibration, that is, inverse images of points of the base, do not need to be homeomorphic to each other\. Still these fibers turn out to have some resemblance to each other\. (Anatoly Fomenko 124) - Definition\. We will say that a topological space S is weakly homotopy equivalent to a topological space T if, for CW complexes X, there exist bijections \.X; S/ $ \.X; T/, natural with respect to X (Anatoly Fomenko 124) - More precisely, for every CW complex X there is fixed a bijection 'X W \.X; S/ ! \.Y; T/ such that for every continuous \(or cellular; it makes no difference in view of the cellular approximation theorem\) map f W X ! Y, the diagram \.X; ' X 'X X; x x ? ? ? X x X x ? ? ? ? \.Y; S/ ' Y ? 'Y ! \.Y; T/ is commutative (Anatoly Fomenko 124) - but it is not true that weakly homotopy equivalent spaces can always be connected by a weak homotopy equivalence\. (Anatoly Fomenko 125) - According to Definition 1 in Sect\. 3\.3, usual homotopy equivalences are established by continuous maps\. Weak homotopy equivalences are established by continuous maps sometimes, but not always\. Namely, a continuous map 'W S ! T is called a weak homotopy equivalence if ' W \.X; S/ ! \.X; T/ is a bijection for every CW complex X\. It is obvious that if there is a weak homotopy equivalence 'W S ! T, then S and T are weakly homotopy equivalent \(just put 'X D ' \); (Anatoly Fomenko 125) - \.1/ A continuous map between path connected spaces with base points 'W \.S; s0 / ! \.T; t0 / is a weak homotopy equivalence if and only if ' W n \.S; s0 / ! n \.T; t0 / is an isomorphism for all n 1\. \.2/ Every topological space is weakly homotopy equivalent to a CW complex, and this CW complex is unique up to a homotopy equivalence\. (Anatoly Fomenko 125) - Proposition\. If CW complexes X and Y are weakly homotopy equivalent, then they are homotopy equivalent\. (Anatoly Fomenko 125) - but it is not true that weakly homotopy equivalent spaces can always be connected by a weak homotopy equivalence\. (Anatoly Fomenko 125) - According to Definition 1 in Sect\. 3\.3, usual homotopy equivalences are established by continuous maps\. Weak homotopy equivalences are established by continuous maps sometimes, but not always\. Namely, a continuous map 'W S ! T is called a weak homotopy equivalence if ' W \.X; S/ ! \.X; T/ is a bijection for every CW complex X\. It is obvious that if there is a weak homotopy equivalence 'W S ! T, then S and T are weakly homotopy equivalent \(just put 'X D ' \); (Anatoly Fomenko 125) - \.1/ A continuous map between path connected spaces with base points 'W \.S; s0 / ! \.T; t0 / is a weak homotopy equivalence if and only if ' W n \.S; s0 / ! n \.T; t0 / is an isomorphism for all n 1\. \.2/ Every topological space is weakly homotopy equivalent to a CW complex, and this CW complex is unique up to a homotopy equivalence\. (Anatoly Fomenko 125) - Proposition\. If CW complexes X and Y are weakly homotopy equivalent, then they are homotopy equivalent\. (Anatoly Fomenko 125) - Theorem\. If \.E; B; p/ is a Serre fibration, then for any points x0 ; x1 from the same path component of B, the fibers p 1 \.x0 /; p 1 \.x1 / are weakly homotopy equivalent\. If \.E; B; p/ is a strong Serre fibration, then p 1 \.x0 /; p 1 \.x1 / are homotopy equivalent (Anatoly Fomenko 126) - Theorem\. If \.E; B; p/ is a Serre fibration, then for any points x0 ; x1 from the same path component of B, the fibers p 1 \.x0 /; p 1 \.x1 / are weakly homotopy equivalent\. If \.E; B; p/ is a strong Serre fibration, then p 1 \.x0 /; p 1 \.x1 / are homotopy equivalent (Anatoly Fomenko 126) - We say that continuous maps f W X ! Y and f 0 W X 0 ! Y 0 are homotopy equivalent if there are homotopy equivalences 'W X ! X 0 and W Y ! Y 0 which make the diagram X f X Y ? ? ? ? ? Y ? ? ? ? ? y X f 0 y f0 ! Y 0 homotopy commutative \( ı f f 0 ı '\)\. (Anatoly Fomenko 127) - Theorem\. For every continuous map, there exists a strong Serre fibration homotopy equivalent to this map\. (Anatoly Fomenko 127) - We say that continuous maps f W X ! Y and f 0 W X 0 ! Y 0 are homotopy equivalent if there are homotopy equivalences 'W X ! X 0 and W Y ! Y 0 which make the diagram X f X Y ? ? ? ? ? Y ? ? ? ? ? y X f 0 y f0 ! Y 0 homotopy commutative \( ı f f 0 ı '\)\. (Anatoly Fomenko 127) - Theorem\. For every continuous map, there exists a strong Serre fibration homotopy equivalent to this map\. (Anatoly Fomenko 127) - X, we take the space of pairs \.x; s/, where x 2 X and s Proof of Theorem\. For e er X ! Y and the is a path if Y beginning at th the point f \.x/\. The projection p\.f /W e X ! X are defined by the formulas Œp\.f /\.x; s/ D s\.1/ homotopy equivalen and Œ'\.f /\.x; s/ D s\. (Anatoly Fomenko 128) - Remark\. This theorem is dual \(in the sense of duality considered in Lecture 4\) to the following simple statement: For every continuous map f W X ! Y there exists an at Y Y homotopy equivalent to f ; (Anatoly Fomenko 128) - Y Y the cylinder Cyl\.f / (Anatoly Fomenko 128) - Because of this duality, the X X constructed in the proof is sometimes called the cocylinder of the map f \. (Anatoly Fomenko 128) - X, we take the space of pairs \.x; s/, where x 2 X and s Proof of Theorem\. For e er X ! Y and the is a path if Y beginning at th the point f \.x/\. The projection p\.f /W e X ! X are defined by the formulas Œp\.f /\.x; s/ D s\.1/ homotopy equivalen and Œ'\.f /\.x; s/ D s\. (Anatoly Fomenko 128) - Remark\. This theorem is dual \(in the sense of duality considered in Lecture 4\) to the following simple statement: For every continuous map f W X ! Y there exists an at Y Y homotopy equivalent to f ; (Anatoly Fomenko 128) - Y Y the cylinder Cyl\.f / (Anatoly Fomenko 128) - Because of this duality, the X X constructed in the proof is sometimes called the cocylinder of the map f \. (Anatoly Fomenko 128) - e Now replace in the homotopy sequence of the pair \.E; F/ the groups i \.E; F/ by the isomorphic groups i \.B/\. We get an exact sequence n \.F; e0 / ! n \.E; e0 / ! n \.B; b0 / ! n 1 \.F; e0 / ! : : : ! 1 \.B; b0 / ! 0 \.F; e0 / ! 0 \.E; e0 / ! 0 \.B; b0 / consisting only of absolute homotopy groups \(not all of them are groups, as we know\)\. This sequence is called the homotopy sequence of the fibration\. (Anatoly Fomenko 129) - e Now replace in the homotopy sequence of the pair \.E; F/ the groups i \.E; F/ by the isomorphic groups i \.B/\. We get an exact sequence n \.F; e0 / ! n \.E; e0 / ! n \.B; b0 / ! n 1 \.F; e0 / ! : : : ! 1 \.B; b0 / ! 0 \.F; e0 / ! 0 \.E; e0 / ! 0 \.B; b0 / consisting only of absolute homotopy groups \(not all of them are groups, as we know\)\. This sequence is called the homotopy sequence of the fibration\. (Anatoly Fomenko 129) - Some other applications of the exactness of the homotopy sequence of a fibration are contained in the following exercises\. EXERCISE 9\. Analyze the homotopy sequence of a covering\. Deduce from it the major results of Sects\. 6\.6 and 6\.8 (Anatoly Fomenko 130) - EXERCISE 10\. Deduce from the homotopy sequence of the Hopf fibration pW S2nC1 ! CPn \(Example 3 of Sect\. 9\.1\) that r \.CPn / is Z for r D 2 and zero for 3 r 2n 1; in particular, CP1 has only one nontrivial homotopy group: 2 \.CP1 / Š Z\. (Anatoly Fomenko 130) - EXERCISE 11\. Using the path fibration from Sect\. 9\.4, prove that n \.X/ Š nC1 \.X/ for all X and n 0\. (Anatoly Fomenko 130) - EXERCISE 12\. Prove that if the base of a Serre fibration is contractible, then the inclusion of \(any\) fiber in the total space induces an isomorphism of homotopy groups\. Prove that if the base of a Serre fibration is connected and one of the fibers is contractible, then the projection induces an isomorphism of homotopy groups of the total space and the base\. (Anatoly Fomenko 130) - these statements mean, respectively, that the inclusion map F ! E and the projection E ! B are weak homotopy equivalences\. (Anatoly Fomenko 130) - F ! E and the projection E ! B are weak homotopy equivalences\. EXERCISE 13\. Prove that if all the homotopy groups of the base and the fiber are finite, then so are homotopy groups of the total space, and the orders of the homotopy groups of the total space do not exceed the product of orders of (Anatoly Fomenko 130) - Some other applications of the exactness of the homotopy sequence of a fibration are contained in the following exercises\. EXERCISE 9\. Analyze the homotopy sequence of a covering\. Deduce from it the major results of Sects\. 6\.6 and 6\.8 (Anatoly Fomenko 130) - EXERCISE 10\. Deduce from the homotopy sequence of the Hopf fibration pW S2nC1 ! CPn \(Example 3 of Sect\. 9\.1\) that r \.CPn / is Z for r D 2 and zero for 3 r 2n 1; in particular, CP1 has only one nontrivial homotopy group: 2 \.CP1 / Š Z\. (Anatoly Fomenko 130) - EXERCISE 11\. Using the path fibration from Sect\. 9\.4, prove that n \.X/ Š nC1 \.X/ for all X and n 0\. (Anatoly Fomenko 130) - EXERCISE 12\. Prove that if the base of a Serre fibration is contractible, then the inclusion of \(any\) fiber in the total space induces an isomorphism of homotopy groups\. Prove that if the base of a Serre fibration is connected and one of the fibers is contractible, then the projection induces an isomorphism of homotopy groups of the total space and the base\. (Anatoly Fomenko 130) - these statements mean, respectively, that the inclusion map F ! E and the projection E ! B are weak homotopy equivalences\. (Anatoly Fomenko 130) - F ! E and the projection E ! B are weak homotopy equivalences\. EXERCISE 13\. Prove that if all the homotopy groups of the base and the fiber are finite, then so are homotopy groups of the total space, and the orders of the homotopy groups of the total space do not exceed the product of orders of (Anatoly Fomenko 130) - corresponding homotopy groups of the base and the fiber\. Formulate and prove a similar statement concerning finitely generated groups and their ranks\. (Anatoly Fomenko 131) - EXERCISE 14\. Prove that if a Serre fibration \.E; B; p/ has a section \(that is, a continuous map sW B ! E such that p ı s D idB \) or if F is a retract of E, then n \.E/ Š n \.B/ ˚ n \.F/ for n 2 (Anatoly Fomenko 131) - EXERCISE 15\. Prove that if the fiber of the Serre fibration E; B; p is contractible in E, then n \.B/ Š n \.E/ ˚ n 1 \.F/ for all n 2\. (Anatoly Fomenko 131) - In Sect\. 8\.6, we promised to construct for a topological pair \.X; A/ a space Y with an isomorphism n \.X; A/ Š n 1 \.Y/ (Anatoly Fomenko 131) - Following Sect\. 9\.7, construct a \(strong\) Serre fibration p 0 W A0 ! X homotopy equivalent to the inclusion map A ! X and denote by Y a fiber of this fibration (Anatoly Fomenko 131) - This observation provides a canonical map n 1 \.Y/ ! n \.X; A/ (Anatoly Fomenko 131) - whose rows are homotopy sequences of the fibration \.A0 ; X; p0 / and the pair \.X; A/\. It follows from the five-lemma that the maps n 1 \.Y/ ! n \.X; A/ are isomorphisms (Anatoly Fomenko 131) - For a spheroid gW \.Sn 1 ; y0 / ! \.Y; const/ we define a spheroid GW Dn D CSn 1 ! X of the pair \.X; A/ by the formula G\.s; t/ D Œg\.s/\.t/; s 2 Sn 1 ; t 2 I\. It is obvious that these canonical maps are included in the commutative diagram (Anatoly Fomenko 131) - corresponding homotopy groups of the base and the fiber\. Formulate and prove a similar statement concerning finitely generated groups and their ranks\. (Anatoly Fomenko 131) - EXERCISE 14\. Prove that if a Serre fibration \.E; B; p/ has a section \(that is, a continuous map sW B ! E such that p ı s D idB \) or if F is a retract of E, then n \.E/ Š n \.B/ ˚ n \.F/ for n 2 (Anatoly Fomenko 131) - EXERCISE 15\. Prove that if the fiber of the Serre fibration E; B; p is contractible in E, then n \.B/ Š n \.E/ ˚ n 1 \.F/ for all n 2\. (Anatoly Fomenko 131) - In Sect\. 8\.6, we promised to construct for a topological pair \.X; A/ a space Y with an isomorphism n \.X; A/ Š n 1 \.Y/ (Anatoly Fomenko 131) - Following Sect\. 9\.7, construct a \(strong\) Serre fibration p 0 W A0 ! X homotopy equivalent to the inclusion map A ! X and denote by Y a fiber of this fibration (Anatoly Fomenko 131) - This observation provides a canonical map n 1 \.Y/ ! n \.X; A/ (Anatoly Fomenko 131) - whose rows are homotopy sequences of the fibration \.A0 ; X; p0 / and the pair \.X; A/\. It follows from the five-lemma that the maps n 1 \.Y/ ! n \.X; A/ are isomorphisms (Anatoly Fomenko 131) - For a spheroid gW \.Sn 1 ; y0 / ! \.Y; const/ we define a spheroid GW Dn D CSn 1 ! X of the pair \.X; A/ by the formula G\.s; t/ D Œg\.s/\.t/; s 2 Sn 1 ; t 2 I\. It is obvious that these canonical maps are included in the commutative diagram (Anatoly Fomenko 131) - Thus, the correspondence f 7! †f gives rise to a homomorphism q \.X/ ! qC1 \.†X/\. This homomorphism is called the suspension homomorphism and is also denoted by †\. In particular, for every q and n, there arises a homomorphism †W q \.Sn / ! qC1 \.SnC1 /: Theorem \(Freudenthal\)\. This homomorphism is an isomorphism if q < 2n1 and is an epimorphism if q D 2n 1\. (Anatoly Fomenko 133) - If X is an n-connected CW complex, then †W q \.X/ ! qC1 \.†X/ is an isomorphism for q < 2n C 1 and an epimorphism for q D 2n C 1\. (Anatoly Fomenko 133) - Thus, the correspondence f 7! †f gives rise to a homomorphism q \.X/ ! qC1 \.†X/\. This homomorphism is called the suspension homomorphism and is also denoted by †\. In particular, for every q and n, there arises a homomorphism †W q \.Sn / ! qC1 \.SnC1 /: Theorem \(Freudenthal\)\. This homomorphism is an isomorphism if q < 2n1 and is an epimorphism if q D 2n 1\. (Anatoly Fomenko 133) - If X is an n-connected CW complex, then †W q \.X/ ! qC1 \.†X/ is an isomorphism for q < 2n C 1 and an epimorphism for q D 2n C 1\. (Anatoly Fomenko 133) - an isotopy \(a homotopy consisting of homeomorphisms\) (Anatoly Fomenko 134) - \(This argument, and actually the whole proof of Freudenthal’s theorem, is based on the fact that polyhedra of dimensions p and q cannot be linked in a space of dimension > p C q C 1\. For example, two disjoint closed polygonal lines can be linked in R3 , but not in R4 ; see Fig\. 52\.\) (Anatoly Fomenko 134) - an isotopy \(a homotopy consisting of homeomorphisms\) (Anatoly Fomenko 134) - \(This argument, and actually the whole proof of Freudenthal’s theorem, is based on the fact that polyhedra of dimensions p and q cannot be linked in a space of dimension > p C q C 1\. For example, two disjoint closed polygonal lines can be linked in R3 , but not in R4 ; see Fig\. 52\.\) (Anatoly Fomenko 134) - Theorem \(Hopf\)\. n \.Sn / Š Z\. Proof\. For n D 1; 2, we already know this \(see Sect\. 6\.3 for n D 1 and Sect\. 9\.9 for n D 2\)\. For n 3, we have an isomorphism †W n 1 \.Sn 1 / ! n \.Sn /\. (Anatoly Fomenko 136) - Theorem \(Hopf\)\. n \.Sn / Š Z\. Proof\. For n D 1; 2, we already know this \(see Sect\. 6\.3 for n D 1 and Sect\. 9\.9 for n D 2\)\. For n 3, we have an isomorphism †W n 1 \.Sn 1 / ! n \.Sn /\. (Anatoly Fomenko 136) - If S n were contractible, the group n \.X/ would have been zero for any X\. (Anatoly Fomenko 137) - Homeomorphisms have degrees ˙1\. A suspension over a map S n ! Sn of degree d is a map SnC1 ! SnC1 of the same degree d\. (Anatoly Fomenko 137) - w we will describe a way of computing the degree of a map f W Sn ! Sn \. A point y 2 Sn is called a regular value of f if there is a neighborhood U in y homeomorphic to a ball D n such that f 1 \.U/ is a disjoint union of open sets U˛ such that f maps every U˛ homeomorphically onto U (Anatoly Fomenko 137) - If y 2 Sn is a regular value of f , then the inverse image f 1 \.y/ is finite [otherwise, f 1 \.y/ contains limit points, and no neighborhood of a limit point of f 1 \.y/ can be homeomorphically mapped onto a neighborhood of y]\. For every point z 2 f 1 \.y/, the map f either preserves or reverses the orientation [in the smooth case this is determined by the sign of the Jacobian of f at z; (Anatoly Fomenko 137) - If S n were contractible, the group n \.X/ would have been zero for any X\. (Anatoly Fomenko 137) - Homeomorphisms have degrees ˙1\. A suspension over a map S n ! Sn of degree d is a map SnC1 ! SnC1 of the same degree d\. (Anatoly Fomenko 137) - w we will describe a way of computing the degree of a map f W Sn ! Sn \. A point y 2 Sn is called a regular value of f if there is a neighborhood U in y homeomorphic to a ball D n such that f 1 \.U/ is a disjoint union of open sets U˛ such that f maps every U˛ homeomorphically onto U (Anatoly Fomenko 137) - If y 2 Sn is a regular value of f , then the inverse image f 1 \.y/ is finite [otherwise, f 1 \.y/ contains limit points, and no neighborhood of a limit point of f 1 \.y/ can be homeomorphically mapped onto a neighborhood of y]\. For every point z 2 f 1 \.y/, the map f either preserves or reverses the orientation [in the smooth case this is determined by the sign of the Jacobian of f at z; (Anatoly Fomenko 137) - Theorem\. If y is a regular value of f , then 1 deg f D #fz 2 f \.y/ j f preserves the orientation at zg #fz 2 f 1 \.y/ \.y/ z2f 1 \.y/ rev "\.z/; where "\.z/ is 1 if f preserves the orientation in the neighborhood of z and is 1 otherwise\. (Anatoly Fomenko 138) - Thus, the homotopy groups of spheres are arranged into stabilizing series of groups nCk \.Sn / with a fixed k: : : : † † ! nCk \.Sn / † † ! nCkC1 \.SnC1 / † † ! nCkC2 \.SnC2 / † † ! : : : with the stabilization occurring in the term 2kC2 \.SkC2 /: : : : † † ! 2kC1 \.SkC1 / epi epi ! 2kC2 \.SkC2 / iso iso ! 2kC3 \.SkC3 / iso iso ! : : : : The groups nCk \.Sn / with n k C 2 do not depend on n\. They are called stable, and for them the notation kS kS is used; the group 2kC1 \.SkC1 / is called metastable\. So far, we have almost no information on the homotopy groups of spheres; what we know is contained in the following table\. (Anatoly Fomenko 138) - Theorem\. If y is a regular value of f , then 1 deg f D #fz 2 f \.y/ j f preserves the orientation at zg #fz 2 f 1 \.y/ \.y/ z2f 1 \.y/ rev "\.z/; where "\.z/ is 1 if f preserves the orientation in the neighborhood of z and is 1 otherwise\. (Anatoly Fomenko 138) - Thus, the homotopy groups of spheres are arranged into stabilizing series of groups nCk \.Sn / with a fixed k: : : : † † ! nCk \.Sn / † † ! nCkC1 \.SnC1 / † † ! nCkC2 \.SnC2 / † † ! : : : with the stabilization occurring in the term 2kC2 \.SkC2 /: : : : † † ! 2kC1 \.SkC1 / epi epi ! 2kC2 \.SkC2 / iso iso ! 2kC3 \.SkC3 / iso iso ! : : : : The groups nCk \.Sn / with n k C 2 do not depend on n\. They are called stable, and for them the notation kS kS is used; the group 2kC1 \.SkC1 / is called metastable\. So far, we have almost no information on the homotopy groups of spheres; what we know is contained in the following table\. (Anatoly Fomenko 138) - In this table, slanted arrows denote †, the letter H means Hopf isomorphism, and the letters i and e mean, respectively, isomorphism and epimorphism\. We can add that since †W 3 \.S2 / ! 4 \.S3 / is an epimorphism and 3 \.S2 / D Z, the group 4 \.S3 / must be cyclic, and so must be the groups 4 \.S2 / and 1S \. (Anatoly Fomenko 139) - In conclusion, we remark that stable homotopy groups do not exist only for spheres\. For any topological space X, we can consider a sequence k \.X/ † † ! kC1 \.†X/ † † ! kC2 \.† X/ † † ! kC3 \.† X/ † † ! : : : : This sequence has a “limit” \(algebraists call it the direct limit \), but we actually do not need it, since this sequence always stabilizes at the term 2kC2 \.†kC2 X/\. (Anatoly Fomenko 139) - The product S m Sn of two spheres has a CW decomposition into four cells, of dimensions 0; m; n; and m C n\. The union of the first three cells is the bouquet S m \_ Sn \. The attaching map of the fourth cell, SmCn 1 ! Sm \_ Sn , is called the Whitehead map\. (Anatoly Fomenko 139) - In this table, slanted arrows denote †, the letter H means Hopf isomorphism, and the letters i and e mean, respectively, isomorphism and epimorphism\. We can add that since †W 3 \.S2 / ! 4 \.S3 / is an epimorphism and 3 \.S2 / D Z, the group 4 \.S3 / must be cyclic, and so must be the groups 4 \.S2 / and 1S \. (Anatoly Fomenko 139) - In conclusion, we remark that stable homotopy groups do not exist only for spheres\. For any topological space X, we can consider a sequence k \.X/ † † ! kC1 \.†X/ † † ! kC2 \.† X/ † † ! kC3 \.† X/ † † ! : : : : This sequence has a “limit” \(algebraists call it the direct limit \), but we actually do not need it, since this sequence always stabilizes at the term 2kC2 \.†kC2 X/\. (Anatoly Fomenko 139) - The product S m Sn of two spheres has a CW decomposition into four cells, of dimensions 0; m; n; and m C n\. The union of the first three cells is the bouquet S m \_ Sn \. The attaching map of the fourth cell, SmCn 1 ! Sm \_ Sn , is called the Whitehead map\. (Anatoly Fomenko 139) - Together, they form a map S m \_ Sn ! X, and the composition of this map with w is a spheroid hW SmCn 1 ! X\. (Anatoly Fomenko 140) - Thus, we get an operation which assigns to ˛ 2 m \.X; x0 / and ˇ 2 n \.X; x0 / some element of mCn 1 \.X; x0 /; (Anatoly Fomenko 140) - this element is called the Whitehead product of ˛ and ˇ and is denoted as Œ˛; ˇ\. (Anatoly Fomenko 140) - EXERCISE 5\. Prove that if m; n; k > 1, then \.1/mkCn Œ˛; Œˇ; C \.1/nmCk Œˇ; Œ ; ˛ C \.1/knCm Œ ; Œ˛; ˇ D 0 \(“super-Jacobi identity”\)\. (Anatoly Fomenko 140) - Together, they form a map S m \_ Sn ! X, and the composition of this map with w is a spheroid hW SmCn 1 ! X\. (Anatoly Fomenko 140) - Thus, we get an operation which assigns to ˛ 2 m \.X; x0 / and ˇ 2 n \.X; x0 / some element of mCn 1 \.X; x0 /; (Anatoly Fomenko 140) - this element is called the Whitehead product of ˛ and ˇ and is denoted as Œ˛; ˇ\. (Anatoly Fomenko 140) - EXERCISE 5\. Prove that if m; n; k > 1, then \.1/mkCn Œ˛; Œˇ; C \.1/nmCk Œˇ; Œ ; ˛ C \.1/knCm Œ ; Œ˛; ˇ D 0 \(“super-Jacobi identity”\)\. (Anatoly Fomenko 140) - COMMENT\. Although the higher homotopy groups are commutative, the Whitehead product may be regarded as a substitute for a commutator in these groups\. The properties above create for homotopy groups \(more rigorously, for the direct sum pro 1 nD2 n \) a structure similar to that of a Lie superalgebra\. (Anatoly Fomenko 141) - XERCISE 6\. Prove that the suspension over the Whitehead product †Œ˛; ˇ 2 mCn \.†X/ is 0\. This implies \(and, actually, is implied by\) the fact that †\.Sm Sn / is homotopy equivalent to S mC1 \_ SnC1 \_ SmCnC1 ; why? (Anatoly Fomenko 141) - EXERCISE 7\. Let n 2 n \.Sn / be the class of the identity spheroid and 2 2 3 \.S2 / be the class of the Hopf map\. Prove that Œ2 ; 2 D 22 \. (Anatoly Fomenko 141) - EXERCISE 8\. Prove that if X is a topological group, or an H-space, then Œ˛; ˇ D 0 for any ˛ 2 m \.X/; ˇ 2 n \.X/\. \( (Anatoly Fomenko 141) - Theorem \(Difficult Part of Freudenthal’s Theorem\)\. The kernel of the homomorphism †W 2n 1 \.Sn / ! 2n \.SnC1 / is a cyclic group generated by the Whitehead square Œn ; n of the class n \. (Anatoly Fomenko 141) - Notice that in combination with Exercise 6, this theorem shows that 4 \.S3 / Š Z2 , and hence 1S Š Z2 \. By the way, the alternative 4 \.S3 / D Z2 or 0 follows directly from \(rather easy\) Exercises 6 and 7\. (Anatoly Fomenko 141) - Notice also that if n is even, then the cyclic group generated by Œn ; n is infinite \(we will prove this in Lecture 16\)\. For n odd, this group is Z2 or 0 \(this follows from Exercise 2\) (Anatoly Fomenko 141) - COMMENT\. Although the higher homotopy groups are commutative, the Whitehead product may be regarded as a substitute for a commutator in these groups\. The properties above create for homotopy groups \(more rigorously, for the direct sum pro 1 nD2 n \) a structure similar to that of a Lie superalgebra\. (Anatoly Fomenko 141) - XERCISE 6\. Prove that the suspension over the Whitehead product †Œ˛; ˇ 2 mCn \.†X/ is 0\. This implies \(and, actually, is implied by\) the fact that †\.Sm Sn / is homotopy equivalent to S mC1 \_ SnC1 \_ SmCnC1 ; why? (Anatoly Fomenko 141) - EXERCISE 7\. Let n 2 n \.Sn / be the class of the identity spheroid and 2 2 3 \.S2 / be the class of the Hopf map\. Prove that Œ2 ; 2 D 22 \. (Anatoly Fomenko 141) - EXERCISE 8\. Prove that if X is a topological group, or an H-space, then Œ˛; ˇ D 0 for any ˛ 2 m \.X/; ˇ 2 n \.X/\. \( (Anatoly Fomenko 141) - Theorem \(Difficult Part of Freudenthal’s Theorem\)\. The kernel of the homomorphism †W 2n 1 \.Sn / ! 2n \.SnC1 / is a cyclic group generated by the Whitehead square Œn ; n of the class n \. (Anatoly Fomenko 141) - Notice that in combination with Exercise 6, this theorem shows that 4 \.S3 / Š Z2 , and hence 1S Š Z2 \. By the way, the alternative 4 \.S3 / D Z2 or 0 follows directly from \(rather easy\) Exercises 6 and 7\. (Anatoly Fomenko 141) - Notice also that if n is even, then the cyclic group generated by Œn ; n is infinite \(we will prove this in Lecture 16\)\. For n odd, this group is Z2 or 0 \(this follows from Exercise 2\) (Anatoly Fomenko 141) - Van Kampen’s theorem, which has no satisfactory generalizations to higher-dimensional homotopy groups (Anatoly Fomenko 143) - Theorem\. Let X be a path connected topological space, and let f W Sn ! X be a continuous map\. Let Y D X [f DnC1 \. The homomorphism i \.X; x0 / ! i \.Y; x0 /; \(\) where x0 2 X Y is an arbitrarily chosen base point, induced by the inclusion map X ! Y is an isomorphism if i < n; if i D n, then it is an epimorphism whose kernel is generated by all classes of the form u# Œf , where u is a path joining f \.y0 / with x0 \(here y0 is a base point of S \) and Œf 2 n \.X; f \.y0 / is the homotopy class of the spheroid f \. (Anatoly Fomenko 143) - Van Kampen’s theorem, which has no satisfactory generalizations to higher-dimensional homotopy groups (Anatoly Fomenko 143) - Theorem\. Let X be a path connected topological space, and let f W Sn ! X be a continuous map\. Let Y D X [f DnC1 \. The homomorphism i \.X; x0 / ! i \.Y; x0 /; \(\) where x0 2 X Y is an arbitrarily chosen base point, induced by the inclusion map X ! Y is an isomorphism if i < n; if i D n, then it is an epimorphism whose kernel is generated by all classes of the form u# Œf , where u is a path joining f \.y0 / with x0 \(here y0 is a base point of S \) and Œf 2 n \.X; f \.y0 / is the homotopy class of the spheroid f \. (Anatoly Fomenko 143) - Theorem\. Let X; Y be CW complexes\. \(1\) If X is p-connected and Y is q-connected where p; q 1, then n \.X \_ Y/ D n \.X/ ˚ n \.Y/ for n p C q\. \(2\) For any n, n \.X \_ Y/ contains a direct summand isomorphic to n \.X/ ˚ n \.Y/\. (Anatoly Fomenko 144) - n \.XY/ D n \.X/˚n \.Y/\. (Anatoly Fomenko 144) - Theorem\. Let X; Y be CW complexes\. \(1\) If X is p-connected and Y is q-connected where p; q 1, then n \.X \_ Y/ D n \.X/ ˚ n \.Y/ for n p C q\. \(2\) For any n, n \.X \_ Y/ contains a direct summand isomorphic to n \.X/ ˚ n \.Y/\. (Anatoly Fomenko 144) - n \.XY/ D n \.X/˚n \.Y/\. (Anatoly Fomenko 144) - EXERCISE 1\. For X; Y; p; and q as in the theorem, prove that pCqC1 \.X \_ Y/ is isomorphic to pCqC1 \.X/ ˚ pCqC1 \.Y/ ˚ ŒpC1 \.X/ ˝ qC1 \.Y/, where the last summand is embedded into pCqC1 \.X \_Y/ be means of the map ˛ ˝ˇ 7! Œi ˛; j ˇ\. In particular, 3 \.S2 \_ S2 / Š Z ˚ Z ˚ Z\. (Anatoly Fomenko 145) - Remark\. There is a result called the Hilton–Milnor theorem stating that the homotopy groups of an arbitrary bouquet of spheres, S n1 \_ \_ Snr , are generated by elements of homotopy groups of spheres S mi and their Whitehead products (Anatoly Fomenko 145) - we assume now that the CW complex considered is \.n 1/-connected where n > 1\. Then we can assume that X has only one vertex and has no cells of dimension 1; : : : ; n1\. (Anatoly Fomenko 145) - In this case, skn X is homeomorphic to by elements the bouquet i i S in i of n-dimensional spheres corresponding to n-dimensional cells of X \(the homeomorphism is established by characteristic maps of the n-dimensional cells\)\. Thus, n \.skn \.X// Š n \. lis i S i / D ha i Z \( (Anatoly Fomenko 145) - Theorem\. Let X be a CW complex with one vertex and with no other cells of dimension < n\. The group n \.X/ has a system of generators corresponding to ndimensional cells \(the classes of characteristic maps of n-dimensional cells\) and defining system of relations corresponding to \.n C 1/-dimensional cells [the classes of attaching maps of \.n C 1/-dimensional cells are equated to zero]\. (Anatoly Fomenko 145) - EXERCISE 1\. For X; Y; p; and q as in the theorem, prove that pCqC1 \.X \_ Y/ is isomorphic to pCqC1 \.X/ ˚ pCqC1 \.Y/ ˚ ŒpC1 \.X/ ˝ qC1 \.Y/, where the last summand is embedded into pCqC1 \.X \_Y/ be means of the map ˛ ˝ˇ 7! Œi ˛; j ˇ\. In particular, 3 \.S2 \_ S2 / Š Z ˚ Z ˚ Z\. (Anatoly Fomenko 145) - Remark\. There is a result called the Hilton–Milnor theorem stating that the homotopy groups of an arbitrary bouquet of spheres, S n1 \_ \_ Snr , are generated by elements of homotopy groups of spheres S mi and their Whitehead products (Anatoly Fomenko 145) - we assume now that the CW complex considered is \.n 1/-connected where n > 1\. Then we can assume that X has only one vertex and has no cells of dimension 1; : : : ; n1\. (Anatoly Fomenko 145) - In this case, skn X is homeomorphic to by elements the bouquet i i S in i of n-dimensional spheres corresponding to n-dimensional cells of X \(the homeomorphism is established by characteristic maps of the n-dimensional cells\)\. Thus, n \.skn \.X// Š n \. lis i S i / D ha i Z \( (Anatoly Fomenko 145) - Theorem\. Let X be a CW complex with one vertex and with no other cells of dimension < n\. The group n \.X/ has a system of generators corresponding to ndimensional cells \(the classes of characteristic maps of n-dimensional cells\) and defining system of relations corresponding to \.n C 1/-dimensional cells [the classes of attaching maps of \.n C 1/-dimensional cells are equated to zero]\. (Anatoly Fomenko 145) - EXERCISE 2\. Prove the following relative version of the theorem\. Let \.X; A/ be a CW pair with connected A such that X A contains no cells of dimension < n, where n 3\. Then the first nontrivial group of the pair \.X; A/, that is, the group n \.X; A/, is generated as a 1 \.A/-module by n-dimensional cells in X A with relations corresponding to \.n C 1/-dimensional cells in X A\. (Anatoly Fomenko 146) - EXERCISE 4\. Let \.X; A/ be a CW pair with simply connected A, and let all cells in X A have dimensions n 2\. Prove that the natural map n \.X; A/ ! n \.X=A/ is an isomorphism\. (Anatoly Fomenko 146) - Remark\. This proposition has a generalization: If A is k-connected and all cells in X A have dimensions n 2, then the natural map q \.X; A/ ! q \.X=A/ is an isomorphism for g n C k 1 and an epimorphism for g D n C k\. (Anatoly Fomenko 146) - EXERCISE 2\. Prove the following relative version of the theorem\. Let \.X; A/ be a CW pair with connected A such that X A contains no cells of dimension < n, where n 3\. Then the first nontrivial group of the pair \.X; A/, that is, the group n \.X; A/, is generated as a 1 \.A/-module by n-dimensional cells in X A with relations corresponding to \.n C 1/-dimensional cells in X A\. (Anatoly Fomenko 146) - EXERCISE 4\. Let \.X; A/ be a CW pair with simply connected A, and let all cells in X A have dimensions n 2\. Prove that the natural map n \.X; A/ ! n \.X=A/ is an isomorphism\. (Anatoly Fomenko 146) - Remark\. This proposition has a generalization: If A is k-connected and all cells in X A have dimensions n 2, then the natural map q \.X; A/ ! q \.X=A/ is an isomorphism for g n C k 1 and an epimorphism for g D n C k\. (Anatoly Fomenko 146) - Theorem\. Let X and Y be CW complexes, and let f W X ! Y be s continuous map If f W n \.X; x0 / ! n \.Y; f \.x0 // is an isomorphism for all n and x0 , then f is a homotopy equivalence\. (Anatoly Fomenko 147) - Theorem\. Let X and Y be CW complexes, and let f W X ! Y be s continuous map If f W n \.X; x0 / ! n \.Y; f \.x0 // is an isomorphism for all n and x0 , then f is a homotopy equivalence\. (Anatoly Fomenko 147) - It follows from Whitehead’s theorem that if all the homotopy groups of some \(nonempty, connected\) CW complex are trivial, then this CW complex is contractible \(homotopy equivalent to a point\) (Anatoly Fomenko 149) - However, in a general case coincidence of homotopy groups is not sufficient for a homotopy equivalence; it is required additionally that the isomorphism between homotopy groups is established by some continuous map (Anatoly Fomenko 149) - EXERCISE 5\. Show that the spaces S and S 3 CP1 have equal homotopy groups, but are not homotopy equivalent\. (Anatoly Fomenko 149) - EXERCISE 6\. Show that the spaces S m RPn and Sn RPm \.m ¤ n/ have equal homotopy groups, but are not homotopy equivalent\. (Anatoly Fomenko 149) - Theorem\. For every topological space X there exists a CW complex Y with a weak homotopy equivalence f W Y ! X\. [Such a pair \.Y; f / is called a cellular approximation of X\.] (Anatoly Fomenko 149) - It follows from Whitehead’s theorem that if all the homotopy groups of some \(nonempty, connected\) CW complex are trivial, then this CW complex is contractible \(homotopy equivalent to a point\) (Anatoly Fomenko 149) - However, in a general case coincidence of homotopy groups is not sufficient for a homotopy equivalence; it is required additionally that the isomorphism between homotopy groups is established by some continuous map (Anatoly Fomenko 149) - EXERCISE 5\. Show that the spaces S and S 3 CP1 have equal homotopy groups, but are not homotopy equivalent\. (Anatoly Fomenko 149) - EXERCISE 6\. Show that the spaces S m RPn and Sn RPm \.m ¤ n/ have equal homotopy groups, but are not homotopy equivalent\. (Anatoly Fomenko 149) - Theorem\. For every topological space X there exists a CW complex Y with a weak homotopy equivalence f W Y ! X\. [Such a pair \.Y; f / is called a cellular approximation of X\.] (Anatoly Fomenko 149) - EXERCISE 8\. Prove that topological spaces X and Y are weakly homotopy equivalent if and only if there exist maps X f Z g ! Y; where Z is a CW complex and f ; g are weak homotopy equivalences\. (Anatoly Fomenko 150) - 1\.7 Eilenberg–MacLane Spaces \(K\.; n/s\) Theorem\. Let n be a positive integer, and let be a group which is supposed to be commutative if n > 1\. Then there exists a CW complex X such that q \.X/ D ; if q D n; 0; if q ¤ n: \(compare with the end of Lecture 4\.\) Such spaces are called Eilenberg–MacLane spaces or spaces of type K\.; n/\. People sometimes say that X is a K\.; n/\. Proof of Theorem (Anatoly Fomenko 150) - EXERCISE 8\. Prove that topological spaces X and Y are weakly homotopy equivalent if and only if there exist maps X f Z g ! Y; where Z is a CW complex and f ; g are weak homotopy equivalences\. (Anatoly Fomenko 150) - 1\.7 Eilenberg–MacLane Spaces \(K\.; n/s\) Theorem\. Let n be a positive integer, and let be a group which is supposed to be commutative if n > 1\. Then there exists a CW complex X such that q \.X/ D ; if q D n; 0; if q ¤ n: \(compare with the end of Lecture 4\.\) Such spaces are called Eilenberg–MacLane spaces or spaces of type K\.; n/\. People sometimes say that X is a K\.; n/\. Proof of Theorem (Anatoly Fomenko 150) - Remark\. The construction in the proof is very far from being explicit: We do not know the groups nC1 \.XnC1 /; nC2 \.XnC2 /; : : : and have no technical means to compute them\. (Anatoly Fomenko 151) - his will make the construction natural \(functorial\), but it will become tremendously inconvenient\. This makes especially interesting the relatively few known explicit constructions of K\.; n/s\. (Anatoly Fomenko 151) - The space CP1 is a space of type K\.Z; 2/\. This is the only case when a K\.; n/ with n > 2 has a geometrically explicit construction (Anatoly Fomenko 151) - S 1 is a K\.Z; 1/\. (Anatoly Fomenko 151) - RP1 has the type K\.Z2 ; 1/\. (Anatoly Fomenko 151) - The infinite-dimensional lens space L 1 m 1 1 D S =Zm , where the generator T of the group Zm acts in S C1 by the 2i=m 2i=m formula T\.z1 ; z2 ; : : : / D \.z1 e ; z2 e ; : : : /, and is a space of type K\.Zm ; 1/\. (Anatoly Fomenko 151) - Since K\.1 ; n/ K\.2 ; n/ D K\.1 2 ; n/, constructions \(2\)–\(4\) give us spaces of type K\.; 1/ for every finitely generated Abelian group \. (Anatoly Fomenko 151) - There are lots of known spaces of type K\.; 1/ with non-Abelian , for example, all classical surfaces, except S 2 and RP (Anatoly Fomenko 151) - EXERCISE 9\. Prove that the space of all unordered sets of n points in R1 \(or S1 \) is a K\.Sn ; 1/\. (Anatoly Fomenko 151) - EXERCISE 10\. Prove that the space of all unordered sets of n points in the plane is a K\.; 1/ for a certain group \. This group is called the Artin n-thread braid group\. (Anatoly Fomenko 151) - EXERCISE 11\. Do the same for the space of ordered n-point subsets of the plane\. \(The group arising is called the group of pure braids\. It is better to do this exercise before the previous one\.\) (Anatoly Fomenko 151) - EXERCISE 12\. Prove that a complete nonpositively curved Riemannian manifold is a space of type K\.; 1/ for some \. \(The proof is based on the fact that in a simply connected complete negatively curved Riemannian manifold every two points are connected by a unique geodesic\.\) (Anatoly Fomenko 151) - Remark\. A complement to a knot in S 3 is also a space of type K\.; 1/, but it is not (Anatoly Fomenko 151) - likely that the reader is able to prove it with the technical means currently at hand\. (Anatoly Fomenko 151) - Remark\. The construction in the proof is very far from being explicit: We do not know the groups nC1 \.XnC1 /; nC2 \.XnC2 /; : : : and have no technical means to compute them\. (Anatoly Fomenko 151) - his will make the construction natural \(functorial\), but it will become tremendously inconvenient\. This makes especially interesting the relatively few known explicit constructions of K\.; n/s\. (Anatoly Fomenko 151) - The space CP1 is a space of type K\.Z; 2/\. This is the only case when a K\.; n/ with n > 2 has a geometrically explicit construction (Anatoly Fomenko 151) - S 1 is a K\.Z; 1/\. (Anatoly Fomenko 151) - RP1 has the type K\.Z2 ; 1/\. (Anatoly Fomenko 151) - The infinite-dimensional lens space L 1 m 1 1 D S =Zm , where the generator T of the group Zm acts in S C1 by the 2i=m 2i=m formula T\.z1 ; z2 ; : : : / D \.z1 e ; z2 e ; : : : /, and is a space of type K\.Zm ; 1/\. (Anatoly Fomenko 151) - Since K\.1 ; n/ K\.2 ; n/ D K\.1 2 ; n/, constructions \(2\)–\(4\) give us spaces of type K\.; 1/ for every finitely generated Abelian group \. (Anatoly Fomenko 151) - There are lots of known spaces of type K\.; 1/ with non-Abelian , for example, all classical surfaces, except S 2 and RP (Anatoly Fomenko 151) - EXERCISE 9\. Prove that the space of all unordered sets of n points in R1 \(or S1 \) is a K\.Sn ; 1/\. (Anatoly Fomenko 151) - EXERCISE 10\. Prove that the space of all unordered sets of n points in the plane is a K\.; 1/ for a certain group \. This group is called the Artin n-thread braid group\. (Anatoly Fomenko 151) - EXERCISE 11\. Do the same for the space of ordered n-point subsets of the plane\. \(The group arising is called the group of pure braids\. It is better to do this exercise before the previous one\.\) (Anatoly Fomenko 151) - EXERCISE 12\. Prove that a complete nonpositively curved Riemannian manifold is a space of type K\.; 1/ for some \. \(The proof is based on the fact that in a simply connected complete negatively curved Riemannian manifold every two points are connected by a unique geodesic\.\) (Anatoly Fomenko 151) - Remark\. A complement to a knot in S 3 is also a space of type K\.; 1/, but it is not (Anatoly Fomenko 151) - likely that the reader is able to prove it with the technical means currently at hand\. (Anatoly Fomenko 151) - EXERCISE 13\. Prove that K\.; n/ D K\.; n 1/\. [This shows that every K\.; n/ with an Abelian is an H-space, and even a homotopy commutative H-space\. Actually, every K\.; n/ with an Abelian can be constructed as an Abelian topological group\.] (Anatoly Fomenko 152) - Theorem\. Any two spaces of type K\.; n/ are weakly homotopy equivalent\. Hence, any two CW complexes of type K\.; n/ are homotopy equivalent\. (Anatoly Fomenko 152) - Moreover, if X is a K\.; n/ and Y is a K\.; n/ and also a CW complex, then for every homomorphism 'W ! there exists a homotopically unique continuous map f W Y ! X such that f W n \.Y/ ! n \.X/ is '\. (Anatoly Fomenko 152) - Remark\. This property of K\.; n/s that their \(weak\) homotopy type is determined by their homotopy groups is not generalizable to spaces with multiple homotopy groups\. (Anatoly Fomenko 152) - \. There exist CW complexes X; Y such that each has two nontrivial homotopy groups, and these groups for X and Y are the same, but, however, X and Y are not homotopy equivalent\. In Chap\. III we will be able not only to find such examples, but even to provide a sort of a classification for them\. (Anatoly Fomenko 152) - EXERCISE 13\. Prove that K\.; n/ D K\.; n 1/\. [This shows that every K\.; n/ with an Abelian is an H-space, and even a homotopy commutative H-space\. Actually, every K\.; n/ with an Abelian can be constructed as an Abelian topological group\.] (Anatoly Fomenko 152) - Theorem\. Any two spaces of type K\.; n/ are weakly homotopy equivalent\. Hence, any two CW complexes of type K\.; n/ are homotopy equivalent\. (Anatoly Fomenko 152) - Moreover, if X is a K\.; n/ and Y is a K\.; n/ and also a CW complex, then for every homomorphism 'W ! there exists a homotopically unique continuous map f W Y ! X such that f W n \.Y/ ! n \.X/ is '\. (Anatoly Fomenko 152) - Remark\. This property of K\.; n/s that their \(weak\) homotopy type is determined by their homotopy groups is not generalizable to spaces with multiple homotopy groups\. (Anatoly Fomenko 152) - \. There exist CW complexes X; Y such that each has two nontrivial homotopy groups, and these groups for X and Y are the same, but, however, X and Y are not homotopy equivalent\. In Chap\. III we will be able not only to find such examples, but even to provide a sort of a classification for them\. (Anatoly Fomenko 152) - In conclusion, we will discuss two constructions for CW complexes which affect their homotopy groups in a prescribed way\. The first of them is known to us, and we have used it several times without naming it: It is capping homotopy groups\. Namely, if X is a CW complex, then for any number n we can construct a CW complex X 0 which contains X and has homotopy groups q \.X 0 q \.X/; if q n; 0; if q > nI moreover, the inclusion map X ! X 0 induces isomorphisms for all homotopy groups of dimensions n\. (Anatoly Fomenko 153) - n\. This is achieved by multiple attaching cells of dimensions > n C 1\. This capping operation is homotopically unique, as the following exercise shows\. (Anatoly Fomenko 153) - Let n \.X/ be the first nontrivial homotopy group of a CW complex X\. Then the capping operation gives rise to a homotopically unique map \(embedding\) X ! K\.n \.X/; n/\. We turn this map into a homotopy equivalent \(strong\) Serre fibration \(see Sect\. 9\.7\) and denote the fiber of this fibration as XjnC1 \. This space which is defined canonically up to a homotopy equivalence is sometimes called the first and sometimes the \.n C 1/st killing space of X (Anatoly Fomenko 153) - In conclusion, we will discuss two constructions for CW complexes which affect their homotopy groups in a prescribed way\. The first of them is known to us, and we have used it several times without naming it: It is capping homotopy groups\. Namely, if X is a CW complex, then for any number n we can construct a CW complex X 0 which contains X and has homotopy groups q \.X 0 q \.X/; if q n; 0; if q > nI moreover, the inclusion map X ! X 0 induces isomorphisms for all homotopy groups of dimensions n\. (Anatoly Fomenko 153) - n\. This is achieved by multiple attaching cells of dimensions > n C 1\. This capping operation is homotopically unique, as the following exercise shows\. (Anatoly Fomenko 153) - Let n \.X/ be the first nontrivial homotopy group of a CW complex X\. Then the capping operation gives rise to a homotopically unique map \(embedding\) X ! K\.n \.X/; n/\. We turn this map into a homotopy equivalent \(strong\) Serre fibration \(see Sect\. 9\.7\) and denote the fiber of this fibration as XjnC1 \. This space which is defined canonically up to a homotopy equivalence is sometimes called the first and sometimes the \.n C 1/st killing space of X (Anatoly Fomenko 153) - EXERCISE 16\. Prove that for every connected CW complex X, the canonical map Xj2 ! X is homotopy equivalent to the universal covering\. (Anatoly Fomenko 154) - EXERCISE 17\. Prove that S 2 j3 S3 ; generalization: CPn j3 D CPn j2nC1 S2nC1 \. (Anatoly Fomenko 154) - Thus, we have two constructions of extinguishing homotopy groups, which are, actually, dual in the sense of Lecture 4\. We can kill homotopy groups of a space X above a certain dimension, and X is canonically mapped into the new space\. Or, we can kill homotopy groups of X below a certain dimension, and the new space is canonically mapped into X\. (Anatoly Fomenko 154) - EXERCISE 16\. Prove that for every connected CW complex X, the canonical map Xj2 ! X is homotopy equivalent to the universal covering\. (Anatoly Fomenko 154) - EXERCISE 17\. Prove that S 2 j3 S3 ; generalization: CPn j3 D CPn j2nC1 S2nC1 \. (Anatoly Fomenko 154) - Thus, we have two constructions of extinguishing homotopy groups, which are, actually, dual in the sense of Lecture 4\. We can kill homotopy groups of a space X above a certain dimension, and X is canonically mapped into the new space\. Or, we can kill homotopy groups of X below a certain dimension, and the new space is canonically mapped into X\. (Anatoly Fomenko 154) - The main geometric idea of homology is as follows\. Spheroids are replaced by cycles ; an n-dimensional cycle is, roughly, an n-dimensional surface, maybe a sphere, but it may be something different, say, a torus\. The relation of being homotopic is replaced by a relation of being homological : Two n-dimensional cycles are homological if they cobound a piece of surface of dimension n C 1\. How do we define cycles and those pieces of surfaces which they bound, the so-called chains? One can try to present them as continuous maps of some standard objects, spheres and something else \(k-dimensional manifolds?\)\. But this leads to severe difficulties, especially in dimensions > 2\. It is easier to define cycles and chains as the union of standard “bricks\.” The role of these bricks is assumed by “singular simplices\.” (Anatoly Fomenko 155) - The main geometric idea of homology is as follows\. Spheroids are replaced by cycles ; an n-dimensional cycle is, roughly, an n-dimensional surface, maybe a sphere, but it may be something different, say, a torus\. The relation of being homotopic is replaced by a relation of being homological : Two n-dimensional cycles are homological if they cobound a piece of surface of dimension n C 1\. How do we define cycles and those pieces of surfaces which they bound, the so-called chains? One can try to present them as continuous maps of some standard objects, spheres and something else \(k-dimensional manifolds?\)\. But this leads to severe difficulties, especially in dimensions > 2\. It is easier to define cycles and chains as the union of standard “bricks\.” The role of these bricks is assumed by “singular simplices\.” (Anatoly Fomenko 155) - There are also common notations Ker @n D Zn \.X/ and Im @nC1 D Bn \.X/\. Thus, Hn \.X/ D Zn \.X/=Bn \.X/\. Elements of the groups Zn \.X/ and Bn \.X/ are called, respectively, cycles and boundaries\. \(Thus, every boundary is a cycle, but the converse is, generally, false\.\) If the difference of two cycles is a boundary, then these cycles are called homologous\. Thus, the homology group is the (Anatoly Fomenko 157) - is the group of classes of homologous cycles \(which may be called homology classes \)\. (Anatoly Fomenko 157) - There are also common notations Ker @n D Zn \.X/ and Im @nC1 D Bn \.X/\. Thus, Hn \.X/ D Zn \.X/=Bn \.X/\. Elements of the groups Zn \.X/ and Bn \.X/ are called, respectively, cycles and boundaries\. \(Thus, every boundary is a cycle, but the converse is, generally, false\.\) If the difference of two cycles is a boundary, then these cycles are called homologous\. Thus, the homology group is the (Anatoly Fomenko 157) - is the group of classes of homologous cycles \(which may be called homology classes \)\. (Anatoly Fomenko 157) - Proposition 1\. The homology H H en \.X/ of the reduced singular complex \(called the reduced homology of X\) is related to the usual homology as follows\. If X is not empty, then H H if n ¤ 0; e e \.X/; H 0 \.X/ ˚ Z; if n D 0I if X is empty, then the only nonzero reduced homology group of X is H H (Anatoly Fomenko 158) - Back to algebra\. If C D fCn ; @n g and C 0 D fCn0 ; @0n g are two chain complexes, then a chain map, or a homomorphism 'W C ! C 0 , is defined as a sequence of group homomorphisms 'n W Cn ! Cn0 which make the diagram @ nC2 ! CnC1 @ nC1 @ @n @ n 1 @n 1 n n ? ? ? ? ? ? ? ? ? ? n? n? ? ? ? @ 0 nC2 @nC2 0 ! Cn nC1 @ 0 nC1 y @nC1 ! Cn0 n0 @ @0n ! Cn0 n0 1 @ 0 n 1 1 commutative\. (Anatoly Fomenko 158) - Proposition 1\. The homology H H en \.X/ of the reduced singular complex \(called the reduced homology of X\) is related to the usual homology as follows\. If X is not empty, then H H if n ¤ 0; e e \.X/; H 0 \.X/ ˚ Z; if n D 0I if X is empty, then the only nonzero reduced homology group of X is H H (Anatoly Fomenko 158) - Back to algebra\. If C D fCn ; @n g and C 0 D fCn0 ; @0n g are two chain complexes, then a chain map, or a homomorphism 'W C ! C 0 , is defined as a sequence of group homomorphisms 'n W Cn ! Cn0 which make the diagram @ nC2 ! CnC1 @ nC1 @ @n @ n 1 @n 1 n n ? ? ? ? ? ? ? ? ? ? n? n? ? ? ? @ 0 nC2 @nC2 0 ! Cn nC1 @ 0 nC1 y @nC1 ! Cn0 n0 @ @0n ! Cn0 n0 1 @ 0 n 1 1 commutative\. (Anatoly Fomenko 158) - en \.X/ ! Hen \.Y/ \(with the same obvious properties\)\. Again back to algebra\. Let C D fCn ; @n g and C 0 D fCn0 ; @0n g be two chain complexes and ' D f'n g; D f n gW C ! C 0 be two chain maps\. A chain homotopy 0 between ' and is a sequence D D fDn W Cn ! Cn nC1 g satisfying the identities Dn 1 ı @n C @ 0nC1 ı Dn D n 'n (Anatoly Fomenko 159) - For the reader’s convenience \(or inconvenience?\) we show all the maps involved in this definition in one diagram \(which, certainly, is not commutative\): If chain maps '; can be connected by a chain homotopy, they are called \(chain \) homotopic\. (Anatoly Fomenko 159) - Proposition 2\. If chain maps '; W C ! C 0 are homotopic, then the induced homology maps ' ; W Hn \.C/ ! Hn \.C 0 / are equal\. (Anatoly Fomenko 159) - EXERCISE 2\. A complex \.C/ is called contractible if the identity map idW C ! C is homotopic to the zero map 0W C ! C\. A complex \.C/ is called acyclic if Hn \.C/ D 0 for all n\. \(Warmup\) Prove that a contractible complex is acyclic\. (Anatoly Fomenko 159) - en \.X/ ! Hen \.Y/ \(with the same obvious properties\)\. Again back to algebra\. Let C D fCn ; @n g and C 0 D fCn0 ; @0n g be two chain complexes and ' D f'n g; D f n gW C ! C 0 be two chain maps\. A chain homotopy 0 between ' and is a sequence D D fDn W Cn ! Cn nC1 g satisfying the identities Dn 1 ı @n C @ 0nC1 ı Dn D n 'n (Anatoly Fomenko 159) - For the reader’s convenience \(or inconvenience?\) we show all the maps involved in this definition in one diagram \(which, certainly, is not commutative\): If chain maps '; can be connected by a chain homotopy, they are called \(chain \) homotopic\. (Anatoly Fomenko 159) - Proposition 2\. If chain maps '; W C ! C 0 are homotopic, then the induced homology maps ' ; W Hn \.C/ ! Hn \.C 0 / are equal\. (Anatoly Fomenko 159) - EXERCISE 2\. A complex \.C/ is called contractible if the identity map idW C ! C is homotopic to the zero map 0W C ! C\. A complex \.C/ is called acyclic if Hn \.C/ D 0 for all n\. \(Warmup\) Prove that a contractible complex is acyclic\. (Anatoly Fomenko 159) - Finally, we will establish a connection between chain homotopies considered here with homotopies between continuous maps\. \(This connection is actually a justification for the term “chain homotopy\.”\) Namely, we will show how a homotopy between continuous maps f ; gW X ! Y determines a chain homotopy between the maps f# ; g# of singular complexes\. (Anatoly Fomenko 160) - Consider the faces j ˛i \.0 i n; 0 j n C 1/ (Anatoly Fomenko 160) - Finally, we will establish a connection between chain homotopies considered here with homotopies between continuous maps\. \(This connection is actually a justification for the term “chain homotopy\.”\) Namely, we will show how a homotopy between continuous maps f ; gW X ! Y determines a chain homotopy between the maps f# ; g# of singular complexes\. (Anatoly Fomenko 160) - Consider the faces j ˛i \.0 i n; 0 j n C 1/ (Anatoly Fomenko 160) - Corollary 2\. A homotopy equivalence f W X ! Y induces for all n isomorphisms f W Hn \.x/ 2\. Š ! Hn \.Y/\. In particular, homotopy equivalent spaces have isomorphic homology groups\. (Anatoly Fomenko 161) - \(Question: And what about weak homotopy equivalence? The answer is in Lecture 14\.\) (Anatoly Fomenko 161) - Corollary 2\. A homotopy equivalence f W X ! Y induces for all n isomorphisms f W Hn \.x/ 2\. Š ! Hn \.Y/\. In particular, homotopy equivalent spaces have isomorphic homology groups\. (Anatoly Fomenko 161) - \(Question: And what about weak homotopy equivalence? The answer is in Lecture 14\.\) (Anatoly Fomenko 161) - en \.pt/ D 0 for all n\. e0 \.pt/ D 0; this shows that H A space whose homology is the same as that of pt is called acyclic\. Corollary \(of homotopy invariance of homology\)\. Contractible spaces are acyclic\. (Anatoly Fomenko 162) - The converse is not true; fans of the function sin 1 x will appreciate an example in Fig\. 61\. There are more interesting examples, say, the Poincaré sphere with one point deleted\. (Anatoly Fomenko 162) - Fig\. 61 A noncontractible acyclic space (Anatoly Fomenko 162) - en \.pt/ D 0 for all n\. e0 \.pt/ D 0; this shows that H A space whose homology is the same as that of pt is called acyclic\. Corollary \(of homotopy invariance of homology\)\. Contractible spaces are acyclic\. (Anatoly Fomenko 162) - The converse is not true; fans of the function sin 1 x will appreciate an example in Fig\. 61\. There are more interesting examples, say, the Poincaré sphere with one point deleted\. (Anatoly Fomenko 162) - Fig\. 61 A noncontractible acyclic space (Anatoly Fomenko 162) - Theorem\. If X is path connected, then H0 \.X/ D Z\. (Anatoly Fomenko 163) - EXERCISE 4\. Prove that if f W X ! Y is a continuous map between two path connected spaces, then f W H0 \.X/ ! H0 \.Y/ is an isomorphism\. (Anatoly Fomenko 163) - This shows that Cn \.X/ D pac belongs ˛ Cn \.X˛ /, where the X ˛ s are path components of X, and also Zn \.X/ D Cn ˛ Zn \.X ˛ /; Bn \.X/ D wh ˛ Bn \.X˛ /; Hn \.X/ D co L ˛ Hn \.X˛ /\. In particular, the two previous computations L L imply the following\. \(1\) For an arbitrary X, H0 \.X/ is a free Abelian group generated by the path components of X; \(2\) If the space X is discrete, then Hn \.X/ D 0 for any n ¤ 0\. (Anatoly Fomenko 163) - Let \.X; A/ be a topological pair; that is, A is a subset of a space X\. Then Cn \.A/ Cn \.X/\. The group Cn \.X; A/ D Cn \.X/=Cn \.A (Anatoly Fomenko 163) - / is called the groups of \(relative \) singular chains of the pair \.X; A/ or of X modulo A\. Obviously, Cn \.X; A/ is a free Abelian group generated by singular simplices f W n ! X such that f \.n / 6 A (Anatoly Fomenko 163) - Theorem\. If X is path connected, then H0 \.X/ D Z\. (Anatoly Fomenko 163) - EXERCISE 4\. Prove that if f W X ! Y is a continuous map between two path connected spaces, then f W H0 \.X/ ! H0 \.Y/ is an isomorphism\. (Anatoly Fomenko 163) - This shows that Cn \.X/ D pac belongs ˛ Cn \.X˛ /, where the X ˛ s are path components of X, and also Zn \.X/ D Cn ˛ Zn \.X ˛ /; Bn \.X/ D wh ˛ Bn \.X˛ /; Hn \.X/ D co L ˛ Hn \.X˛ /\. In particular, the two previous computations L L imply the following\. \(1\) For an arbitrary X, H0 \.X/ is a free Abelian group generated by the path components of X; \(2\) If the space X is discrete, then Hn \.X/ D 0 for any n ¤ 0\. (Anatoly Fomenko 163) - Let \.X; A/ be a topological pair; that is, A is a subset of a space X\. Then Cn \.A/ Cn \.X/\. The group Cn \.X; A/ D Cn \.X/=Cn \.A (Anatoly Fomenko 163) - / is called the groups of \(relative \) singular chains of the pair \.X; A/ or of X modulo A\. Obviously, Cn \.X; A/ is a free Abelian group generated by singular simplices f W n ! X such that f \.n / 6 A (Anatoly Fomenko 163) - The homology groups of this complex are denoted Hn \.X; A/ and are called relative homology groups\. One can say that Hn \.X; A/ is the quotient Zn \.X; A/=Bn \.X; A/ of the group of relative cycles over the group of relative boundaries\. Here a relative cycle is a singular chain of X whose boundary lies in A, and a relative boundary is a chain of X which becomes a boundary after adding a chain from A\. \(Obviously, relative boundaries are relative cycles\.\) (Anatoly Fomenko 164) - The homology groups of this complex are denoted Hn \.X; A/ and are called relative homology groups\. One can say that Hn \.X; A/ is the quotient Zn \.X; A/=Bn \.X; A/ of the group of relative cycles over the group of relative boundaries\. Here a relative cycle is a singular chain of X whose boundary lies in A, and a relative boundary is a chain of X which becomes a boundary after adding a chain from A\. \(Obviously, relative boundaries are relative cycles\.\) (Anatoly Fomenko 164) - \(We have to disappoint a reader who expects an exact “homology sequence of a fibration” relating homology groups of the total space, the base, and the fiber of a fibration\. The relations between homology and fibrations are more complicated, and we will thoroughly study them in the subsequent chapters of this book\.\) (Anatoly Fomenko 165) - \(We have to disappoint a reader who expects an exact “homology sequence of a fibration” relating homology groups of the total space, the base, and the fiber of a fibration\. The relations between homology and fibrations are more complicated, and we will thoroughly study them in the subsequent chapters of this book\.\) (Anatoly Fomenko 165) - \(1\) The inclusion X ! X [ CA, where X [ CA is obtained from X by attaching the cone over A, induces for every n an isomorphism Hn \.X; A/ Š Hn \.X [ CA; CA/ D Hn \.X [ CA; v/ D H (Anatoly Fomenko 166) - Part \(2\) follows from part \(1\) because of the homotopy equivalence X [ CA X=A for Borsuk pairs (Anatoly Fomenko 166) - In Sect\. 9\.10, we showed how relative homotopy groups can be presented as absolute homotopy groups of a certain space\. Here we do the same for homology groups, and it is obvious that for homology the construction is much simpler than for homotopy\. (Anatoly Fomenko 166) - If \.X; A/ is a Borsuk pair \(see Sect\. 5\.6\), for example, a CW pair \(see again Sect\. 5\.6\), then p W Hn \.X; A/ ! Hn \.X=A; a/ D H [where pW X ! X=A is the projection and a D p\.A/] is an isomorphism for all n\. (Anatoly Fomenko 166) - The proof of the theorem is based on the so-called refinement lemma, whose proof is based on the so-called transformator lemma\. Both lemmas \(especially, the first\) have considerable independent value (Anatoly Fomenko 166) - \(1\) The inclusion X ! X [ CA, where X [ CA is obtained from X by attaching the cone over A, induces for every n an isomorphism Hn \.X; A/ Š Hn \.X [ CA; CA/ D Hn \.X [ CA; v/ D H (Anatoly Fomenko 166) - Part \(2\) follows from part \(1\) because of the homotopy equivalence X [ CA X=A for Borsuk pairs (Anatoly Fomenko 166) - In Sect\. 9\.10, we showed how relative homotopy groups can be presented as absolute homotopy groups of a certain space\. Here we do the same for homology groups, and it is obvious that for homology the construction is much simpler than for homotopy\. (Anatoly Fomenko 166) - If \.X; A/ is a Borsuk pair \(see Sect\. 5\.6\), for example, a CW pair \(see again Sect\. 5\.6\), then p W Hn \.X; A/ ! Hn \.X=A; a/ D H [where pW X ! X=A is the projection and a D p\.A/] is an isomorphism for all n\. (Anatoly Fomenko 166) - The proof of the theorem is based on the so-called refinement lemma, whose proof is based on the so-called transformator lemma\. Both lemmas \(especially, the first\) have considerable independent value (Anatoly Fomenko 166) - EXERCISE 15\. Let \.X; A/ be a topological pair, and let B A\. The inclusion map \.X B; A B/ ! \.X; A/ induces a homomorphism Hn \.X B; A B/ ! Hn \.X; A/ called an excision homomorphism\. Prove that if B Int A, then the excision homomorphism is an isomorphism (Anatoly Fomenko 170) - EXERCISE 16\. Let X D A[B; A\B D C\. We suppose that the excision homomorphisms Hn \.B; C/ ! Hn \.X; A/ and Hn \.A; C/ ! Hn \.X; B/ are isomorphisms\. Then the homomorphisms Hn \.X/ j j ! Hn \.X; A/ exc: 1 exc: ! Hn \.B; C/ @ @ ! Hn 1 \.C/ Hn \.X/ j j ! Hn \.X; B/ exc: 1 exc: ! Hn \.A; C/ @ @ ! Hn 1 \.C/ are the same, and we denote them as n\. The sequence ! Hn \.C/ ˛ ˛n ! Hn \.A/ ˚ Hn \.B/ ˇ n ˇn ! Hn \.X/ n n ! Hn 1 \.C/ ! : : : ; where ˛n is the difference (Anatoly Fomenko 170) - n is the difference of the homomorphisms induced by the inclusions C ! A and C ! B and ˇn is the sum of the homomorphisms induced by the inclusions A ! X and B ! X, is called the Mayer–Vietoris homology sequence or the homology sequence of the triad \.XI A; B (Anatoly Fomenko 170) - EXERCISE 15\. Let \.X; A/ be a topological pair, and let B A\. The inclusion map \.X B; A B/ ! \.X; A/ induces a homomorphism Hn \.X B; A B/ ! Hn \.X; A/ called an excision homomorphism\. Prove that if B Int A, then the excision homomorphism is an isomorphism (Anatoly Fomenko 170) - EXERCISE 16\. Let X D A[B; A\B D C\. We suppose that the excision homomorphisms Hn \.B; C/ ! Hn \.X; A/ and Hn \.A; C/ ! Hn \.X; B/ are isomorphisms\. Then the homomorphisms Hn \.X/ j j ! Hn \.X; A/ exc: 1 exc: ! Hn \.B; C/ @ @ ! Hn 1 \.C/ Hn \.X/ j j ! Hn \.X; B/ exc: 1 exc: ! Hn \.A; C/ @ @ ! Hn 1 \.C/ are the same, and we denote them as n\. The sequence ! Hn \.C/ ˛ ˛n ! Hn \.A/ ˚ Hn \.B/ ˇ n ˇn ! Hn \.X/ n n ! Hn 1 \.C/ ! : : : ; where ˛n is the difference (Anatoly Fomenko 170) - n is the difference of the homomorphisms induced by the inclusions C ! A and C ! B and ˇn is the sum of the homomorphisms induced by the inclusions A ! X and B ! X, is called the Mayer–Vietoris homology sequence or the homology sequence of the triad \.XI A; B (Anatoly Fomenko 170) - heorem 2\. For any topological space X and any n, H H (Anatoly Fomenko 172) - Remark\. From the point of view of the Eckmann–Hilton duality \(Lecture 4\), this isomorphism is dual to n \.X/ D n 1 \.X/\. Freudenthal’s theorem \(Lecture 10\) is dual to a relation between the homology groups of X and X (Anatoly Fomenko 172) - heorem 2\. For any topological space X and any n, H H (Anatoly Fomenko 172) - Remark\. From the point of view of the Eckmann–Hilton duality \(Lecture 4\), this isomorphism is dual to n \.X/ D n 1 \.X/\. Freudenthal’s theorem \(Lecture 10\) is dual to a relation between the homology groups of X and X (Anatoly Fomenko 172) - Theorem 2\. If \.X˛ ; x˛ / are base point spaces which are Borsuk pairs, then for any m, H H ˛2A X˛ ˛2A H H (Anatoly Fomenko 173) - Theorem 2\. If \.X˛ ; x˛ / are base point spaces which are Borsuk pairs, then for any m, H H ˛2A X˛ ˛2A H H (Anatoly Fomenko 173) - Recall that a continuous map of S n into S n has a degree, an integer which characterizes its homotopy class \(Sect\. 10\.3\)\. A continuous map gW ˛2A S ˛n ! ˇ2B S ˇn ˇn \(where S ˛n ; Sˇn are copies of the sphere Sn \) has a whole matrix of degrees fd˛ˇ j ˛ 2 A; ˇ 2 Bg, where d˛ˇ is the degree of the map S i˛ i˛ ! S ˛n g ! S ˇn pˇ ! Sn ; (Anatoly Fomenko 174) - In particular, the map Hn \.Sn / f f ! Hn \.Sn / Z Z induced by the map f W Sn ! Sn of degree d is the multiplication by d\. (Anatoly Fomenko 174) - Recall that a continuous map of S n into S n has a degree, an integer which characterizes its homotopy class \(Sect\. 10\.3\)\. A continuous map gW ˛2A S ˛n ! ˇ2B S ˇn ˇn \(where S ˛n ; Sˇn are copies of the sphere Sn \) has a whole matrix of degrees fd˛ˇ j ˛ 2 A; ˇ 2 Bg, where d˛ˇ is the degree of the map S i˛ i˛ ! S ˛n g ! S ˇn pˇ ! Sn ; (Anatoly Fomenko 174) - In particular, the map Hn \.Sn / f f ! Hn \.Sn / Z Z induced by the map f W Sn ! Sn of degree d is the multiplication by d\. (Anatoly Fomenko 174) - Pre-lemma\. The space X n =X n 1 is homeomorphic to the bouquet ˛2An S ˛n ˛n ; (Anatoly Fomenko 175) - W em \.X n =X n 1 / D H em \. ˛2A S˛n /: The group Cn \.X/ D Hn \.X n ; X n 1 / is called the groups of cellular chains of X\. The cellular differential or cellular boundary operator @ D @n W Cn \.X/ ! Cn 1 \.X/ is defined as the connecting homomorphism (Anatoly Fomenko 175) - from the homology sequence of the triple \.X n ; X n 1 ; X n 2 / (Anatoly Fomenko 175) - Pre-lemma\. The space X n =X n 1 is homeomorphic to the bouquet ˛2An S ˛n ˛n ; (Anatoly Fomenko 175) - W em \.X n =X n 1 / D H em \. ˛2A S˛n /: The group Cn \.X/ D Hn \.X n ; X n 1 / is called the groups of cellular chains of X\. The cellular differential or cellular boundary operator @ D @n W Cn \.X/ ! Cn 1 \.X/ is defined as the connecting homomorphism (Anatoly Fomenko 175) - from the homology sequence of the triple \.X n ; X n 1 ; X n 2 / (Anatoly Fomenko 175) - There are two important things concerning cellular complexes\. First, it is far from being as big as the singular complex; for example, for finite CW complexes the cellular chain groups are finitely generated\. Moreover, not only the cellular chain groups, but also the cellular boundary operators have an explicit description that is easy to deal with\. Second, we will prove that the homology of the cellular complex is the same as the homology of the singular complex (Anatoly Fomenko 177) - There are two important things concerning cellular complexes\. First, it is far from being as big as the singular complex; for example, for finite CW complexes the cellular chain groups are finitely generated\. Moreover, not only the cellular chain groups, but also the cellular boundary operators have an explicit description that is easy to deal with\. Second, we will prove that the homology of the cellular complex is the same as the homology of the singular complex (Anatoly Fomenko 177) - en \.\.X en \.X =\.X e// \(which is One can say that the orientation of e is a choice of a generator in H H (Anatoly Fomenko 179) - be presented as finite integral linear combinations of Thus, chains in Cn \.X/ can oriented n-dimensional cells, be ki ei \. (Anatoly Fomenko 179) - en \.\.X en \.X =\.X e// \(which is One can say that the orientation of e is a choice of a generator in H H (Anatoly Fomenko 179) - be presented as finite integral linear combinations of Thus, chains in Cn \.X/ can oriented n-dimensional cells, be ki ei \. (Anatoly Fomenko 179) - heo (Anatoly Fomenko 180) - Theorem\. Let e be an oriented \.nC1/-dimensional cell of X regarded as an element of CnC1 \.X/\. Then X @nC1 \.e/ D Œe W f f ; (Anatoly Fomenko 180) - where the sum is taken over all n-dimensional cells of X with fixed orientations (Anatoly Fomenko 180) - e; X n / @ @ ! Hn \.X n ; X n f / Š Z: The choice of the isomorphisms with Z corresponds to the orientations of the cells e; f \. Every homomorphism Z ! Z is a multiplication by some integer\. This integer is called the incidence number of the oriented ce (Anatoly Fomenko 180) - Having this in mind, we can give our theorem an aggressively tautological form: The boundary of a cell is the sum of cells which appear in the boundary of this cell with coefficients equal to the multiplicity of their appearance in this boundary\. (Anatoly Fomenko 181) - Theorem 1\. If the number of n-dimensional cells of a CW complex X is N, then the group Hn \.X/ is generated by at most N generators; in particular, the nth Betti number Bn \.X/ does not exceed N\. For example, if X does not have n-dimensional cells at all, then Hn \.X/ D 0; in particular, if X is finite dimensional, then Hn \.X/ D 0 for all n > dim X\. \(Compare with homotopy groups!\) (Anatoly Fomenko 181) - Algebraic Lemma \(Euler–Poincaré\)\. Let @ nC2 ! CnC1 @ nC1 ! Cn @ @n ! Cn 1 @ n 1 @n 1 be a complex with the “total group” n Cn finitely generated\. Let cn be the rank of the group Cn and hn be the rank of the homology group Hn \. Then n \.1/n cn D n \.1/n hn : (Anatoly Fomenko 181) - Thus, the number n \.1/n cn does not depend on the CW structure; it is determined by the topology \(actually, be the homotopy type\) of X\. This number is called the Euler characteristic of X and is traditionally denoted by \.X/\. (Anatoly Fomenko 181) - Theorem 1\. If the number of n-dimensional cells of a CW complex X is N, then the group Hn \.X/ is generated by at most N generators; in particular, the nth Betti number Bn \.X/ does not exceed N\. For example, if X does not have n-dimensional cells at all, then Hn \.X/ D 0; in particular, if X is finite dimensional, then Hn \.X/ D 0 for all n > dim X\. \(Compare with homotopy groups!\) (Anatoly Fomenko 181) - Algebraic Lemma \(Euler–Poincaré\)\. Let @ nC2 ! CnC1 @ nC1 ! Cn @ @n ! Cn 1 @ n 1 @n 1 be a complex with the “total group” n Cn finitely generated\. Let cn be the rank of the group Cn and hn be the rank of the homology group Hn \. Then n \.1/n cn D n \.1/n hn : (Anatoly Fomenko 181) - Thus, the number n \.1/n cn does not depend on the CW structure; it is determined by the topology \(actually, be the homotopy type\) of X\. This number is called the Euler characteristic of X and is traditionally denoted by \.X/\. (Anatoly Fomenko 181) - Theorem 2 \(Excision Theorem\)\. Let X be a CW complex and let A; B be CW subcomplexes of X such that A [ B D X\. Then \(for every n\) Hn \.X; A/ D Hn \.B; A \ B/: Indeed, X=A and B=\.A \ B/ are the same as CW complexes\. (Anatoly Fomenko 182) - Theorem 3 \(Mayer–Vietoris Sequence\)\. Let X be a CW complex and let A; B be CW subcomplexes of X such that A [ B D X\. Then there exists an exact sequence ! Hn \.A \ B/ ! Hn \.A/ ˚ Hn \.B/ ! Hn \.X/ ! Hn 1 \.A \ B/ ! : : : \(see the description of maps in Exercise 14 of Lecture 12\)\. (Anatoly Fomenko 182) - Theorem 2 \(Excision Theorem\)\. Let X be a CW complex and let A; B be CW subcomplexes of X such that A [ B D X\. Then \(for every n\) Hn \.X; A/ D Hn \.B; A \ B/: Indeed, X=A and B=\.A \ B/ are the same as CW complexes\. (Anatoly Fomenko 182) - Theorem 3 \(Mayer–Vietoris Sequence\)\. Let X be a CW complex and let A; B be CW subcomplexes of X such that A [ B D X\. Then there exists an exact sequence ! Hn \.A \ B/ ! Hn \.A/ ˚ Hn \.B/ ! Hn \.X/ ! Hn 1 \.A \ B/ ! : : : \(see the description of maps in Exercise 14 of Lecture 12\)\. (Anatoly Fomenko 182) - The real case is more complicated, since RPn has cells e0 ; e1 ; e2 ; : : : Œ; en if n is finite]\. (Anatoly Fomenko 183) - Proof\. The attaching map f W Si ! RPi is the standard twofold covering\. The inverse image of \(actually, any\) point of RPi consists of two points, and the restrictions of f to neighborhoods of these points are related by the antipodal map S i ! Si \. (Anatoly Fomenko 183) - 13\.8 Some Calculations A: Spheres 171 We already know the homology of spheres, but let us calculate them again for practice in the technique based on cellular complexes\. The sphere Sn has a CW structure with two cells, of dimensions 0 and n (Anatoly Fomenko 183) - The real case is more complicated, since RPn has cells e0 ; e1 ; e2 ; : : : Œ; en if n is finite]\. (Anatoly Fomenko 183) - Proof\. The attaching map f W Si ! RPi is the standard twofold covering\. The inverse image of \(actually, any\) point of RPi consists of two points, and the restrictions of f to neighborhoods of these points are related by the antipodal map S i ! Si \. (Anatoly Fomenko 183) - This antipodal map preserves the orientation if i is odd and reverses the orientation if i is even\. Thus, the contributions of these two points in ŒeiC1 ; ei have the same sign if i is odd and have different signs if i is even (Anatoly Fomenko 184) - EXERCISE 10\. Find the Euler characteristics of all finite-dimensional projective spaces\. (Anatoly Fomenko 184) - The Betti numbers are as follows\. For i odd, Bi \.G\.n; k// D 0; for i even, this is the number of Young diagrams of i 2 cells contained in the k \.n k/ rectangl (Anatoly Fomenko 184) - This antipodal map preserves the orientation if i is odd and reverses the orientation if i is even\. Thus, the contributions of these two points in ŒeiC1 ; ei have the same sign if i is odd and have different signs if i is even (Anatoly Fomenko 184) - EXERCISE 10\. Find the Euler characteristics of all finite-dimensional projective spaces\. (Anatoly Fomenko 184) - The Betti numbers are as follows\. For i odd, Bi \.G\.n; k// D 0; for i even, this is the number of Young diagrams of i 2 cells contained in the k \.n k/ rectangl (Anatoly Fomenko 184) - Classical surfaces with holes are homotopy equivalent to bouquets of circles (Anatoly Fomenko 185) - so we will consider classical surfaces without holes\. The cellular complex for such a surface has the form @2 @2 @1 @1 C2 C1 C0 ; where the number of the summands Z in C1 is 2g; 2g C 1; or 2g C 2 if our surface is a sphere with g handles, a projective plane with g handles, or a Klein bottle with g handles, respectively\. (Anatoly Fomenko 185) - projective plane with g handles, or a Klein bottle with g handles, respectively\. The differential @1 is zero \(every one-dimensional cell has equal endpoints\)\. To find @2 , we consider the construction of the classical surface from a polygon (Anatoly Fomenko 185) - H0 \.X/ D Z Z 8̂ 8̂ 8̂ 8̂ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ always; always; Z ˚ ˚ Z 2g ; if X is a sphere with g handles; Z ˚ ˚ Z 2g ˚Z2 ; if X is a projective plane with g handles; Z ˚ ˚ Z 2gC1 ˚Z2 ; if X is a Klein bottle with g handles; H2 \.X/ D ˆ ˆ : Z; if X is a sphere with handles; 0 in all other cases (Anatoly Fomenko 185) - Classical surfaces with holes are homotopy equivalent to bouquets of circles (Anatoly Fomenko 185) - so we will consider classical surfaces without holes\. The cellular complex for such a surface has the form @2 @2 @1 @1 C2 C1 C0 ; where the number of the summands Z in C1 is 2g; 2g C 1; or 2g C 2 if our surface is a sphere with g handles, a projective plane with g handles, or a Klein bottle with g handles, respectively\. (Anatoly Fomenko 185) - projective plane with g handles, or a Klein bottle with g handles, respectively\. The differential @1 is zero \(every one-dimensional cell has equal endpoints\)\. To find @2 , we consider the construction of the classical surface from a polygon (Anatoly Fomenko 185) - H0 \.X/ D Z Z 8̂ 8̂ 8̂ 8̂ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ always; always; Z ˚ ˚ Z 2g ; if X is a sphere with g handles; Z ˚ ˚ Z 2g ˚Z2 ; if X is a projective plane with g handles; Z ˚ ˚ Z 2gC1 ˚Z2 ; if X is a Klein bottle with g handles; H2 \.X/ D ˆ ˆ : Z; if X is a sphere with handles; 0 in all other cases (Anatoly Fomenko 185) - every continuous map is homotopic to a cellular map\. (Anatoly Fomenko 186) - Thus, one can say that the cellular theory can be used as a substitute for the singular theory\. But without the singular theory \(which is topologically invariant from the very beginning\) we would have had to prove that homeomorphic CW complexes have isomorphic homology group (Anatoly Fomenko 186) - A cellular complex appears especially attractive when a CW structure is actually a triangulation (Anatoly Fomenko 186) - We consider a triangulated space X with an additional structure \(a substitute for fixing characteristic maps\): We suppose that the set of vertices of X is ordered, or, at least, vertices of every simplex are ordered in such a way that the ordering of vertices of a face of a simplex is always compatible with the ordering of vertices of this simplex\. We refer to such triangulations as ordered triangulations (Anatoly Fomenko 186) - It can be described very directly: Chains are integral linear combinations of simplices \(remember P the ordering!\), and the boundary is given by the very familiar formula @ i k i si D ordering!\), and the j k \.1/ s P P j i k i j \.1/ j si , where the si are simplices of our triangulation and the j si are their faces\. (Anatoly Fomenko 187) - The classical definition of homology created the necessity of proving a topological invariance theorem: Homeomorphic triangulated spaces have isomorphic homology groups\. (Anatoly Fomenko 187) - main conjecture \) of combinatorial topology: Any two triangulations of a topological space have simplicially equivalent subdivisions\. (Anatoly Fomenko 187) - But the Hauptvermutung turns out to be false: The first counterexample was found by J\. Milnor in 1961, and many other counterexamples were constructed later, in particular for simply connected smooth manifolds\. (Anatoly Fomenko 187) - There was an attempt to deduce the topological invariance of classical homology from the so-called Hauptvermutung \(German f (Anatoly Fomenko 187) - man for main con (Anatoly Fomenko 187) - les were constructed later, in particular for simply connected smooth manifolds\. The whole problem of topological invariance disappeared mysteriously when singular homology was defined\. The first definition of singular homology was (Anatoly Fomenko 187) - y was given by O\. Veblen in the late 1920s but became broadly known some 10 years later\. (Anatoly Fomenko 187) - The initial proof, due to J\. Alexander, was long and complicated \(hundreds of pages in old topology textbooks\)\. There (Anatoly Fomenko 187) - The connection between homology and homotopy groups is seen always from the preliminary description of homology in the beginning of Lecture 12: Spheroids are cycles and homotopical spheroids are homological cycles (Anatoly Fomenko 190) - This suggests that there must be a natural map from homotopy groups into homology groups\. This map, called the Hurewicz homomorphism, i (Anatoly Fomenko 190) - we also will show examples which should serve as a warning to a reader who expects too much of this connection\. (Anatoly Fomenko 190) - Theorem\. If f W X ! Y is a weak homotopy equivalence, then f W Hn \.X/ ! Hn \.Y/ is an isomorphism for all n\. (Anatoly Fomenko 190) - Recall that according to another result from Sect\. 11\.4, a map is a weak homotopy equivalence if and only if it induces an isomorphism in homotopy groups (Anatoly Fomenko 190) - Corollary\. If a continuous map induces an isomorphism between homotopy groups, then it also induces an isomorphism between homology groups\. (Anatoly Fomenko 190) - EXERCISE 1\. Prove that the spaces S2 and S3 CP1 have isomorphic homotopy groups but nonisomorphic homology groups\. Same for the spaces S m RPn and Sn RPm with m ¤ n; m ¤ 1; n ¤ 1\. (Anatoly Fomenko 191) - EXERCISE 2\. Prove that the spaces S1 S1 and S1 \_ S1 \_ S2 have isomorphic homology groups but nonisomorphic homotopy groups\. (Anatoly Fomenko 191) - EXERCISE 3\. Prove that the Hopf map S3 ! S2 induces a trivial homomorphism in reduced homology groups but a nontrivial homomorphism in homotopy groups\. (Anatoly Fomenko 191) - EXERCISE 4\. Prove that the projection map S1 S1 ! \.S1 S1 /=\.S1 \_ S1 / D S2 induces a trivial homomorphism in homotopy groups but a nontrivial homomorphism in reduced homology groups\. (Anatoly Fomenko 191) - Theorem \(Hurewicz\)\. Let 0 \.X; x0 / D D n 1 \.X; x0 / D 0, where n 2\. Then H 1 \.X/ D D Hn 1 \.X; x0 / D 0 and hW n \.X; x0 / ! Hn \.X; x0 / is an isomorphism\. (Anatoly Fomenko 192) - Corollary \(The Inverse Hurewicz Theorem\)\. If X is simply connected and H 2 \.X/ D D Hn 1 \.X/ D 0 \.n 2/, then 2 \.X/ D D n 1 \.X/ D 0 and hW n \.X/ ! Hn \.X/ is an isomorphism\. (Anatoly Fomenko 192) - Together these theorems mean that the first nontrivial homotopy and homology groups of a simply connected space occur in the same dimension and are isomorphic\. (Anatoly Fomenko 192) - EXERCISE 6\. Prove that a simply connected CW complex with the same homology groups as Sn is homotopy equivalent to Sn \. [Hint: Apply Whitehead’s theorem to a spheroid Sn ! X representing a generator of the group n \.X/ Š Z\.] Do the same for the bouquet of spheres of the same dimensions\. (Anatoly Fomenko 192) - Remark\. Thus, we see that the triviality of the homotopy groups, as well as the triviality of the homology groups, implies the homotopy triviality \(contractibility\) of a simply connected CW complex\. At the same time, we have the examples which show that neither the triviality of induced homotopy groups homomorphisms nor the triviality of induced homology homomorphisms secures homotopy triviality of a continuous map\. (Anatoly Fomenko 192) - EXERCISE 7\. Prove that the composition S proj: 1 S1 S1 ! \.S1 S1 S1 /= Hopf sk2 \.S1 S1 S1 / D S3 ! Hopf ! S2 induces a trivial map of both homotopy and homology groups but is not homotopic to a constant map\. (Anatoly Fomenko 192) - Theorem \(Poincaré\)\. For an arbitrary path connected space X, the Hurewicz homomorphism hW 1 \.X/ ! H1 \.X/ is an epimorphism whose kernel is the commutator subgroup Œ1 \.X/; 1 \.X/ of the group 1 \.X/\. Thus, H 1 \.X/ Š 1 \.X/=Œ1 \.X/; 1 \.X/: (Anatoly Fomenko 193) - EXERCISE 9\. Show that a loop f W S1 ! X determines an element of the kernel of the map hW 1 \.X/ ! H1 \.X/ \(“homologous to zero”\) if and only if it can be extended 1 to the map into X of the disk \(with the boundary S1 \) with handle (Anatoly Fomenko 193) - an be extended 1 to the map into X of the disk \(with the boundary S1 \) with handles\. Moreover, the minimal number of these handles is equal to the minimal number of commutators in 1 \.X/ whose product is Œf (Anatoly Fomenko 193) - EXERCISE 10\. The space XAb is called an Abelianization, or Quillenization, of a path connected space X if the fundamental group of XAb is Abelian and there exists a continuous map X ! XAb inducing an isomorphism Hn \.X/ ! Hn \.XAb / for every n\. Prove that X possesses an Abelianization if and only if Œ 1 \.X/; 1 \.X/ D Œ1 \.X/; Œ1 \.X/; 1 \.X/ ; that is, if every element of Œ1 \.X/; 1 \.X/ can be presented as a product of commutators of elements of 1 \.X/ with elements of Œ1 \.X/; 1 \.X/\. (Anatoly Fomenko 193) - But there are several remarkable examples of the Abelianization, two of which we will mention\. The first was discovered in 1971 by M\. Barratt, D\. Kahn, and S\. Priddy: The Abelianization of the space X D K\.S1 ; 1/, where S1 D [n Sn is the group of finite permutations of the set Z>0 , is XAb D \.1 S1 /0 D [n \.n Sn /0 \(the subscript 0 indicates that we consider only one component of the set\)\. Another example belongs to G\. Segal \(1973\) and (Anatoly Fomenko 193) - Remark\. Our definition of an Abelianization is a simplified version of a more common definition in which the space XAb is assumed simple \(see Sect\. 8\.2\), or even an H-space \(see Exercise 2 in Sect\. 8\.2\) or even a loop space \(s (Anatoly Fomenko 193) - This enhanced definition of an Abelianization plays an important technical role in one of the versions of constructing an algebraic K-functor\. The problem of the existence of an Abelianization in this sense is much more complicated, and there are no general theorems about it\. B (Anatoly Fomenko 193) - states that if X D K\.B\.1/; 1/ where B\.1/ is the infinite braid group and hence X is the set of \(unordered\) countable subsets of the plane consisting, for some N \(depending on the subset\), of points \.n C 1; 0/; \.n C 2; 0/; : : : and n more points different from each other and from the points listed above, then XAb is 2 S3 \. In both cases, the space X has a complicated fundamental group and trivial higher homotopy groups, and the space XAb has a simple fundamental group \(Z2 in the first case and Z in the second case\) and complicated, so far unknown, homotopy groups (Anatoly Fomenko 194) - y groups\. (Anatoly Fomenko 194) - \. For further details, see Barratt and Priddy [20], Segal [74], and Fuchs [37]\. (Anatoly Fomenko 194) - EXERCISE 11\. Prove that any two-dimensional homology class of an arbitrary space X can be represented by a sphere with handles; that is, for every ˛ 2 H2 \.X/, there exist a sphere with handles S and a continuous map f W S ! X such that the map f W H2 \.S/ ! H2 \.X/ takes the canonical generator of H2 \.S/ D Z into ˛\. (Anatoly Fomenko 194) - Proof The proof can be obtained from the proof of the theorem in Sect\. 14\.2 by modifications characteristic for a transition from the absolute case to a relative case\. (Anatoly Fomenko 194) - he relative Hurewicz homomorphism hW n \.X; A/ ! Hn \.X; A/ is defined similarly to the absolute one\. (Anatoly Fomenko 194) - The rela (Anatoly Fomenko 194) - Theorem\. Let \.X; A/ be a topological pair such that the space X is path connected and A is simply connected\. Let n 3\. \(1\) Suppose that 2 \.X; A/ D D n 1 \.X; A/ D 0\. Then H1 \.X; A/ D H2 \.X; A/ D D Hn 1 \.X; A/ D 0 and hW n \.X; A/ ! Hn \.X; A/ is an isomorphism\. \(2\) Suppose that H2 \.X; A/ D D Hn 1 \.X; A/ D 0\. Then 2 \.X; A/ D D n 1 \.X; A/ D 0 and hW n \.X; A/ ! Hn \.X; A/ is an isomorphism\. (Anatoly Fomenko 194) - EXERCISE 12\. If A is not simply connected, then part \(1\) of the theorem remains true with the following modification: Hn \.X; A/ is isomorphic to n \.X; A/ factorized over the natural action of 1 \.A/\. (Anatoly Fomenko 195) - Theorem\. Let X and Y be simply connected spaces, and let f W X ! Y be a continuous map such that f W 2 \.X/ ! 2 \.Y/ is an epimorphism\. \(1\) If the homomorphism f W m \.X/ ! m \.Y/ is an isomorphism for m < n and an epimorphism for m D n, then the same is true for f W Hm \.X/ ! Hm \.Y/\. \(2\) The same with and H swapped\. (Anatoly Fomenko 195) - Proof\. We may assume that f is an embedding, so \.Y; X/ is a topological pair\. The exactness of homotopy and homology sequences of this pair yields a translation of conditions and claims of the theorem into the language of relative homotopy and homology groups (Anatoly Fomenko 195) - Corollary\. If a continuous map f W X ! Y between simply connected topological spaces induces an epimorphism f W 2 \.X/ ! 2 \.Y/ and isomorphisms f W Hm \.X/ ! H m \.Y/ for all m, then f is a weak homotopy equivalence \(a homotopy equivalence, if X and Y are CW complexes\)\. (Anatoly Fomenko 195) - One can apply to the singular or cellular complex of a topological space the standard algebraic operations ˝ G and Hom\.; G/\. In this way, we obtain new complexes which also have homologies; these homologies are called homology and cohomology of the space with coefficients \(values\) in G (Anatoly Fomenko 195) - The group of n-dimensional singular chains of X with coefficients in G is denoted as Cn \.XI G/; obviously, Cn \.XI G/ D Cn \.X/ ˝ G\. Our previous group of chains, Cn \.X/, is, in this notation, Cn \.XI Z/ (Anatoly Fomenko 196) - The group of n-dimensional cochains of X with coefficients in G is denoted as Cn \.XI G/; obviously, Cn \.XI G/ D Hom\.C (Anatoly Fomenko 196) - m\.C ˝ The value of a cochain c on a chain a is denoted as hc; ai; thus, c; Pn \.X/ i gi fi ;˛ G/\. D Th i c\.fi /g i \. A generalization: if a bilinear multiplication \(pairing\) G 1 G2 ! G3 P is given, then for c 2 Cn \.XI G1 / and a 2 Cn \.XI G2 / there arises the “value” hc; ai 2 G3 \. (Anatoly Fomenko 196) - Boundary and coboundary operators @ D @n W Cn \.XI G/ ! Cn 1 \.XI G/; ı D ı n W Cn \.XI G/ ! CnC1 \.XI G/ are defined by the formulas n n X X X X @ gi fi D gi \.1/j j fi ; \.ıc/\.f / D \.1/j c\.j f /: i i jD0 jD0 (Anatoly Fomenko 196) - Obviously, for every c 2 Cn \.XI G/ and a 2 CnC1 \.XI G/, hc; @ai D hıc; ai: A simple computation shows that @@ D 0 and ıı D 0 \(the second follows from the first and the formula for h; i above\), and we set KerŒ@n W Cn \.XI G/ ! Cn 1 \.XI G/ H KerŒ@n W Cn \.XI G/ ! Cn 1 \.XI G/ n \.XI G/ D ; ImŒ@ nC1 W CnC1 \.XI G/ ! Cn \.XI G/ KerŒı n W Cn \.XI G/ ! CnC1 \.XI G/ KerŒı n W Cn \.XI G/ ! CnC1 \.XI G/ H n \.XI G/ D : ImŒı n 1 W Cn 1 \.XI G/ ! Cn \.XI G/ (Anatoly Fomenko 196) - The related terminology is homology, cohomology, cycles, cocycles, boundaries\. coboundaries, homological cycles, cohomological cocycles\. (Anatoly Fomenko 196) - A continuous map hW X ! Y induces homology and cohomology homomorphisms, the latter of which acts in the “opposite direction”: h W Hn \.XI G/ ! Hn \.YI G/; h W H n \.YI G/ ! H n \.XI G/ (Anatoly Fomenko 197) - Homology with coefficients and cohomology are homotopy invariant: If g h, then g D h and g D h ; in particular, homology with coefficients and cohomology of homotopy equivalent spaces are the same\. (Anatoly Fomenko 197) - For a disjoint union X D X 1 t t XN , H M M n \.XI G/ D Hn \.Xi I G/; H n \.XI G/ D H n \.XI G/: M i i (Anatoly Fomenko 197) - For infinite disjoint unions, a difference appears between homology and cohomology: Hn \.XI G/ is the direct sum of the groups Hn \.Xi I G/, while H n \.XI G/ is the direct product of the groups H n \.Xi I G/\. (Anatoly Fomenko 197) - The homomorphism ı W H n 1 \.AI G/ ! H n \.XI A/ is defined in the following \(expectable\) way\. (Anatoly Fomenko 198) - or a Borsuk pair \.X; A/, there are isomorphisms H n \.X; AI G/ D H H (Anatoly Fomenko 198) - established by the projection X ! X=A\. (Anatoly Fomenko 198) - For an arbitrary pair there are similar isomorphisms with X=A replaced by X [ CA\. (Anatoly Fomenko 198) - HISTORICAL AND TERMINOLOGICAL REFERENCE\. The homomorphisms @ and ı were discovered, in a particular case, by M\. Bockstein long before exact sequences became commonplace in algebraic topology\. Here is how the Bockstein homomorphism was first described\. Let ˛ 2 Hn \.XI Zm /\. Take a representative a of ˛\. (Anatoly Fomenko 200) - We have constructed “Bockstein homomorphisms” Bm W Hn \.XI Zm / ! Hn 1 \.XI Z/ and bm W Hn \.XI Zm / ! Hn 1 \.XI Zm /: (Anatoly Fomenko 201) - Actually, all of these Bockstein homomorphisms are connecting homomorphisms @ and ı of coefficient sequences induced by the short exact sequences m m 0 ! Z ! Z ! Zm ! 0 and 0 ! Zm ! Zm2 ! Zm ! 0: (Anatoly Fomenko 201) - et A and B be Abelian groups\. Then let B D F1 =F2 , where F1 is a free Abelian group and F2 is a subgroup of F1 which must also be free \(such a presentation exists for any Abelian group\) (Anatoly Fomenko 201) - What are the interrelations between A ˝ F1 ; A ˝ F2 ; and A ˝ B? To answer this question, we need a lemma which can be regarded as the most fundamental property of tensor products\. (Anatoly Fomenko 201) - Lemma 1\. The tensor product operation is right exact\. This means that if the sequence ˛ ˇ ˛ ˇ A ! B ! C ! 0 is exact, then the sequence G˝˛ G˝ˇ G˝˛ G˝ˇ G ˝ A ! G ˝ B ! G ˝ C ! 0 is exact\. (Anatoly Fomenko 201) - Proof\. Recall that, by definition, the tensor product K ˝ L is F\.K L/=R\.K; L/, where F\.K L/ is the free Abelian group generated by the set K L and R\.K; L/ is the subgroup of F\.KL/ generated by elements of the form \.k; `/C\.k0 ; `/\.kCk0 ; `/ and \.k; `/ C \.k; `0 / \.k; ` C `0 /\. The he (Anatoly Fomenko 202) - EXERCISE 6\. Prove that if A \(or B\) is a free Abelian group, then Tor\.A; B/ D 0 (Anatoly Fomenko 203) - EXERCISE 4\. Show that the operation Tor is natural with respect to both arguments; that is, homomorphisms A ! A 0 ; B ! B0 induce a homomorphism Tor\.A; B/ ! Tor\.A 0 ; B0 / with all expectable properties \(for A it is obvious, while for B this requires a construction like the one in the beginning of the proof of the lemma\)\. (Anatoly Fomenko 203) - EXERCISE 5\. Prove a natural isomorphism Tor\.A; B/ ! Tor\.B; A/\. \(This might be harder than one can expect\. The most common idea of proving that is the following\. Consider two presentations A D F1 =F2 ; B D G1 =G2 with free Abelian F 1 ; F2 ; G1 ; G2 , form the complex 0 ! F2 ˝ G2 ! Œ\.F1 ˝ G2 / ˚ \.F2 ˝ G1 / ! F1 ˝ G1 ! 0; and prove that the homology groups H2 ; H1 ; and H0 of this complex are 0; Tor\.A; B/; and Hom\.A; B/\. This provides a definition of Tor symmetric in A; B\.\) (Anatoly Fomenko 203) - EXERCISE 7\. Prove that Tor\.Zm ; Zn / Š Zm ˝ Zn ŒD Zgcd\.m;n/ [this isomorphism is not canonical; it depends on the choice of generators in Zm and Zn ]\. Thus, for finitely generated Abelian groups A; B, Tor\.A; B/ Š Tors A ˝ Tors B \(Tors A D torsion of A, the group of elements of finite order\)\. (Anatoly Fomenko 203) - EXERCISE 8\. For infinitely generated A; B, the last isomorphism, in general, does not hold: Construct an example\. (Anatoly Fomenko 203) - The “dual” operation Ext is defined in a similar way\. First, we dualize Lemma 1: Lemma 3\. If the sequence ˛ ˇ ˛ ˇ A ! B ! C ! 0 is exact, then the sequence Hom\.˛;G/ Hom\.ˇ;G/ Hom\.˛;G/ Hom\.ˇ;G/ Hom\.A; G/ Hom\.B; G/ Hom\.C; G/ 0 is exact\. (Anatoly Fomenko 204) - EXERCISE 10\. Prove that the operation Hom\.G; / is left exact\. This means that if the sequence ˛ ˇ ˛ ˇ 0 ! A ! B ! C is exact, then the sequence Hom\.G;˛/ Hom\.G;ˇ/ Hom\.G;˛/ Hom\.G;ˇ/ 0 ! Hom\.G; A/ ! Hom\.G; B/ ! Hom\.G; C/ is exact\. (Anatoly Fomenko 204) - Let A; B be Abelian groups, and let A D F 1 =F2 , where F1 and F2 are free Abelian groups\. Lemma 3 says that the kernel of the map Hom\.F1 ; B/ ! Hom\.F2 ; B/; f 7! f jF2 is Hom\.A; B/, but this map is not onto\. The cokernel of this map, which is the quotient of Hom\.F2 ; B/ over the image of this map, is taken for Ext\.A; B/\. (Anatoly Fomenko 204) - EXERCISE 14\. The set Ext\.A; B/ has another definition \(due to Yoneda\) as the set of equivalence classes of “extensions” of A by B, that is, short exact sequences 0!B!C!A!0 where C is an Abelian group\. Prove the equivalence of the two definitions of Ext and make up a direct definition of a group structure in the set Ext\.A; B/ described as the set of extensions\. (Anatoly Fomenko 204) - 15\.5 The Universal Coefficients Formula (Anatoly Fomenko 205) - Theorem\. For any X; n; and G, H n \.XI G/ Š \.Hn \.X/ ˝ G/ ˚ Tor\.Hn 1 \.X/; G/ H n \.XI G/ Š \.H n \.X/ ˝ G/ ˚ Tor\.H nC1 \.XI Z/; G/ H n \.XI G/ Š Hom\.Hn \.X/; G/ ˚ Ext\.Hn 1 \.X/; G/: IMPORTANT ADDITION\. The isomorphisms of the theorem are not canonical\. What is canonical are the following three exact sequences: 0 ! Hn \.X/ ˝ G ! Hn \.XI G/ ! Tor\.Hn 1 \.X/; G/ ! 0; 0 ! H n \.XI Z/ ˝ G ! H n \.XI G/ ! Tor\.H nC1 \.XI Z/; G/ ! 0; 0 Hom\.Hn \.X/; G/ H n \.XI G/ Ext\.Hn 1 \.X/; G/ 0: (Anatoly Fomenko 205) - In other words, this homomorphism sends a cohomology class 2 H n \.XI G/ to a homomorphism ˛ 7! h ; ˛i of Hn \.X/ into G\. The fact that this homomorphism is onto yields the following important proposition\. Corollary 1\. For every homomorphism f W Hn \.X/ ! G, there exists a cohomology class 2 H n \.XI G/ such that f \.˛/ D h ; ˛i for every ˛ 2 Hn \.G/\. Remark also that this is defined up to an element of Ext\.Hn \.X/; G/; in particular, if Hn \.X/ and G are finitely generated, then this Ext group is finite, so is defined by f up to adding an element of finite order\. (Anatoly Fomenko 208) - Corollary 2\. If the groups Hn \.X/ are finitely generated, then H n \.XI Z/ Š Free part of Hn \.X/ ˚ Torsion part of Hn 1 \.X/: In particular, H 1 \.XI Z/ is a free Abelian group\. EXERCISE 16\. If K D Q; R, or C, then H n \.XI K/ D Hn \.X/ ˝ K and H n \.XI K/ D Hom\.Hn \.X/; K/: (Anatoly Fomenko 208) - Thus, the transition from the integral coefficients to the rational, real, or complex coefficients kills the torsion\. On the other hand, the Betti numbers of X become the dimension of homology or cohomology with coefficients in Q; R or C\. \(Actually, the same is true for any field of characteristic zero\.\) (Anatoly Fomenko 208) - EXERCISE 18\. Prove that if X is a finite CW complex and K is a field, then \.1/m dim K Hm \.XI K/ does not depend on K and is equal to the Euler characteristic of X (Anatoly Fomenko 208) - 15\.6 Künneth’s Formula (Anatoly Fomenko 209) - Theorem 1\. Let X1 ; X2 be topological spaces\. Then for any n, \(1\) There is a \(noncanonical\) isomorphism H n \.X1 X2 / Š 1 X2 / Š \.Hi \.X L L 1 / ˝ Hj \.X2 // Tor\.Hi \.X1 /; Hj \.X2 //: iCjDn iCjDn 1 \(2\) There is a canonically defined exact sequence L 0! iCjDn \.Hi \.X1 / ˝ Hj \.X2 // ! Hn \.X1 X2 / j \. L ! iCjDn 1 Tor\.Hi \.X1 /; Hj \.X2 // ! 0: (Anatoly Fomenko 209) - The complex arising, nC1 n n 1 nC1 n n 1 : : : ! Tn ! Tn 1 ! : : : ; is called the tensor product of the complexes C and C 0 and is denoted as C ˝ C 0 \. (Anatoly Fomenko 209) - Our next goal is to express the homology of the tensor product of two complexes in terms of homologies of these complexes\. (Anatoly Fomenko 212) - Theorem 2\. If the complexes C; and C 0 are free \(that is, all Cn ; Cn0 are free Abelian groups\), then, for every n, \(1\) There is a \(noncanonical\) isomorphism H n \.C ˝ C 0 / Š C ˝ C 0 / Š Tor\.Hi \.C/; Hj \.C 0 L L \.Hi \.C/ ˝ Hj \.C 0 / Tor\.Hi \.C/; Hj \.C 0 //: iCjDn iCjDn 1 \(2\) There is a canonically defined exact sequence L 0! iCjDn \.Hi \.C/ ˝ Hj \.C 0 // ! Hn \.C ˝ C 0 / Hj \. L ! iCjDn 1 Tor\.Hi \.C/; Hj \.C 0 // ! 0: (Anatoly Fomenko 212) - Remarks\. \(1\) It is not true, in general, that the singular complex of the product X 1 X2 of two topological spaces is isomorphic the tensor product of the singular complexes of X1 and X2 \. But these complexes are homotopy equivalent \(there exists a homotopy equivalence canonically defined up to a homotopy between them\)\. This fact, known as the Eilenberg–Zilber theorem, is proved in many textbooks in topology\. (Anatoly Fomenko 214) - 2\) A comparison of the universal coefficients formula with Künneth’s formula gives the following result \(which may be useful in Chap\. 3\): H M n \.X1 X2 / D Hi \.X1 I Hj \.X2 (Anatoly Fomenko 214) - H i \.X1 I Hj \.X2 //: iCjDn (Anatoly Fomenko 214) - EXERCISE 20\. Find the homology of RP2 RP2 \. \(If the result seems unexpected to you, check it using a direct cellular computation\.\) (Anatoly Fomenko 214) - EXERCISE 21\. Prove that if K is a field, then H L n \.X1 X2 I K/ D iCjDn Hi \.X1 I K/ ˝K Hj \.X2 I K/; L H L n \.X1 X2 I K/ D iCjDn H i \.X1 I K/ ˝K H j \.X2 I K/: (Anatoly Fomenko 214) - P EXERCISE 22\. Bn \.X1 B2 / D iCjDn Bi \.X1 /Bj \.X2 /\. (Anatoly Fomenko 214) - EXERCISE 23\. \.X1 X2 / D \.X1 / \.X2 /: (Anatoly Fomenko 214) - cohomology is immensely more useful because it possesses many naturally defined additional structures\. (Anatoly Fomenko 215) - The simplest way to introduce the cohomological multiplication is as follows\. (Anatoly Fomenko 215) - EXERCISE 1\. Another definition of the homological -product can be obtained from Künneth’s formula: This formula yields a canonical map Hn1 \.X1 /˝Hn2 \.X2 / ! H n 1 Cn2 \.X1 X2 /, (Anatoly Fomenko 215) - he two -products \(usually called cross-products \) are connected by the formul (Anatoly Fomenko 215) - A check shows that @\.a1 a2 / D \.@a1 / a2 C \.1/n1 a1 @a2 , which gives rise to a homological multiplication Œ˛1 2 Hn1 \.X1 I G/; ˛2 2 Hn2 \.X2 I G/ 7! ˛1 ˛2 2 Hn1 Cn2 \.X1 X2 I G/: (Anatoly Fomenko 215) - h 1 2 ; ˛1 ˛2 i D \.1/n1 n2 h 1 ; ˛1 ih 2 ; ˛2 i: (Anatoly Fomenko 215) - At this moment, however, the difference between homology and cohomology becomes important\. For any topological space X, there exists the diagonal map W X ! X X; \.x/ D \.x; x/\. This maps induces homomorphisms W Hn \.XI G/ ! Hn \.X XI G/; W H n \.X XI G/ ! H n \.XI G/I of these homomorphisms; the first one is useless for us now, but the second one provides cohomological multiplication: For 1 2 H n1 \.XI G/; 2 2 H n2 \.XI G/, we put 1 ^ 2 D \. 1 ˝ 2/ 2 H n1 Cn2 \.XI G/: \(The classical notation ^, “cup,” is not very convenient, so often instead of 1 ^ 2 we will simply write 1 2 \.\) (Anatoly Fomenko 216) - However, this way of defining the cohomological product has two important disadvantages\. First, we must still prove the independence of the CW structure\. Second, the diagonal map is not cellular (Anatoly Fomenko 216) - \. To avoid these difficulties we will use the opposite order of the definition\. First, we will define a ^-product \(usually called the cup-product \) by a singular, topologically invariant, construction, and then we will use it to define the cross-product\. (Anatoly Fomenko 216) - Terminological Remark\. The cup-product was initially called the Kolmogorov– Alexander product, after the two remarkable mathematicians who \(independently of each other\) conceived of this operation in the mid-1930s\. U (Anatoly Fomenko 216) - Proposition \(Properties of the Cochain Cup-Product\)\. Let c1 2 Cn1 \.XI G/; c2 2 C n 2 \.XI G/\. Then (Anatoly Fomenko 217) - Theorem \(Properties of the Cohomology Cup-Product\)\. Let 1 2 H n1 \.XI G/; 2 2 H n2 \.XI G/\. Then (Anatoly Fomenko 217) - 16\.3 The Cross-Product: A Construction via the Cup-Product (Anatoly Fomenko 218) - 16\.5 First Application: The Hopf Invariant (Anatoly Fomenko 223) - Theorem\. The group 4n 1 \.S2n / is infinite for any n 1\. Moreover, the Whitehead square Œ2n ; 2n of the generator of 2n \.S2n / has an infinite order in 4n 1 \.S2n /\. (Anatoly Fomenko 223) - The proof of this theorem is based on the Hopf invariant, which is an integer assigned to every element of ' 2 4n 1 \.S2n /\. Its definition is as follows\. (Anatoly Fomenko 223) - Consider a spheroid f W S4n 1 ! S2n and form the space X' D S 2n [f D4n \(aka the cone of f \)\. The space X' depends, up to a homotopy equivalence, only on ' \(which justifies the notation\)\. It has a natural CW structure with three cells of dimensions 0; 2n; and 4n\. Thus, q Z for q D 0; 2n; 4n; q H \.X' I Z/ D 0 for q ¤ 0; 2n; 4n: The groups H 2n \.X' I Z/; H 4n \.X' I Z/ \(isomorphic to Z\) have natural generators \(determined by the canonical orientations of S2n and D4n \), and we denote these generators by a and b\. Since the cup-square a2 D a ^ a has dimension 4n, we have a 2 D hb, where h 2 Z\. The number h D h\.'/ is, by definition, the Hopf invariant of '\. 1 (Anatoly Fomenko 223) - Lemma 1\. The Hopf invariant is additive: h\.' C / D h\.'/ C h\. /\. Lemma 2\. The Hopf invariant is nontrivial; in particular, h\.Œ 2n ; 2n D 2: (Anatoly Fomenko 223) - Remark 5\. Lemma 2 shows that the image of the Hopf homomorphism hW 4n 1 \.S 2n / ! Z is either the whole group Z or the group of even integers\. The choice between these two options is reduced to the question: Does 4n 1 \.S2n / contain an element with the Hopf invariant one? This question has several remarkable equivalent statements\. For example, it is possible to show that Sm possesses an H-space structure if and only if m is odd, that is, m D 2n 1, and 4n 1 \.S2n / contains an element with the Hopf invariant one\. The same condition is necessary and sufficient for the existence in RmC1 of a bilinear multiplication with a unique division\. The combination of Lemma 2 and Exercise 7 in Lecture 10 shows that the Hopf invariant of the Hopf class 2 2 3 \.S2 / equals 1 \(this corresponds to the complex number multiplication in R2 or to the natural group structure in S1 \)\. In 1960, J\. Adams showed that elements with the Hopf invariant one are contained only in 3 \.S2 /; 7 \.S4 /, and 15 \.S8 / \( (Anatoly Fomenko 225) - 16\.6 An Addendum: Other Multiplications A: Homological -Product (Anatoly Fomenko 225) - B: Cap-Product (Anatoly Fomenko 226) - This (Anatoly Fomenko 226) - EXERCISE 12\. Prove that if n1 ; n2 are positive integers, then there is no way to introduce for all X a nonzero bilinear multiplication H n 1 \.XI G/ Hn2 \.XI G/ ! Hn1 Cn2 \.XI G/ natural with respect to continuous maps\. (Anatoly Fomenko 226) - However, it is possible to define a multiplication in homology groups of X if X itself possesses a multiplication making it a topological group or, at least, an Hspace\. The definition is obvious: If W X X ! X is the multiplication in X and ˛ 1 2 Hn1 \.XI G/; ˛2 2 Hn2 \.XI G/ where G is a ring, then ˛1 ˛2 D \.˛1 ˛2 /\. This product is called the Pontryagin–Samelson product\. (Anatoly Fomenko 227) - A Hausdorff topological space with a countable base of open sets \(these topological assumptions are not in the spirit of this book, but we have to impose them, since without them many statements that follow would be plainly wrong\) is called an ndimensional \(topological\) manifold if every point of it possesses a neighborhood homeomorphic to the space Rn or the half-space Rn D f\.x1 ; : : : ; xn / 2 Rn j x n 0g\. A point of an n-dimensional manifold X which has no neighborhood homeomorphic to Rn is called a boundary point\. Boundary points of X form an \.n 1/-dimensional manifold @X called the boundary of X\. Obviously, @X is a manifold without boundary: @@X D ;\. (Anatoly Fomenko 227) - Examples of manifolds: Euclidean spaces, spheres, balls, classical surfaces, projective spaces, Grassmann manifolds, flag manifolds, Lie groups, Stiefel manifolds, products of the spaces listed above, open sets in these spaces, closed domains with smooth boundaries in these spaces, and so on\. (Anatoly Fomenko 229) - If the domains U; V of local coordinate systems f W U ! Rn\. / ; gW V ! Rn\. / \(also called charts \) overlap, then there arises a transition map 1 f 1 g f g f \.U \ V/ ! U \ V ! g\.U \ V/ \ \ Rn Rn ; (Anatoly Fomenko 229) - A set of charts which cover the manifold is called an atlas\. An atlas is called smooth \(analytic\) if such functions are all transition functions between charts of this atlas\. Two smooth \(analytic\) atlases are called smoothly \(analytically\) equivalent if their union is smooth \(analytic\) atlas\. A class of equivalent smooth \(analytic\) atlases is called a smooth \(analytic\) structure on a manifold\. A manifold with a smooth \(analytic\) structure is called a smooth \(analytic\) manifold\. The boundary of a smooth \(analytic\) manifold is, in a natural way, a smooth \(analytic\) manifold (Anatoly Fomenko 229) - There are two fundamental theorems in the theory of smooth manifolds \(also called differential topology \)\. Theorem 1\. Every smooth manifold is diffeomorphic \(that is, homeomorphic with preserving the smooth structure\) to a smooth surface in an Euclidean space\. Theorem 2\. Every compact smooth manifold is homeomorphic to a triangulated subset of an Euclidean space, and the homeomorphism can be made smooth on every simplex of the triangulation\. (Anatoly Fomenko 229) - Remarks\. \(1\) In both theorems, the dimension of the Euclidean space can be as small as twice the dimension of the manifold\. \(2\) Theorem 2 also holds for noncompact manifolds, but the triangulation in this case has to be infinite\. (Anatoly Fomenko 230) - Theorem 1 is proved in many textbooks in differential topology\. Its proof is not hard\. The situation with Theorem 2 is worse\. Since the 1920s, the topologist regarded this fact as obvious\. There are many geometric approaches to this result which look promising (Anatoly Fomenko 230) - g\. For example, take a compact smooth surface in an Euclidean space and decompose this space into a union of small cubes\. If the decomposition satisfies some general position condition with respect to the surface, we can expect that the intersections of the surface with the cubes will be close to convex polyhedra and we can easily triangulate these polyhedra\. Or, choose a random finite subset of the smooth surface which is sufficiently dense, and take the Dirichlet domain; again we should get a subdivision of the surface into smooth polyhedra\. However, numerous attempts to make this proof rigorous turned out to be unsuccessful\. The first flawless proof of this theorem \(actually, of a stronger relative result\) was given in the 1930s by H\. Whitney\. (Anatoly Fomenko 230) - We know two textbook presentations of this proof, in the books Whitney [89] and Munkres [64]\. (Anatoly Fomenko 230) - EXERCISE 1\. Construct a realization as smooth surfaces in Euclidean spaces of projective spaces, Grassmann manifolds, flag manifolds, and Stiefel manifolds\. EXERCISE 2\. Prove that all classical surfaces can be presented as smooth surfaces in Rn with n 4\. (Anatoly Fomenko 230) - EXERCISE 3\. Construct smooth triangulations of classical surfaces; try to minimize the number of simplices needed\. (Anatoly Fomenko 230) - EXERCISE 4\. Prove that the number of n-dimensional simplices adjacent to an \.n 1/-dimensional simplex of a smooth triangulation of an n-dimensional smooth manifold is 2 if this \.n 1/-dimensional simplex is not contained in the boundary, and is 1 otherwise\. (Anatoly Fomenko 230) - An atlas of a smooth manifold is called oriented if for every two overlapping charts the transition map has a positive determinant at every point\. Two oriented atlases determine \(belong to\) the same orientation if their union is an oriented atlas\. (Anatoly Fomenko 230) - A manifold is called orientable \(oriented \) if it possesses \(is furnished by\) an oriented atlas, that is, an orientation\. (Anatoly Fomenko 231) - EXERCISE 6\. Which projective spaces and Grassmann manifolds are orientable? \(Answer : Only real projective spaces and Grassmann manifolds can be nonorientable\. Namely, RPn is orientable if and only if n is odd, and G\.n; k/ is orientable if and only if n is even\.\) (Anatoly Fomenko 231) - EXERCISE 7\. Prove that spheres with handles are orientable and projective planes and Klein bottles are nonorientable; drilling holes does not affect the orientability\. (Anatoly Fomenko 231) - EXERCISE 8\. Prove that a connected orientable manifold of positive dimension has precisely two orientations\. (Anatoly Fomenko 231) - EXERCISE 12\. Prove that every connected nonorientable manifold possesses an orientable twofold covering\. (Anatoly Fomenko 231) - EXERCISE 10\. Prove that a manifold is orientable if and only if a neighborhood of every closed curve on this manifold is orientable\. (Anatoly Fomenko 231) - EXERCISE 11\. Prove that every simply connected manifold is orientable\. (Anatoly Fomenko 231) - Definition\. A triangulated space X is called an n-dimensional pseudomanifold if it satisfies the following three axioms\. (Anatoly Fomenko 232) - Thus, a smoothly triangulated connected smooth manifold without boundary is a pseudomanifold\. The converse is wrong: A pseudomanifold is not always a manifold\. See the simplest example in Fig\. 66\. (Anatoly Fomenko 232) - anifold\. See the simplest example in Fig\. 66\. There are fewer artificial examples of pseudomanifolds topologically different from manifolds: complex algebraic varieties, and Thom spaces of vector bundle (Anatoly Fomenko 232) - Theorem\. Let X be an n-dimensional pseudomanifold\. Then Z; if X is compact and orientable; H n \.X/ D 0 otherwiseI Z 2 ; if X is compact; H n \.XI Z2 / D 0 otherwise: (Anatoly Fomenko 233) - This proof provides a canonical generator for the group Hn \.X/ for a compact oriented pseudomanifold X: This is the homology class of the cycle, which is the sum of all n-dimensional simplices of X with orientations compatible with the orientation of X and with the coefficients all equal to 1\. This homology class is called the fundamental class of X \(and the cycle is called the fundamental cycle \)\. In the orientation-free case, we have fundamental classes and fundamental cycles with coefficients in Z2 (Anatoly Fomenko 233) - : A compact manifold without boundary is called closed (Anatoly Fomenko 233) - EXERCISE 16\. Prove that if X is a connected n-dimensional manifold with nonempty boundary, then Hn \.X/ D Hn \.XI Z2 / D 0\. (Anatoly Fomenko 234) - There arises a natural question regarding the possibility to present a homology class of a topological space as an image of the fundamental class of a manifold\. The answer is negative, for homology classes with coefficients in Z as well as for those with coefficients in Z2 \. (Anatoly Fomenko 234) - A more popular question arises in the topology of manifolds: If Y is a manifold and ˛ 2 Hn \.Y/, then when is it possible to find a closed oriented n-dimensional submanifold X of Y \(we assume that the reader understands what it is\) such that the homomorphism induced by the inclusion map sends ŒX into ˛ \(as people say, X realizes ˛\)? (Anatoly Fomenko 234) - There are many remarkable results regarding submanifold realizations; for example, for any homology class ˛ of a manifold, there exists a number N such that N˛ can be realized by a submanifold\. \(For this result and other results, see the classical paper by Thom [84]\.\) (Anatoly Fomenko 234) - EXERCISE 20\. Prove that the generators of groups H m \.RPn I Z2 /; Hm \.RPn /; H2m \.CPn /; H4m \.HPn / are realized by projective subspaces of RPn ; CPn ; HPn \. (Anatoly Fomenko 234) - Mention in conclusion that if X; Y are oriented pseudomanifolds of the same dimension, and f W X ! Y is a continuous map, then f ŒX D k ŒY, where k is an integer\. This k is called the degree of f and is denoted as deg f ; it is a homotopy invariant\. In the nonoriented case, the degree deg f may be defined as an element of Z2 \. We have already had this notion in the particular case X D Y D Sn (Anatoly Fomenko 234) - EXERCISE 21\. Let f W X ! Y be a \(piecewise\) smooth map between two closed oriented n-dimensional manifolds, and let yW Y be a regular value of this map\. Then there is a neighborhood U of y such that f 1 \.U/ is a disjoint union of a finite collection of sets Ui with all restrictions f jUi being homeomorphisms Ui ! U\. Prove that deg f is the number of i for which this homeomorphism preserves the orientation minus the number of i for which it reverses the orientation (Anatoly Fomenko 235) - The most general definition of a homology manifold is formulated in terms of local homology : For a topological space X, its mth local homology at the point x0 2 X is loc defined as Hm;x 0 \.X/ D Hm \.X; X x0 /\. (Anatoly Fomenko 235) - Definition\. A space X is called an n-dimensional homology manifold if, for any m, H loc m;x 0 \.X/ D H H loc Z; if m D n; H m;x Z; if m D n; 0 \.X/ D : 0; if m ¤ n: (Anatoly Fomenko 235) - For us, the most important will be the case when X is triangulated\. Recall that the star St\.s/ of a simplex s of triangulation is the union of simplices that contain s\. The link Lk\.s/ is the union of faces of simplices that contain s opposite to s\. Figure 67 shows examples of stars and links of a vertex and a one-dimensional simplex of the standard triangulation of the plane\. (Anatoly Fomenko 235) - Proposition 2\. Every connected n-dimensional homology manifold is an n-dimensional pseudomanifold\. (Anatoly Fomenko 236) - Remark 2\. This argumentation shows a difference between pseudomanifolds and homology manifolds\. (Anatoly Fomenko 236) - Remark 4\. A homology manifold is not always a topological manifold\. For example, there are manifolds with the same homology as a sphere, but not simply connected \(the best known example is the Poincaré sphere defined in S5 D f\.z1 ; z2 ; z3 / 2 C3 j jz 1 j2 C jz2 j2 C jz3 j2 D 1g by the equation z51 C z32 C z23 D 0\)\. The suspension over such a manifold is a homology manifold, but no neighborhoods of vertices are homeomorphic to a Euclidean space\. (Anatoly Fomenko 237) - Theorem\. Let X be a compact n-dimensional homology manifold, and 0 m n\. If X is orientable, then for any G, H m \.XI G/ Š H n m \.XI G/: In the general case, H m \.XI Z2 / Š H n m \.XI Z2 /: let In both cases, there are canonical isomorphisms DW H n m \.XI G/ ! Hm \.XI G/ (Anatoly Fomenko 237) - hich act by the formula D\.˛/ D ŒX \_ ˛, where ŒX is the fundamental class \(see Sect\. 17\.2\) and \_ denotes the cap-product \(see Sect\. 16\.6\)\. (Anatoly Fomenko 237) - Corollary\. The Euler characteristic of a closed homology manifold of odd dimension equals 0\. (Anatoly Fomenko 240) - The results of Sect\. 15\.5 give the possibility to restate Poincaré isomorphisms between homology and cohomology as \(noncanonical\) isomorphisms between homology and homology\. Namely, H m \.XI Z2 / Š Hn m \.XI Z2 / (Anatoly Fomenko 241) - for an arbitrary n-dimensional homology manifold X and Free Part of Hm \.X/ Š Free Part of Hn m \.X/ Torsion Part of Hm \.X/ Š Torsion Part of Hn m 1 \.X/ (Anatoly Fomenko 241) - in the oriented case\. It turns out that these noncanonical isomorphisms reflect a very canonical duality called Poincaré duality which is much more classical than Poincaré isomorphisms (Anatoly Fomenko 241) - (Anatoly Fomenko 241) - The integer \.c X 1 ; c2 / D ki `i D hD 1 c1 ; c2 i i is called the intersection number of c1 and c2 \. It follows from the last formula and the properties of Poincaré isomorphism that the intersection number of two cycles depends only on the homology classes of these cycles, and we can speak of intersection numbers of homology classes: (Anatoly Fomenko 241) - A remarkable property of the intersection numbers is their geometric visualizability\. A simplex and its barycentric star transversely intersect each other at one point, so the intersection number of two cycles may be regarded as the number of their intersection points taken with the signs determined by their orientations\. This statement has a convenient differential statement\. (Anatoly Fomenko 242) - Theorem 1\. Let X be a smooth closed oriented n-dimensional manifold, and let ˛ 1 2 Hm \.X/; ˛2 2 Hn m \.X/\. Let Y1 and Y2 be closed oriented submanifolds of X of dimensions m and n m which realize ˛1 and ˛2 in the sense that ˛1 D \.i 1 / ŒY1 and ˛2 D \.i2 / ŒY2 where i1 ; i2 are inclusion maps\. We assume also that Y 1 ; Y2 are in general position \(which means that they intersect in finitely many points and transverse to each other at each of these points\)\. We assign a sign to every intersection point: plus if the orientations of Y1 and Y2 \(in this order\) compose the orientation of X at this point, and minus otherwise\. Then the intersection number \.˛ 1 ; ˛2 / equals to the number of the intersection points of Y1 and Y2 counted with the signs described above\. (Anatoly Fomenko 242) - If X is a complex manifold, that is, its charts are maps into Cn and the transition maps are holomorphic, then X possesses a natural, “complex,” orientation\. The matter is that the Jacobian of a holomorphic map Cn ! Cn regarded as a smooth map R2n ! R2n is equal to the square of the absolute value of the complex Jacobian and, hence, (Anatoly Fomenko 242) - Example\. Natural generators yr ; yn r of the groups H2r \.CPn /; H2\.n r/ \.CPn / have the intersection number 1\. Indeed, they are realized by projective subspaces CPr ; CPn r of CPn which \(in the general position\) intersect in one point\. Regarding the sign, we will make an important remark\. I (Anatoly Fomenko 242) - is always positive\. (Anatoly Fomenko 243) - Moreover, if Y1 ; Y2 are complex \(that is, locally determined by holomorphic equations\) submanifolds of X of complementary dimensions in a general position, then every point in Y1 ; Y2 contributes C1 into the intersection number of the homology classes\. Thus, \.yr ; yn r / D 1, not 1\. (Anatoly Fomenko 243) - EXERCISE 22\. Let be a Young diagram inscribed into a rectangle k \.n k/, and let be the “dual” Young diagram obtained from the complement of in the rectangle by the reflection in the center of the rectangle \(see Fig\. 69\)\. Then the intersection number of the homology classes of CG\.n; k/ corresponding to the Young diagrams ; 0 \(see Sects\. 5\.4\.C and 13\.8\.C\) is 1 if 0 D and is 0 otherwise (Anatoly Fomenko 243) - The fact that the intersection number of two cycles depends only on the homology classes of these cycles is often used (Anatoly Fomenko 243) - EXERCISE 23\. Prove that on any smooth closed orientable surface in R4 D C2 , there exist at least two different points for which the tangent planes are complex lines\. \(Hint : The orientation takes care of the existence of more than one such point\. (Anatoly Fomenko 243) - EXERCISE 24\. Prove that if X1 ; X2 are two closed orientable surfaces in R4 , then there are at least four pairs of points \.x1 2 X1 ; x2 2 X2 / such that the tangent planes to X1 m; x2 at x1 ; x2 are parallel\. (Anatoly Fomenko 243) - Theorem 2\. Let X be compact oriented homology manifold\. \.1/ For every homomorphism f W Hm \.X/ ! Z, there exists a homology class ˛ 2 Hn m \.X/ such that f \.˛/ D \.˛; ˇ/ for every ˇ 2 Hm \.X/\. \.2/ The class ˇ is determined by f uniquely, up to adding an element of finite order\. (Anatoly Fomenko 243) - Notice that in the middle-dimensional homology of an even-dimensional manifold, Theorem 2 has the following, more algebraic restatement\. Theorem 3\. Let X be a connected closed orientable manifold of even dimension 2k, and let Hk0 \.X/ be the free part of Hk \.X/\. Then the integral bilinear form \(the intersection index\) on Hk0 \.X/ is unimodular [that is, the matrix k \.˛i ; ˛j /k where ˛ 1 ; ˛2 ; : : : is a system of generators in Hk0 \.X/ has determinant ˙1]\. This matrix is symmetric if k is even and is skew-symmetric if k is odd\. Since any skew-symmetric matrix of odd order is degenerate, we have the following: Corollary\. The middle Betti number of any closed orientable manifold of dimension 2 mod 4 is even; hence, the Euler characteristic of such a manifold is even\. (Anatoly Fomenko 244) - For nonorientable manifolds neither is true; examples: the first Betti number of the Klein bottle is 1, and the Euler characteristic of the real projective plane is 1\. (Anatoly Fomenko 244) - Theorem 3 demonstrates the importance of the theory of integral unimodular \(det D ˙1\) forms in topology of manifolds, especially of dimensions divisible by 4: For an oriented closed manifold of such dimension, there arises a unimodular integral quadratic form as the intersection form in the middle dimension (Anatoly Fomenko 244) - For example, the famous Pontryagin theorem states that a homotopy type of a simply connected closed four-dimensional manifold is fully determined by this form\. A lot is known about the classification of such forms \(the best source is Milnor and Husemoller [58]\), but the question of which forms can be intersection forms for smooth closed four-dimensional simply connected manifolds is very far from being resolved\. In co (Anatoly Fomenko 244) - 17\.6 Application: The Lefschetz Formula L Let X be a compact topological space with finitely generated homology n Hn \.X/, and let f W X ! X be a continuous map\. The number X L\.f / D \.1/n Tr f n n is called the Lefschetz number of f [here Tr f n denotes the trace of the lattice homomorphism f n W Hn \.X/= Tors Hn \.X/ ! Hn \.X/= Tors Hn \.X/: (Anatoly Fomenko 245) - Obviously, L\.f / is a homotopy invariant of f \. The goal of this section is to establish a relation between the Lefschetz number of f and the behavior of fixed points of f \. (Anatoly Fomenko 245) - This observation alone yields the first, and maybe the most important, application of Lefschetz numbers \(not related to manifolds, the more so to Poincaré duality\)\. Theorem 1\. Let X be a finitely triangulated space, and let f W X ! X be a continuous map\. If f has no fixed points, then L\.f / D 0\. (Anatoly Fomenko 247) - For example, if X is a finite CW complex, then the Lefschetz number of a continuous map f W X ! X can be calculated as the alternated sum of traces of homomorphisms g# W Cn \.X/ ! Cn \.X/ induced by a cellular approximation g of f \. This observation alone yields the first, and maybe the most important, application (Anatoly Fomenko 247) - L be a complex with finitely generated n Cn , and let f D ffn W Cn ! Cn g L be an endomorphism of C\. Let f n W Hn \.C/ ! Hn \.C/ be the induced homology endomorphism\. Then X \.1/n Tr fn D \.1/n Tr f n : (Anatoly Fomenko 247) - et us return to manifolds \(but, for now, not to Poincaré duality\)\. Theorem 2\. Let X be a compact smooth manifold \(not necessarily orientable, and maybe with a nonempty boundary\), and let be a vector field on X\. Suppose that has no zeroes and that on the boundary @X it is directed inside X\. Then \.X/ D 0\. (Anatoly Fomenko 247) - This result implied the immensely popular “hairy ball theorem”: There is no 2 nowhere vanishing vector field on S2 \(one cannot comb a hairy ball\)\. (Anatoly Fomenko 247) - Proof of Theorem 2\. A vector field on X \(with or without zeroes\) determines a “flow” ft W X ! X, and for a sufficiently small positive " the fixed points of f" are zeroes of \. Since f" is homotopic to the identity, L\.f" / D L\.id/ D \.X/, and if has no zeroes, then \.X/ D 0\. (Anatoly Fomenko 247) - \(We will see in Lecture 18 that the converse is also true: If a closed manifold, orientable or not, has zero Euler characteristic, then it possesses a nowhere vanishing vector field\.\) (Anatoly Fomenko 247) - So far, regarding Lefschetz numbers, we were interested only in their being zero or not zero\. But in reality, in the case of manifolds, the Lefschetz number gives some count of fixed points (Anatoly Fomenko 247) - Theorem 3\. Let X be a triangulated compact orientable n-dimensional homology manifold \(we will discuss later how much the orientability is really needed\) and let (Anatoly Fomenko 247) - f W X ! X be a continuous map\. Let FW X ! X X; F\.x/ D \.x; f \.x// be the graph of f , and let W X ! X X be the diagonal map, \.x/ D \.x; x/\. Then \.F ŒX; ŒX/ D L\.f /: (Anatoly Fomenko 248) - The intersection points of F\.X/ and \.X/ correspond precisely to fixed points of f \. (Anatoly Fomenko 248) - First, we need to assume that all the intersections of the graph and the diagonal are transverse\. This condition may be formulated in the language of calculus\. If x0 is a fixed point of a smooth map f W X ! X, then there arises the differential, dx0 f W Tx0 X ! Tx0 X\. The graph and the diagonal are transverse at x0 if the matrix of dx0 f id is nondegenerate, that is, if fx0 f has no eigenvalues equal to 1\. If this condition holds, then every intersection point acquires some sign, and the intersection number, equal to the Lefschetz number by Theorem 3, is the “algebraic number of fixed points\.” The sign can be described as the parity of the number of real eigenvalues of dx0 f less than 1\. A v (Anatoly Fomenko 248) - A very similar thing can be said about the vector fields\. A nondegenerate zero of a vector field can be assign a sign, and then the algebraic number of zeroes of a vector field must be equal to the Euler characteristic of the manifold\. (Anatoly Fomenko 248) - Another extension of the Lefschetz theory may be obtained by admitting, for a manifold considered, a nonempty boundary\. Namely, if X is a compact manifold (Anatoly Fomenko 249) - with the boundary @X, then we can double X by attaching to it a second copy of X to the common boundary of the two copies \(see Fig\. 70\)\. (Anatoly Fomenko 250) - he common boundary of the two copies \(see Fig\. 70\)\. Let f W X ! X be a continuous map without fixed points on @X, and let XX be the double of X\. We can extend f to a map ff W XX ! X XX defining this map on the second half to be the same as on the first half [thus ff \.XX/ is contained in the first half of XX]\. It is obvious that ff has the same fixed points as f and L\.f / D L\.ff /; hence, the statement of the relation of Lefschetz numbers with fixed points holds for compact manifolds with boundary \(orientable or not\)\. (Anatoly Fomenko 250) - EXERCISE 27\. There exists a different approach to the Lefschetz theory\. First we prove Theorem 1: The Lefschetz number of a fixed-point–free map is zero\. Then we consider a map f W X ! X with a nondegenerate fixed point, and, at a neighborhood of this point, we modify both X and f in such a way that the fixed point disappears and the Lefschetz number is changed in a controllable way\. Try to recover the details\. (Anatoly Fomenko 250) - EXERCISE 28\. The n-dimensional torus T n can be regarded as Rn =Zn \. Hence, a linear map Rn ! Rn determined by an integral matrix A can be factorized to some continuous map T n ! T n ; denote it as fA (Anatoly Fomenko 250) - Calculate the Lefschetz number for fA \(the best possible answer expresses this Lefschetz number in terms of the eigenvalues of A\)\. (Anatoly Fomenko 250) - n n EXERCISE 30\. Prove that a map f W T ! T homotopic to fA with A 21 D has 11 infinitely many periodic points\. [A point y 2 Y is called a periodic point of a map gW Y ! Y if gn \.y/ D y for some n\.] (Anatoly Fomenko 250) - Let us return to Poincaré duality\. The duality between Tors Hm \.X/ and Tors Hn m 1 \.X/ is based on secondary intersection numbers, which are defined ahead\. (Anatoly Fomenko 251) - Let X and Y be compact oriented homology manifolds of, possibly, different dimensions m and n, and let f W X ! Y be a continuous map\. Poincaré isomorphism allows us to construct “wrong direction” homology and cohomology homomorphisms D 1 f D f Š W D D Hq \.YI G/!H n q \.YI G/ ! H n q \.XI G/ ! Hm nCq \.XI G/; D f D 1 mCq D f D f Š W H q \.XI G/!Hm q \.XI G/ ! Hm q \.YI G/ ! H n mCq \.YI G/: Š (Anatoly Fomenko 251) - It can be regarded as Š the simplest case of a general construction called “direct image\.” Its analytic sense \(and it belongs rather to analysis than to topology\), at least in the case when f is the projection of a smooth fibration, can be best described by the words “fiberwise integration” \(people familiar with the de Rham theory can easily understand them\)\. Š As to the homology homomorphism f Š \(called the inverse Hopf homomorphism \), it has a transparent geometric sense which is described, in the smooth case, by the following proposition (Anatoly Fomenko 252) - Theorem\. Let a homology class ˛ 2 Hq \.Y/ be represented by a q-dimensional submanifold Z of Y \(that is, ˛ D i ŒZ, where iW Z ! Y is the inclusion map\), and let f be transversely regular with respect to Z \(that is, the composition d y f proj Ty Y dy f ! Tf \.y/ X proj ! Tf \.y/ X=Tf \.y/ Z is onto for every point y 2 f 1 \.Z/\)\. Then f 1 \.Z/ is a \.q C m n/-dimensional submanifold of X which represents the homology class f Š \.˛/ 2 HqCm n \.X/\. (Anatoly Fomenko 252) - Proposition 1\. Let X; Y, and f be as above, and let ˛ 2 Hq \.Y/; ˇ 2 Hm q \.X/\. Then X \.f Š ˛; ˇ/ D Y \.˛; f ˇ/ \( X and Y denote the intersection number in X and Y\)\. (Anatoly Fomenko 252) - Proposition 2\. Let X; Y be compact oriented homological manifolds, and let pW X Y ! Y be the projection\. Then, for any ˛ 2 Hm \.Y/, p Š ˛ D ŒX ˛: (Anatoly Fomenko 253) - GENERALIZATION\. If d ¤ 0, then every homology class of Y multiplied by d belongs to the image of f , and every cohomology class of Y belonging to Ker f is annihilated by the multiplication by d\. For example, there is no map S2 ! S1 S1 of a nonzero degree, but there is a map S1 S1 ! S2 of degree 1: factorization over S1 \_ S1 \. (Anatoly Fomenko 253) - Proposition 3\. Let X; Y be connected compact oriented manifolds of the same dimension n, and let f W X ! Y be a continuous map of degree d\. Then the compositions fŠ f H f m \.Y/ ! Hm \.X/ ! f Hm \.Y/; ! Hm \.X/ ! f f f Š m Š H \.YI Z/ ! H \.XI Z/ ! H m \.YI Z/ m m are both multiplication by d\. (Anatoly Fomenko 253) - Corollary\. If d D ˙1, then f is an epimorphism, and f is a monomorphism\. (Anatoly Fomenko 253) - Theorem 1\. Let Y1 ; Y2 be closed oriented submanifolds of a smooth closed oriented manifold X transverse to each other; the latter means that the inclusion map i1 of Y 1 in X is transversely regular to Y2 \. Then the intersection Z D Y1 \ Y2 D i 1 1 \.Y2 / is a submanifold of X whose dimension k is related to the dimensions n; m1 ; m2 of X; Y1 ; Y2 by the formula k D m1 Cm2 n\. Let ˛1 2 H n m1 \.XI Z/; ˛2 2 H n m2 \.XI Z/; (Anatoly Fomenko 253) - and ˇ 2 H 2n m1 m2 \.XI Z/ be cohomology classes such that homology classes D˛ 1 ; D˛2 , and Dˇ are represented by Y1 ; Y2 , and Z\. Then ˛1 ^ ˛2 D ˇ: (Anatoly Fomenko 254) - This theorem provides a very powerful tool for determining multiplicative structure in cohomology, mainly for manifolds, but actually for all spaces, because of the naturality of the multiplicative structure\. (Anatoly Fomenko 254) - Example\. If q C r n, then the product of canonical generators of the groups H 2q \.CPn I Z/ and H 2r \.CPn I Z/ is the canonical generator of H 2\.qCr/ \.CPn I Z/; indeed, Poincaré isomorphism takes the three generators into the homology classes of projective subspaces of dimensions n q; n r, and n q r, and, in general position, intersection of the first two is the third\. Thus, the ring H \.CPn I Z/ D position, intersection of the first two is the third\. Thus, the ring H n 0 n i H i the n \.CP I Z/ has the following structure: There is 1 2 H 0 n \.CP I Z/ and the L i the 2 n 2q n generator x 2 H \.CP I Z/; the group H \.CP I Z/ with 1 q n is generated by xq \. If n is finite, then xnC1 D 0\. In more algebraic terms, H \.CPn I Z/ is the ring of polynomials of one variable x factorized by the ideal generated by xnC1 , H \.CPn I Z/ D ZŒx=\.xnC1 /; dim x D 2I (Anatoly Fomenko 254) - EXERCISE 36\. Prove that the integral cohomology ring of the sphere Sg2 with g handles is as follows: (Anatoly Fomenko 254) - Describe the multiplicative structure in Z2 cohomology of the projective plane with handles and the Klein bottle with handles\. (Anatoly Fomenko 255) - EXERCISE 37\. Prove that any continuous map CPn ! CPm with n > m induces a trivial map in cohomology of any positive dimension \(with any coefficients\)\. Prove a similar statement for real projective spaces\. (Anatoly Fomenko 255) - EXERCISE 38\. Prove that if g < h, then there are no continuous maps Sg2 ! Sh2 of a nonzero degree\. (Anatoly Fomenko 255) - Theorem 1 shows that the multiplicative structure in cohomology of a closed orientable manifold is rich \(many nonzero products\)\. Actually, we already have a strong statement of this kind: Theorems 2 and 3 of Sect\. 17\.5 show that if X is a compact oriented n-dimensional homology manifold, then for every infinite order class ˛ 2 H m \.XI Z/ there exists a ˇ 2 H n m \.XI Z/ such that h˛ ^ ˇ; ŒXi D 1\. If dim X D 2k and ˛1 ; ˛2 ; : : : is a basis in the free part of H k \.XI Z/, then the matrix kh˛ i ^ ˛j ; ŒXik is unimodular \(that is, its determinant is ˙1\)\. (Anatoly Fomenko 255) - Suppose that a connected triangulated space X is an oriented n-dimensional homology manifold which, however, is not assumed to be compact; that is, the triangulation may be not finite\. In this case we still have a correspondence between \(oriented\) simplices and barycentric stars of complementary dimensions, but no isomorphism between chains and cochains, since chains are supposed to be finite linear combinations of simplices \(or barycentric stars\), and cochains are allowed to take nonzero values on infinitely many simplices\. To construct Poincaré isomorphism, we need to modify the definition either of chains or of cochains\. (Anatoly Fomenko 255) - \. Both modifications are well known in topology; (Anatoly Fomenko 255) - There is also a similar \(dual\) definition of compact or compactly supported cohomology of a locally compact topological space X\. Namely, a cochain c 2 Cn \.XI G/ is called compactly supported if there exists a compact set K X such that c\.f / D 0 for any singular simplex f W n ! X such that f \.n / \ K D ;\. (Anatoly Fomenko 256) - Proposition 1\. Let X be a compact topological space and let A X be a closed subset\. Then there are natural \(make the statement precise: in what sense natural?\) isomorphisms H nopen \.X AI G/ Š Hn \.X; AI G/ and Hcomp n \.X AI G/ Š H n \.X; AI G/: In particular, if X is locally compact and X is the one-point compactification of X, then H nopen \.XI G/ Š H H n \.XI G/ Š H H (Anatoly Fomenko 256) - These results appear the most interesting when n is even: n D 2k\. Consider the fragment @ i H @ i kC1 \.X; @X/ ! Hk \.@X/ ! Hk \.X/ (Anatoly Fomenko 259) - all the dualities are with respect to the intersection number (Anatoly Fomenko 259) - For example, the torus T can be presented as a boundary of an orientable compact three-dimensional manifold in many different ways \(for example, the torus is the boundary of the solid torus\)\. But if T D @X \(where X is a compact orientable three-dimensional manifold\), then the inclusion homomorphism i W H1 \.T/ ! H1 \.X/ must have a one-dimensional kernel, not less and not more \(if X is a solid torus, then i annihilates the homology class of the meridian, but not the homology class of the parallel\)\. (Anatoly Fomenko 259) - \. In the case when k is odd, the form determines a symplectic structure in Hk \.@X/, and the last statement means that Ker i D Im @ is a Lagrangian subspace of H k \.@X/\. (Anatoly Fomenko 260) - ast statement means that Ker i D Im @ is a Lagrangian subspace of H k \.@X/\. This, however, does not impose any condition on the manifold @X\. The case when k is even, however, is very much different\. A real vector space V with a nondegenerate symmetric bilinear form ! can have a subspace W of dimension one half of dim V with a zero restriction !jW if and only if the signature of ! \(the difference between the positive and negative inertia indices\) is zero\. For a compact oriented 4`-dimensional manifold Y, the signature of the form in H2` \.Y/ is called the signature of Y and is denoted as \.Y/\. (Anatoly Fomenko 260) - EXERCISE 44\. Let Y1 and Y2 be two connected orientable closed manifolds of the dimension 4`, and let Y D Y1 #Y2 be the connected sum (Anatoly Fomenko 260) - Prove that \.Y/ D \.Y1 / C \.Y2 /\. (Anatoly Fomenko 260) - Theorem\. If a closed oriented 4`-dimensional manifold Y is a boundary of a compact oriented manifold X, then \.Y/ D 0 [in particular, B2` \.Y/ is even]\. (Anatoly Fomenko 260) - xample\. The manifold CP2` cannot be a boundary of a compact orientable \.4` C 1/-dimensional manifold, because B2` \.CP2` / D 1 is odd\. (Anatoly Fomenko 260) - d\. But the connected sum CP 2` #CP 2` \(see Exercise 44\), which has even middle Betti number, is also not a boundary since its signature is not zero \(it is 2\)\. The same is true for a connected sum of a number of copies of CP2` \. But the connected sum CP2` #\.CP2` / \(where the minus sign stands for the orientation reversion\) has zero signature and may be a boundary\. Actually, it is a boundary (Anatoly Fomenko 260) - EXERCISE 45\. Let Y be a connected closed oriented manifold\. Prove that the manifold Y#\.Y/ is a boundary of some compact manifold\. \(Hint : Drill a hole in Y and then multiply by I\.\) (Anatoly Fomenko 260) - Let A Sn be a simplicial subset of Sn , that is, a union of some simplices of some triangulation of Sn \. The goal of this section is to construct Alexander isomorphisms, (Anatoly Fomenko 261) - LW H Š Š H / ! en and LW H 1 m \.Sn AI G/ Š G/ Š H 1 m \.AI G/; and then to reformulate them as a duality between homology groups of A and Sn A (Anatoly Fomenko 261) - Like Poincaré isomorphism, Alexander isomorphism may be turned into a homology–homology duality, with the role of intersection numbers played by socalled linking numbers\. From the point of view of Alexander isomorphism, the definition of linking numbers is immediately clear\. Let A Sn be as above, and let ˛ 2 Hp \.Sn A/; ˇ 2 Hq \.A/ be two homology classes with p C q D n 1\. Then \.˛; ˇ/ D hL 1 ˛; ˇi is called the linking number of ˛ and ˇ, and the isomorphism L \(rather L 1 \) becomes a duality Š Š Free Hq \.A/ ! Hom\.Free Hp \.Sn A/; Z/; ˇ 7! f˛ 7! \.˛; ˇ/g: (Anatoly Fomenko 262) - Fig\. 72 Definition of the linking number \.a; b/ (Anatoly Fomenko 262) - Thus, linking numbers provide Alexander duality similar to the Poincaré duality\. (Anatoly Fomenko 263) - EXERCISE 50\. Let A be a k-component link \(D the union of k disjoint non-self intersecting closed curves in S3 \)\. Find the homology of S3 A\. (Anatoly Fomenko 263) - EXERCISE 52\. The following is a description of a “secondary multiplicative structure in cohomology” provided by “Massey produc (Anatoly Fomenko 264) - ducts\.” L (Anatoly Fomenko 264) - Check all this and compute the cohomology, with cup-products and Massey products, of the complement of the “Borromeo rings” \(see Fig\. 73\) (Anatoly Fomenko 264) - There exists an extensive theory of “triple linking numbers” and their relations to Massey products \(with further generalizations\); see Milnor [54] and Turaev [87]\. (Anatoly Fomenko 264) - 17\.12 Integral Poincaré Isomorphism for Nonorientable Manifolds These isomorphisms have the form H m \.XI Z/ Š H n m \.XI ZT /; H m \.XI ZT / Š Hn m \.XI Z/: Here X is a connected compact n-dimensional nonorientable homology manifold, and homology and cohomology with coefficients in ZT \(“twisted” integers\) are defined in the following way\. (Anatoly Fomenko 264) - Let X e be the oriented twofold covering of X\. Then there is a canonical orientation reversing involution t W X there is a canonical orientation reversing involution t W X e ! X\. e Ther transformation t# W Cq \.X/ ! Cq \.X/ with the square 1, and a decomposition C q \.X/ (Anatoly Fomenko 264) - where Cq˙ \.X/ X/ C e D fc 2 Cq \.X/ e j t# \. q \.X/; we take the other summand, c/ D ˙ Cq \.X/, ˙cg\. O X/, for C bviously, q \.XI ZT /\. (Anatoly Fomenko 265) - e The homology of this complex is denoted as H q \.XI ZT /, and the corresponding cohomology is taken for H q \.X; ZT / (Anatoly Fomenko 265) - Most problems in homotopy topology consist in a homotopy classification of continuous maps between two topological spaces\. A natural intermediate problem is the question of whether a given continuous map A ! Y can be extended to a continuous map X ! Y for some X A \(with a subsequent classification of such extensions\)\. This is what the obstruction theory was designed for (Anatoly Fomenko 265) - Let X be a CW complex, and let Y be a connected topological space which is assumed homotopically simple \(that is, the action of the fundamental group in all homotopy groups is trivial; later, we will discuss several possibilities of removing or, at least, weakening this condition\)\. Consider the problem of extending a continuous map f W X n ! Y to a continuous map X nC1 ! Y \(where X n ; X nC1 are skeletons\)\. Let e X be a cell of dimension n C 1, and let hW DnC1 ! X be a corresponding characteristic map\. There arises a continuous map fe D f ı hjSn W Sn ! Y\. It is obvious that f can be continuously extended to X n [ e if and only if fe is homotopic to a constant, that is, if fe represents the class 0 2 (Anatoly Fomenko 265) - n \.Y/ (Anatoly Fomenko 265) - Furthermore, the possibility of extension of f to X nC1 is the same as the possibility of its extension to every \.n C 1/-dimensional cell of X\. If we construct, as above, a map fe W Sn ! Y for every e and denote by 'e the class of fe in n \.Y/, we arrive at the following, essentially tautological, statement: A continuous map f W X n ! Y can be extended to a continuous map X nC1 ! Y if and only if every 'e is equal to 0\. (Anatoly Fomenko 265) - The function e 7! 'e can be regarded as an \.n C 1/-dimensional cellular cochain c f of X with coefficients in n \.Y/\. (Anatoly Fomenko 265) - Thus, cf 2 C nC1 \.XI n \.Y//; and f can be extended to X nC1 if and only if cf D 0\. The cochain cf is called the obstruction cochain to the extension of f to X nC1 \. (Anatoly Fomenko 271) - Theorem 1\. The obstruction cochain is a cocycle: ıcf D 0\. (Anatoly Fomenko 271) - e n \.Y// is p# cf , p# ıcf D ıp# cf D ıcf ıp D 0, and hence ıcf D 0\. The cohomology class Cf 2 H nC1 \.XI n \.Y// of the cocycle cf is called the cohomology obstruction, or simply the obstruction to extension of f to X nC1 \. (Anatoly Fomenko 272) - Theorem 2\. The condition Cf D 0 is necessary and sufficient to the existence of extending f jX n 1 to X nC1 \. In other words, Cf =0 if and only if it is possible to extend f to X nC1 after, possibly, a changing f on X n X n 1 \. (Anatoly Fomenko 272) - Let A be a CW subcomplex of a CW complex X, and let the continuous map f be defined on A[X n \. The obstruction cochain cf to an extension of this map to A[X nC1 is contained in C nC1 \.X; AI n \.Y//, it is a cocycle, and its cohomology class Cf 2 H nC1 \.X; AI n \.Y// is called an obstruction\. (Anatoly Fomenko 274) - Theorem\. If f ; gW X ! Y are two continuous maps which agree on X n 1 , then the difference cochain df ;g is a cocycle whose cohomology class Df ;g 2 H n \.XI n \.Y// is equal to 0 if and only if f jX n and g jX n are X n 2 -homotopic\. (Anatoly Fomenko 274) - If we assign to every n-dimensional cell of K\.; n/ the corresponding element of , we get a cochain c 2 C n \.K\.; n/I (Anatoly Fomenko 274) - Theorem\. Let X be a CW complex\. For any Abelian group and for any n > 0, the map \.X; K\.; n// ! H n \.XI /; Œf ! f \.F /; \(\) is a bijection\. (Anatoly Fomenko 275) - Lemma\. c is a cocycle\. (Anatoly Fomenko 275) - The cohomology class F 2 H n \.K\.; n/I / of the cocycle c is called the fundamental cohomology class of K\.; n (Anatoly Fomenko 275) - Corollary 1\. A CW complex of the type K\.; n/ is homotopically unique\. Hence, a topological space of the type K\.; n/ is weakly homotopically unique\. (Anatoly Fomenko 276) - Corollary 2\. For a CW complex X, there is a group isomorphism H 1 \.XI Z/ Š \.X; S1 / \(where S1 is regarded as an Abelian topological group\)\. (Anatoly Fomenko 276) - Theorem 1 \(Hopf\)\. For every n-dimensional CW complex X, there is a bijection H n \.XI Z/ $ \.X; Sn /; Œf 7! f \.s/; where s D 1 2 Z D H n \.Sn I Z/\. (Anatoly Fomenko 276) - Theorem 2 \(Hopf\)\. Let an n-dimensional CW complex X contain as a CW subcomplex a sphere Sn 1 \. This sphere is a retract of X if and only if the inclusion homomorphism H n 1 \.XI Z/ ! H n 1 \.Sn 1 I Z/ is an epimorphism\. (Anatoly Fomenko 278) - et D \.E; B; F; p/ be a locally trivial fibration\. We assume that the fiber F is homotopically simple \(for example, simply connected\), and the base B is simply connected\. (Anatoly Fomenko 279) - Let (Anatoly Fomenko 279) - Assume that the base B is a CW complex and that there given a section sW Bn ! E [which means that p ı s D id] over the nth skeleton of the base\. We are going to describe an obstruction to extending this section to BnC1 (Anatoly Fomenko 279) - We get a cochain cs 2 C nC1 \.BI n \.F//\. This is the obstruction cochain to extending s to BnC1 (Anatoly Fomenko 279) - Obstructions to extending maps may be regarded as particular cases of obstructions to extending sections\. Namely, a continuous map f W X ! Y can be represented by the graph FW X ! XY; F\.x/ D \.x; f \.x//, which, in turn, is a section of the trivial fibration \.X Y; X; Y; p/, where pW X Y ! X is the projection of the product onto a factor\. Obstructions to extending a map are the same as obstructions to extending its graph\. (Anatoly Fomenko 279) - Suppose that 0 \.F/ D 1 \.F/ D D n 1 \.F/ D 0, and n \.F/ ¤ 0\. 0 Then there are no obstructions to extending a section from B0 \(where it obviously exists\) to B1 ; : : : ; Bn 1 and the first obstruction emerges in H nC1 \.BI n \.F//: It is the obstruction to extending the section from Bn 1 to Bn \. This obstruction could depend, however, on the sections on the previous skeletons; however, the next proposition states that it is not the case\. (Anatoly Fomenko 279) - Proposition 1\. Let 0 \.F/ D 1 \.F/ D D n 1 \.F/ D 0, and let s; s 0 W Bn ! E be two sections\. Then Cs D Cs0 2 H nC1 \.BI n \.F//\. (Anatoly Fomenko 280) - Proposition 1 shows that the first obstruction to extending a section to the nth skeleton of the base is determined by the fibration, so we obtain a well-defined class C\./ 2 H nC1 \.BI n \.F// \(recall that n is the number of the first nontrivial homotopy group of F\); this class is called the characteristic class of ; we will also use the term primary characteristic class to distinguish it from numerous characteristic classes of vector bundles, (Anatoly Fomenko 280) - One can say that a fibration as above has a section over the nth skeleton of the base if and only if its characteristic class is zero\. (Anatoly Fomenko 280) - Let X be a connected closed oriented n-dimensional manifold and let T be the manifold of all nonzero tangent vectors of M\. The projection pW T ! X \(which assigns to a tangent vector the tangency point\) gives rise to a locally trivial fibration X D \.T; X; Rn 0; p/\. Since the fiber is homotopy equivalent to Sn 1 , there arises a characteristic class C\. X / 2 H n \.XI Z/ (Anatoly Fomenko 281) - Proposition 2\. hC\. X /; ŒXi D \.X/\. (Anatoly Fomenko 281) - Corollary\. A connected closed orientable manifold possesses a nowhere vanishing vector field if and only if \.X/ D 0\. (Anatoly Fomenko 281) - EXERCISE 9\. Let X be a CW complex with 0 \.X/ D 1 \.X/ D D n 1 \.X/ D 0; n \.X/ ¤ 0\. Prove that the characteristic class of the Serre fibration EX ! X with the fiber X which belongs to H n \.XI nC1 \.X// D H n \.XI n \.X// is just the fundamental class of X\. (Anatoly Fomenko 281) - A real ndimensional vector bundle with the base B is a locally trivial fibration with the base B and the fiber homeomorphic to Rn with an additional structure: Each fiber is furnished by a structure of an n-dimensional vector space, in such a way that the vector space operations \.; x/ 7! x and \.x; y/ 7! x C y depend continuously on the fiber, in the sense that the arising maps R E ! E and f\.x; y/ 2 E E j p\.x/ D p\.y/g ! E \(where E is the total space and p is the projection of the fibration\) are continuous\. (Anatoly Fomenko 282) - Let E e be the set of all bases in all fibers of the fibration; there is a natural topology in E\. e The fibration is orientable if and only if E e has two \(not one\) components; a choice of one of these components is an orientation of the fibration\. (Anatoly Fomenko 282) - For vector bundles of all three kinds there are natural definitions of equivalences, restrictions \(over subspaces of the base\) and induced bundles \(by a continuous map of some space into the base\)\. A trivial bundle is a bundle equivalent \(in its class\) to the projection bundle B Rn ! B or B Cn ! B (Anatoly Fomenko 282) - Important Example\. The Hopf or tautological vector bundle over RPn is the onedimensional vector bundle whose total space is the set of pairs \.`; x/, where ` 2 RPn is a line in RnC1 and x 2 ` is a point on this line [topology in this set is defined by the inclusion into RPn RnC1 ]\. Precisely in the same way, the Hopf, or tautological, one-dimensional complex vector bundle over CPn is defined\. An obvious generalization of this construction provides tautological vector bundles over the Grassmannians G\.m; n/; GC \.m; n/, and CG\.m; n/ (Anatoly Fomenko 282) - : If x1 ; : : : ; xn is a complex basis in a fiber of , then x 1 ; ix1 ; : : : ; xn ; ixn is a real basis in the same (Anatoly Fomenko 282) - space, and the orientation of this basis does not depend on the choice of the complex basis x1 ; : : : ; xn [this follows from the fact that the image of the natural embedding cW GL\.n; C/ ! GL\.2n; R/ consists of matrices with positive determinant; the last statement follows from the fact that GL\.n; C/ is connected, or, more convincingly, from the formula det\.cA/ D j det A (Anatoly Fomenko 283) - t Aj2 ; (Anatoly Fomenko 283) - f 1 ; 2 are two vector bundles of the same type \(real, complex, oriented\) and with the same base, then the \(direct or Whitney\) sum 1 ˚2 and the tensor product 1 ˝2 are defined as vector bundles with the same base whose fibers are, respectively, direct sums or tensor products of the fibers of the bundles 1 and (Anatoly Fomenko 283) - 2 \. Here is a more formal definition of the sum \(here and below, K denotes R or C\)\. Let 1 D \.E1 ; B1 ; Kn1 ; p1 /; 2 D \.E2 ; B2 ; Kn2 ; p2 / be two vector bundles \(the bases may not be the same\)\. Put 1 2 D \.E1 E2 ; B1 B2 ; Kn1 Cn2 ; p1 p2 /; this is a vector bundle over B1 B2 of dimension n1 C n2 \. If B1 D B2 D B, then we define 1 ˚ 2 as the restriction of 1 2 to the diagonal B B B\. Another formal definition: Let B1 D B2 D B and let p 2 1 D \.E; iagonal B B B\. Another formal definition: E; p/ be the bundle over E2 induced by Let B1 D B2 D B and let p2 1 D \.E;e E2 ; n1 Cn2 1 \. Then 1 ˚ 2 D \.E; B; K p/ (Anatoly Fomenko 283) - wo vector bundles of the same type, but, possibly, of different dimensions, are called stably equivalent if they become equivalent after adding trivial bundles\. (Anatoly Fomenko 284) - To make up a more formal definition, notice that a standard trivial n-dimensional bundle B Kn ! B is usually denoted simply as n\. With this notation, stab , 9m; nW ˚ n ˚ m: (Anatoly Fomenko 284) - D: Linear Maps Between Vector Bundles, Subbundles, and Quotient Bundles (Anatoly Fomenko 284) - A linear map of a vector bundle 1 D \.E1 ; B1 ; Kn1 ; p1 / into a vector bundle 2 D \.E2 ; B2 ; Kn2 ; p2 / \(as before, K denotes R or C\) is a pair of continuous maps FW E1 ! E 2 ; f W B1 ! B2 such that f ı p1 D p2 ı F and for every x 2 B, the appropriate restriction of F is a linear map p 1 1 \.x/ ! p2 1 \.f \.x//\. x//\. 0 The s bundle D \.E; B; Kn ; p/ is a vector bundle 0 D E0 ; B; Kn ubbundle 0 of a vector ; pjE0 with E0 E whose fibers are subspaces of the fibers of \. The inclusion map E 0 ! E and the identity map B ! B compose a linear map \(inclusion\) 0 ! \. If 0 is a subbundle of , then a fiberwise factorization creates a quotient bundle = 0 \. More formally, the total space of = 0 is obtained from E by a factorization over the equivalence relation: x1 x2 if p\.x1 / D p\.x2 / and x2 x1 2 E 0 \. There is an obvious linear map \(projection\) ! = 0 \. (Anatoly Fomenko 284) - This presentation of a vector bundle is called the coordinate presentation\. (Anatoly Fomenko 285) - For every y 2 Ui \ Uj , there arises a composition ' 1 j 'i Kn ! p 'i \.y/ ! Kn I the function which assigns this composition to y is a continuous map 'ij W Ui \ Uj ! G where G D GL\.n; K/ (Anatoly Fomenko 285) - Fix an open covering fUi g of the base B such that the restrictions jUi are all trivial vector bundles; let 'i W p 1 \.Ui / ! Kn be a trivializatio (Anatoly Fomenko 285) - ion, (Anatoly Fomenko 285) - An obvious generalization of the so presented vector bundles consists in specifying a topological group G and a G-space F (Anatoly Fomenko 285) - `functions 'ij W Ui \ Uj ! G with properties \(i\)–\(iii\) just listed\. In the disjoint union i \.Ui F/, make, for every i; j; y 2 Ui \ Uj , an identification \.y; f / 2 Uj F \.y; 'ij \.y/f / 2 Ui F ; the space arising we take for E\. The projections Ui F ! Ui B form a projection pW E ! F, and there arises a locally trivial fibration \.E; B; F; p/ with a certain additional structure similar to a structure of a vector bundle\. Such fibrations are called fiber bundles \(or Steenrod fibrations\); according to this terminology, G is the structure group, and F is the standard fiber\. (Anatoly Fomenko 285) - ssume that the functions 'ij take values not in the group GL\.n; R/; GL\.n; C/ or GLC \.nI R/, but in some subgroup of one of these groups, say, in O\.n/; SO\.n/, or U\.n/\. It is clear that the fiber bundles arising have an adequate description as real, complex, or oriented vector bundles with an additional structure, for the examples above, with an Euclidean or Hermitian structure, in every fiber\. If the subgroup is the group of block diagonal matrices, GL\.p; K/ GL\.q; K/ GL\.nI K/; n D p C q, then the fiber bundle arising is the usual n-dimensional vector bundle presented as the sum of two vector bundles, of dimensions p and q\. (Anatoly Fomenko 285) - q\. In a similar way, we can present vector bundles with a fixed nonvanishing section, or with a fixed subbundle, and so o (Anatoly Fomenko 285) - Take an arbitrary G and put F D G with the left translation action; the fibrations arising are called principal\. (Anatoly Fomenko 285) - An algebraically more convenient approach consists in defining a tangent vector of X at x as a linear map vW C 1 \.X/ ! R \(C 1 \.X/ is the space of real C 1 -functions\) such that v\.fg/ D v\.f /g\.x/ C f \.x/v\.g/ \(in other words, tangent vectors are identified with directional derivatives\)\. Finally, if X is presented as a smooth surface in an Euclidean space, then a tangent vector to X is simply a tangent vector to this surface\. (Anatoly Fomenko 286) - The set of tangent vectors to an n-dimensional manifold X at a point x is an n-dimensional vector space which is denoted as Tx X\. The union of all spaces Tx X possesses a natural topology and, moreover, a structure of a 2n-dimensional smooth manifold; this manifold is denoted as TX\. The natural projection TX ! X makes TX a total space of a vector bundle over X; this vector bundle is called the tangent bundle of X and is denoted as \.X/\. (Anatoly Fomenko 286) - /\. A section of a tangent bundle is a vector field on the manifold\. A manifold whose tangent bundle is trivial is called parallelizable ; a manifold is parallelizable if it is possible to choose bases in all tangent spaces depending continuously of a point or, equivalently, if there exist n D dim X vector fields on X which are linearly independent at every point\. (Anatoly Fomenko 286) - For example, the circle is parallelizable, the torus is parallelizable, while the two-dimensional sphere is not parallelizable\. The three-dimensional sphere is parallelizable: If it is presented as the space of unit quaternions, then the basis at the space Tx S3 is formed by quaternions ix; jx; kx where i; j; k are quaternion units\. If you replace quaternions by octonions, 7 you will prove that the sphere S7 is parallelizable (Anatoly Fomenko 286) - There is a remarkable fact that no spheres besides S1 ; S3 ; S7 are parallelizable: This is one of the versions of the Frobenius conjecture proven by Adam (Anatoly Fomenko 286) - Notice that the problem of parallelization of spheres is equivalent to the problem of existence of spheroids with the invariant Hopf equal to one (Anatoly Fomenko 286) - EXERCISE 8\. Prove that the orientability of a manifold X \(in the sense of Sect\. 17\.1\) is equivalent to the orientability of the tangent bundle \.X/\. (Anatoly Fomenko 287) - The quotient bundle \.X/jY = \.Y/ is called the normal bundle of Y in X and is denoted as X \.Y/ or \.Y/\. The word “normal” is an indication of the fact that if X is a submanifold of an Euclidean space, then the total space of \.Y/ may be regarded as consisting of vectors at points of Y which are tangent to X and normal to Y\. (Anatoly Fomenko 287) - If Y is a submanifold of a manifold X, then there arise two vector bundles with the base Y: \.Y/ and \.X/jY , and \.Y/ \.X/jY \(a tangent vector to a submanifold is also a tangent vector to the manifold (Anatoly Fomenko 287) - Notice that the construction of normal bundles with all properties listed can be applied not only to submanifolds, that is, to embeddings of a manifold Y to a manifold X, but also to immersions W Y ! X; the only significant change is that the restriction bundle \.X/jY should be replaced by the induced bundle \.X/\. (Anatoly Fomenko 287) - EXERCISE 9\. Deduce from the last equality that normal bundles of a manifold corresponding to different embeddings or immersions of this manifold to Euclidean spaces \(possibly, of different dimensions\) are stably equivalent\. (Anatoly Fomenko 287) - EXERCISE 10\. Prove that the normal bundle to an n-dimensional oriented surface embedded \(or immersed\) into the \.n C 1/-dimensional Euclidean space is trivial\. (Anatoly Fomenko 287) - Deduce from this that the tangent bundle to such a surface \(for example, to an arbitrary sphere with handles\) is stably trivial \(that is, stably equivalent to a trivial bundle\)\. A manifold whose tangent bundle is stably trivial is called stably parallelizable\. Obviously, a manifold is stably parallelizable if and only if its normal bundle is stably trivial\. (Anatoly Fomenko 287) - FYI \(this is not an exercise\)\. A closed connected manifold is stably parallelizable if and only if it is parallelizable in the complement to a point\. A noncompact connected manifold if stably parallelizable if and only if it is parallelizable\. A manifold is stably parallelizable if and only if it is orientable and admits an immersion in the Euclidean space of the dimension bigger by 1\. (Anatoly Fomenko 287) - EXERCISE 11\. Let be the Hopf bundle over RPn \. Prove that \.RPn / ˚ 1 ˚ ˚ D \.n C 1/: ˚ ˚ (Anatoly Fomenko 287) - A characteristic class c of n-dimensional vector bundles on the chosen type with values in q-dimensional cohomology with the coefficients in G is a function which assigns to every n-dimensional vector bundle of the chosen type with a CW base B a cohomology class c\./ 2 H q \.BI G/ such that if f W B0 ! B is a continuous map of another CW complex into B, then c\.f / D f c\./\. Here f on the left-hand side of the formula means the inducing operation for vector bundles, and on the right-hand side it means the induced cohomology homomorphism\. (Anatoly Fomenko 288) - The term “characteristic class” is not new for us: In Sect\. 18\.5, we called the first obstruction to extending a section of a locally trivial fibration a characteristic class \(or a primary characteristic class\) of this fibration, and the equality c\.f / D f c\./ held for that characteristic classes\. However, that construction cannot be applied to vector bundles directly, because their fiber is contractibl (Anatoly Fomenko 288) - ble\. \(Recall that the coefficient domain for the characteristic classes of Sect\. 18\.5 is the first nontrivial homotopy group of the fiber\.\) W (Anatoly Fomenko 288) - \) What we still can do is to apply the construction to some fibration which can be constructed from the given vector bundl (Anatoly Fomenko 288) - le\. (Anatoly Fomenko 288) - dle\. An ample variety of such fibrations is delivered by the construction of an associated fibration\. (Anatoly Fomenko 288) - we construct the total space E of a new fibration as i \.Ui F/ j Œ\.y; f / 2 Uj F Œ\.y; 'ij \.y/f / 2 Ui F for all y 2 Ui \ Uj ; f 2 F: The fibration \.E; B; F; p/ [where pW E ! B is the projection \.y; f / 7! y] is the associated \(by the given vector bundle\) fibration with the standard fiber F\. (Anatoly Fomenko 288) - However, usually we will not need this general construction: Almost always, we will restrict ourselves to one particular case of it, which is described ahead\. Let D \.E; B; Rn ; p/, or \.E; B; Cn ; p/, be a given vector bundle, and let 1 k n\. Put E k D f\.x1 ; : : : ; xk / 2 E E j p\.x1 / D D p\.xk /I x 1 ; : : : ; xk are linearly independentg: (Anatoly Fomenko 288) - There is an obvious projection pk W Ek ! B, and there arises a locally trivial fibration k D \.Ek ; B; Rk ; pk / where Rk is the space of all linearly independent k-frames in Rn or Cn \. (Anatoly Fomenko 289) - E 1 is E B, where B is embedded into E as the zero section, and R1 is Rn 0 or Cn 0\. (Anatoly Fomenko 289) - The fibers are noncompact spaces which would have better been replaced by homotopy equivalent classical manifolds: Stiefel manifolds and spheres\. This can be done with the help of the following simple lemma\. Lemma\. If a vector bundle has a CW base, then it is possible to introduce in all fibers an Euclidean or Hermitian structure which depends continuously on the point of the base; moreover, this can be done in a homotopically unique way\. (Anatoly Fomenko 289) - The set of all Euclidean \(Hermitian\) structures in fibers of a vector bundle is a total space of a fibration whose fiber is the space of all Euclidean \(Hermitian\) structures in a given vector space \( (Anatoly Fomenko 289) - Obviously, the fiber of this fibration is contractible \(it is a convex subset of the space of all symmetric bilinear \(Hermitian\) forms in this vector space\. This fibration has a section \(all the obstructions are zeroes\) and this section is homotopically unique \(all difference cochains are zeroes\)\. (Anatoly Fomenko 289) - Using these Euclidean or Hermitian structures in the fibers, we can replace the fibration k into the fibration k0 whose total space is the space of all orthonormal \(unitary\) frames in the fibers of \. The fiber of k0 is the Stiefel manifold V\.n; k/ or CV\.n; k/; in particular, 10 is the fibration whose fiber is the sphere Sn 1 \(S2n 1 in the complex case\); this fibration is called spherical\. (Anatoly Fomenko 289) - \) Thus, there arises the first obstruction to extending a section of 10 , and this first obstruction is an element of H n \.BI Z/\. Regarded as a characteristic class of the bundle , this element is called the Euler class of ; the notation: e\./ (Anatoly Fomenko 289) - Let be an n-dimensional oriented \(real\) vector bundle with the CW base B\. Consider the corresponding spherical fibration 10 \. It is easy to see that the orientability of the bundle implies the orientability of the fibration 10 ; that is, the fibration 10 is homologically simple\. \( (Anatoly Fomenko 289) - Lemma\. Let 1 k < n\. Then \(i\) i \.V\.n; k// D 0 for i < n k: (Anatoly Fomenko 289) - Z; if k D 1 or n k is evenI \(ii\) n k \.V\.n; k// Š Z2 in all other cases: (Anatoly Fomenko 290) - The space V\.n; 2/ is the space T1 Sn 1 of unit tangent vectors to the sphere Sn 1 , the fibration S n 2 Sn V\.n; 2/ ! Sn 1 is the natural fibration of the space of unit tangent vecto (Anatoly Fomenko 290) - the resulting element of n 2 \.Sn 2 / is the value of the obstruction to extending a vector field on Sn 1 \. As proved in Sect\. 18\.5 \(see Proposition 2\), this value is the Euler characteristic of Sn 1 , that is, 2 for n odd and 0 for n even (Anatoly Fomenko 290) - The lemma shows that the first obstruction to extending a section of fibration k0 \(or k \) takes value in H n kC1 \.BI Z or Z2 /\. Reduced modulo 2, this obstruction is a characteristic class of with the values in H j \.BI Z2 /; j D n k C 1\. This class is called the jth Stiefel–Whitney class of and is denoted as wj \./\. We also put w i \./ D 0 for i > dim and w0 \./ D 1 2 H 0 \.BI Z2 /\. (Anatoly Fomenko 291) - Lemma\. Let 1 k < n\. Then 0 for i < 2\.n k/ C 1; i \.CV\.n; k// Š Z for i D 2\.n k/ C 1: (Anatoly Fomenko 291) - Let be an n-dimensional complex vector bundle with a CW base B\. The lemma shows that the first obstruction to extending a section in the fibration k0 \(or k \) is a class cj \./ 2 H 2j \.BI Z/ where j D nk C1\. We get a characteristic class of complex vector bundles which is called the jth Chern class\. (Anatoly Fomenko 291) - Finally, if is again an n-dimensional vector bundle, then we put pj \./ D \.1/j c2j \.C/ 2 H 4j \.BI Z/ and call the classes pj \./ Pontryagin classes of the bundle (Anatoly Fomenko 291) - It is possible to define Pontryagin classes directly: We can associate with an n-dimensional vector bundle a fibration whose standard bundle is the space of all systems of n 2jC2 vectors of rank > n 2j; the first obstruction to extending sections in this fibration is pj \./ (Anatoly Fomenko 291) - EXERC (Anatoly Fomenko 291) - CISE EXERCISE 12\. 13\. Prove that w1 \./ D 0 if and only if the bundle is orientable\. (Anatoly Fomenko 291) - \. Prove that w1 \./ D 0 if and only if the bundle is orientable\. Prove that if is an n-dimensional complex vector bundle, then e\.R/ D cn \./; w2j \.R/ D 2 cj \./; w2jC1 \.R/ D 0: (Anatoly Fomenko 291) - et be an n-dimensional oriented vector bundle with a CW base B\. Then there exists a nowhere vanishing section of over the \.n 1/st skeleton Bn 1 of B\. We can extend this section to Bn , but it may have zeroes over n-dimensional cells\. If we assume these zeroes to be transverse intersections with the zero section, then we can count the “algebraic number” of these zeroes \(that is, we assign a C or sign to every zero\), and a function which assigns this number to every cell is an n-dimensional integral cellular cocycle\. Its cohomology class is the Euler class e\./ \(this is the construction of the first obstruction\)\. (Anatoly Fomenko 292) - If is not assumed oriented, then the previous construction gives a cohomology class modulo 2, and this is wn \./\. (Anatoly Fomenko 292) - hen the previous construction gives a cohomology class modulo 2, and this is wn \./\. We can construct in this way the other Stiefel– Whitney classes\. (Anatoly Fomenko 292) - s\. Namely, let us assume that has an Euclidean structure \(in the fibers\), and consider again a nowhere vanishing section of over Bn 1 \. Let us try to construct a second nowhere vanishing section of orthogonal to the first section\. This can be done over Bn 2 , but if we want to extend the second section to Bn 1 , we have to admit that it will have zeroes over \.n1/-dimensional cells\. Assuming these zeroes transverse, we can count their number modulo 2 in every \.n 1/-dimensional cell, and in this way we get an \.n 1/-dimensional cellular cocycle with coefficients in Z2 , and the cohomology class of this cocycle is wn 1 \./\. Then we construct a third section orthogonal to the first two, it can be made nowhere vanishing over Bn 3 , but to extend this third section to Bn 2 , we have to admit transverse zeroes over \.n 2/dimensional cells, and in this way we obtain a cocycle representing wn 2 \./\. And so on\. The (Anatoly Fomenko 292) - The Chern classes of complex vector bundles may be constructed in a similar way; (Anatoly Fomenko 292) - A: The Classification Theorem In Sect\. 19\.1\.A, we mentioned tautological bundles over Grassmannians\. They will be of primary importance now\. (Anatoly Fomenko 292) - Recall that the total space of the tautological bundle \(which we denote as or n \) over the Grassmannian G\.1; n/ is the space of pairs \.; x/ where 2 G\.1; n/ is an n-dimensional subspace of RN and x 2 R1 ; the projection acts as \.; x/ 7! \. (Anatoly Fomenko 292) - Theorem\. Let X be a finite CW complex\. Then \(i\) For every n-dimensional vector bundle over X, there exists a continuous map f W X ! G\.1; n/ such that f D \. \(ii\) This map f is unique up to a homotopy; that is, if f1 f2 , then f1 f2 \(the second means a homotopy\)\. \(iii\) Conversely, if f1 f2 , then f1 f2 \. (Anatoly Fomenko 293) - Corollary\. The correspondence f 7! f establishes a bijection between the set \.X; G\.1; n// of homotopy classes of continuous maps X ! G\.1; n/ and equivalence classes of n-dimensional vector bundles with the base X\. (Anatoly Fomenko 293) - First, notice that since X is compact and G\.1; n/ D Pro ! ! lim G\.N; n/, a continuous map X ! G\.1; n/ is the same as a continuous map X ! G\.N; n/ \(with sufficiently large N\) composed with the inclusion map G\.N; n/ ! G\.1; n/\. (Anatoly Fomenko 293) - The space G\.1; n/ is called a classifying space for real n-dimensional vector bundles, and is called a universal bundle ; a similar terminology is applied to GC \.1; n/ and CG\.1; n/\. (Anatoly Fomenko 294) - For a topological group G, there exists a principal fibration \(see Sect\. 19\.1\.E\) \.EG; BG; G; pG / with a cellular base and contractible space EG; for a given G, a principal fibration with these properties is unique up to a homotopy equivalence\. The space BG is called the classifying space for G; in particular, BGL\.n; R/ D BO\.n/ D G\.1; n/; BGLC \.n; R/ D BSO\.n/ D G C \.1; n/; BGL\.n; C/ D BU\.n/ D CG\.1; n/\. If F is a space with a faithful action of G, then, for a finite CW complex X, there is a bijection between the set of equivalence classes of Steenrod bundles over X with the structure group G and the standard fiber F and the set \.X; BG/ of homotopy classes of continuous maps X ! BG\. This construction belongs to J\. Milnor [55]\. It has further generalizations to the cases when G is not a topological group, but an H-space or a topological groupoid (Anatoly Fomenko 295) - So (Anatoly Fomenko 295) - Some definitions and theorems of the previous sections can be clarified with the help of the classification theorem (Anatoly Fomenko 295) - The definition of the sum of vector bundles can be done in the following way: If f W X ! G\.N; n/ and gW X ! G\.M; m/ are two continuous maps, then there arises a map f ˚ gW X ! G\.M C N; m C n/; \.f ˚ g/\.x/ D f \.x/ ˚ g\.x/ RN ˚ RM , and f ˚ g D \.f ˚ g/ , which gives an alternative construction of the sum of vector bundles\. The same for tensor products: We consider a map f ˝ gW X ! RNM ; f ˝ g\.x/ D f \.x/ ˝ g\.x/ RN ˝ RM D RNM , and f ˝ g D \.f ˝ g/ , which can be regarded as a definition of a tensor product of vector bundles \(same with complex vector bundles\)\. In a similar way, for a vector bundle , we can define S r ; ƒr ; , etc\. (Anatoly Fomenko 295) - Theorem\. The group of q-dimensional characteristic classes of n-dimensional real (Anatoly Fomenko 295) - vector bundles with coefficients in G is isomorphic to the group H q \.G\.1; n/I G (Anatoly Fomenko 295) - GENERALIZATION\. Characteristic classes of Steenrod fibrations with the structure group G taking values in the q-dimensional cohomology of the base with coefficients in A correspond bijectively to elements of H q \.BGI A/\. (Anatoly Fomenko 296) - \(i\) Every characteristic class of n-dimensional real vector bundles with coefficients in Z2 is a polynomial of the Stiefel–Whitney classes w 1 ; : : : ; wn , and different polynomials are different characteristic classes\. (Anatoly Fomenko 296) - Every characteristic class of n-dimensional complex vector bundles with coefficients in Z is a polynomial of the Chern classes c1 ; : : : ; cn , and different polynomials are different characteristic classes\. (Anatoly Fomenko 296) - Every characteristic class of n-dimensional real vector bundles with coefficients in Q, or R, or C is a polynomial of the \(images with respect to the inclusion of Z into the coefficient domain\) of the Pontryagin classes p 1 ; : : : ; pŒn=2 , and different polynomials are different characteristic classes\. (Anatoly Fomenko 296) - Every characteristic class of n-dimensional orientable vector bundles with coefficients in Q, or R, or C is a polynomial of the \(images with respect (Anatoly Fomenko 296) - o the inclusion of Z into the coefficient domain\) of the Pontryagin classes p 1 ; : : : ; pŒn=2 and, if n is even, the Euler class e, and different polynomials are different characteristic classes\. (Anatoly Fomenko 297) - 19\.5 The Most Important Properties of the Euler, Stiefel–Whitney, Chern, and Pontryagin Classes (Anatoly Fomenko 297) - For the Hopf \(tautological\) bundle over RPn \.n 2/, 0 ¤ w1 \./ H 1 \.RPn I Z2 / D Z2 and wi \./ D 0 for i > 1\. (Anatoly Fomenko 297) - For arbitrary real vector bundles ; with \(the same\) CW base, w X i \. ˚ / D wp \./wq \./: (Anatoly Fomenko 297) - Remark\. Statements \(1\) and \(2\) are often considered as axioms for Stiefel–Whitney classes: (Anatoly Fomenko 297) - Proof of Part \(1\) is immediate\. The restriction of to RP1 D S1 is the Möbius bundle, and obviously it has no nowhere vanishing section\. Thus, has no section over the first skeleton, which means that the first obstruction w1 \./ 2 H 1 \.RPn I Z2 / is not zero\. (Anatoly Fomenko 297) - Stiefel–Whitney classes invariant with respect to stable equivalence, which is the same as the statement wi \. ˚ 1/ D wi \. (Anatoly Fomenko 298) - It is convenient to write the formulas from \(2\) and \.20 / as w\. ˚ / D w\./w\./; w\. / D w\./ w\./ where w is the formal sum 1 C w1 C w2 C : : : \. (Anatoly Fomenko 299) - We begin with a computation of the Stiefel–Whitney classes for a very important example\. Proposition\. Consider the vector bundle over the space n „ ƒ‚ … RP1 1 1 1 n RP1 ƒ‚ „ … : Let x1 ; : : : ; xn 2 of H \.RP1 RP1 I Z2 /\. Then w i \. / D ei \.x1 ; : : : ; xn /; (Anatoly Fomenko 299) - where ei is ith elementary symmetric polynomial\. (Anatoly Fomenko 300) - It is well known in algebra that every symmetric polynomial in n variables with coefficients in an arbitrary integral domain R has a unique presentation as a polynomial in the elementary symmetric polynomial; the uniqueness statement means that no nonzero polynomial in e1 ; : : : ; en is equal to zero (Anatoly Fomenko 300) - But this number is precisely the number of q-dimensional cells in the standard \(Schubert\) CW decomposition of G\.1; n/, which, in turn, does not exceed dimZ2 H q \.G\.1; n/I Z2 /, that is, the dimension of the space of characteristic classes\. Thus, all these numbers and dimensions are the same (Anatoly Fomenko 300) - To establish a relation between characteristic classes it is sufficient to check it for splitting bundles, that is, for bundles isomorphic to sums of one-dimensional bundles\. T (Anatoly Fomenko 300) - splitting bundles, that is, for bundles isomorphic to sums of one-dimensional bundles\. This proposition is known under the name of the splitting principle (Anatoly Fomenko 300) - We see also that a nonzero characteristic class with coefficients in Z2 of n-dimensional vector bundles takes a nonzero value on the bundle ˚ ˚ \. This provides a method of finding relations between characteristic classes: A relation holds if it holds for ˚ ˚ \. Usually, this statement is formulated in a seemingly weaker, but actually equivalent form: To es (Anatoly Fomenko 300) - Formulas expressing the Stiefel–Whitney classes of the bundles ˝; ƒk ; Sk , and so on via the Stiefel–Whitney classes of and \(and the dimensions of and \) exist, but more complicated and less convenient, than the formulas for the Stiefel– Whitney classes of the sum \(or direct product\) (Anatoly Fomenko 301) - Lemma\. Let ; be one-dimensional real vector bundle over the same CW base\. Then w1 \. ˝ / D w1 \./ C w1 \./: (Anatoly Fomenko 301) - D: Properties of the Euler, Chern, and Pontryagin Classes For the Euler classes, a multiplication formula e\. ˝ / D e\./e\./ holds\. (Anatoly Fomenko 305) - All the major properties of the Stiefel–Whitney classes can be repeated with appropriate changes for the Chern classe (Anatoly Fomenko 305) - Like Stiefel–Whitney classes, the Chern classes are invariant with respect to stable equivalence\. The computation of the Chern classes of tensor product, exterior powers, and symmetric powers of complex vector bundles repeats the computations in Sect\. 19\.5\.C\. (Anatoly Fomenko 305) - The \(nonhomogeneous\) characteristic class ch with coefficients in Q defined by the formula ch D ch0 C ch1 C ch2 C 2 H even \.XI Q/ is called the Chern character\. (Anatoly Fomenko 306) - For a complex vector bundle with the base X, put 1 ch r \./ D Qr \.c1 \./; : : : ; cr \.// 2 H 2r \.XI Q/ (Anatoly Fomenko 306) - Prove that ch\. ˚ / D ch\./ C ch\./ and ch\. ˝ / D ch\./ ch\./: (Anatoly Fomenko 306) - For the Pontryagin classes, the multiplication formulas and all the other formulas are deduced from the corresponding formulas for the Chern classes and hold “modulo 2-torsion”; for example, X 2 pi \. ˚ / pp \./pq \./ D 0: (Anatoly Fomenko 306) - EXERCISE 17\. Prove that stably equivalent bundles have equal Pontryagin classes\. (Anatoly Fomenko 306) - In conclusion, we give two more formulas expressing the Stiefel–Whitney and Chern classes via the Euler class\. (Anatoly Fomenko 306) - P Claim: The homology class i ˛i ŒXi 2 Hn k \.BI / is the Poincaré dual of the first obstruction to extending a section in our fibration\. (Anatoly Fomenko 307) - A: Geometric Interpretation of the First Obstruction (Anatoly Fomenko 307) - Suppose also that we were able to construct a section over B X where X is a submanifold of B \(possibly, with singularities of codimension 2\) of dimension n k or a such submanifolds which are connected and transversally union of a finite number of S intersect each other, X D X (Anatoly Fomenko 307) - Since there is a section over si , and the fibration is trivial in a proximity of xi , we obtain a continuous map Sk 1 ! p \.xi / which determines, since the fibration and the fiber are homotopically simple, an element ˛i 2 k 1 \.F/ D \. (Anatoly Fomenko 307) - For a closed oriented manifold X, the value of the Euler class of the tangent bundle e\.X/ D e\. X / on the fundamental class ŒX is equal to the Euler characteristic (Anatoly Fomenko 307) - This implies that a closed manifold possesses a nonvanishing vector field if and only if its Euler characteristic is zero (Anatoly Fomenko 307) - EXERCISE 19\. Prove that a closed manifold X \(orientable or not\) possesses a continuous family of tangent lines \(equivalently: The tangent bundle \.X/ possesses a one-dimensional subbundle\) if and only if \.X/ D 0\. (Anatoly Fomenko 309) - Let sW B ! E be a section of in a general position with the zero section\. Show that the intersection B \ s\.B/ \(we assume that B is embedded into E as the zero section\) represents the homology class of B which is the Poincaré dual of the Euler class e\./ of (Anatoly Fomenko 309) - the Euler class of the normal bundle of a manifold embedded into an Euclidean space or a sphere is zero\. (Anatoly Fomenko 309) - Let Y be a closed oriented submanifold of a closed oriented manifold X, (Anatoly Fomenko 309) - and let X \.Y/ D \. \.X//jY = \.Y/ be the corresponding normal bundle (Anatoly Fomenko 309) - Prove the formula D\.e\.X \.Y// D i Š ŒY; (Anatoly Fomenko 309) - EXERCISE 22\. The last statement does not hold for immersion (Anatoly Fomenko 309) - mmersions (Anatoly Fomenko 309) - \. Show, in particular, that if f is an immersion of a closed oriented manifold of even dimension n into R2n with transverse self-intersections, then the algebraic number of the self-intersection points (Anatoly Fomenko 309) - is equal to one half of the “normal Euler number,” that is, of the value of the Euler class of the normal bundle on the fundamental class of the manifold (Anatoly Fomenko 309) - Example: Construct an immersion of S2 into R4 with one transverse self-intersection \(such a two-dimensional figure-eight\) and find the Euler class of the corresponding normal vector bundl (Anatoly Fomenko 309) - The Stiefel–Whitney classes of the tangent bundle of a smooth manifold X are called the Stiefel–Whitney classes of X and are denoted as wi \.X/\. (Anatoly Fomenko 309) - X/\. [In a similar way, people consider the Pontryagin classes pi \.X/ of a smooth manifold X and the Chern classes ci \.X/ of a complex manifold X\.] (Anatoly Fomenko 309) - ] Since the normal bundle of (Anatoly Fomenko 309) - a smooth manifold embedded into a Euclidean space does not depend, up to a stable equivalence, on the embedding, we can speak of the “normal Stiefel–Whitney classes,” wi \.X/, of a smooth manifold X (Anatoly Fomenko 310) - w p \.X/wq \.Y/ D 0 for i > 0; (Anatoly Fomenko 310) - pCqDi or w D w 1 \(we already remarked in (Anatoly Fomenko 310) - marked in the end of Sect\. 19\.5\.C that w is invertible in the cohomology ring\)\. (Anatoly Fomenko 310) - let Y X be the set of points where this map is not a submersion \(the rank of the differential is less than q\)\. (Anatoly Fomenko 310) - EXERCISE 25\. If an n-dimensional manifold X possesses an immersion into RnCq , then wi \.X/ D 0 for i > q (Anatoly Fomenko 310) - EXERCISE 26\. If an n-dimensional manifold X possesses an embedding into RnCq , then wi \.X/ D 0 for i q\. \( (Anatoly Fomenko 310) - kC1 EXERCISE 27\. Prove that if 2k n < 2kC1 , then RPn has no immersion in R2 2 2kC1 1 and no embedding in R \. (Anatoly Fomenko 310) - Remark 1\. Thus, if n D 2k , the n-dimensional manifold RPn cannot be embedded into R2n 1 \. This is a very rare phenomenon\. The classical Whitney theorem asserts that an n-dimensional manifold \(with a positive n\) can always be embedded into R 2n \(this result should not be confused with an earlier theorem of Whitney stating that any smooth map of an n-dimensional manifold into any manifold of dimension 2n C 1 can be smoothly approximated by smooth embeddings\); embeddings into (Anatoly Fomenko 310) - R 2n 1 are almost always possible (Anatoly Fomenko 311) - For a nonexistence of such an embedding, it is necessary and sufficient that n is a power of 2, and there exists a one-dimensional cohomology class with coefficients in Z2 whose nth power is not zero \(these conditions imply the nonorientability\)\. (Anatoly Fomenko 311) - Remark 2\. Further information concerning embeddability of \(real and complex\) projective spaces into Euclidean spaces can be obtained with the help of K-theory (Anatoly Fomenko 311) - EXERCISE 29\. Find Stiefel–Whitney numbers of classical surfaces\. (Anatoly Fomenko 311) - For example, two-dimensional manifolds have two Stiefel– Whitney numbers: w11 ŒX and w2 ŒX\. (Anatoly Fomenko 311) - he values of the cohomology classes of the form wi1 \.X/ : : : wir \.X/ with i 1 : : : ir D n on the fundamental class of closed n-dimensional manifold \(they are residues modulo 2\) are called Stiefel–Whitney numbers of the manifold X; (Anatoly Fomenko 311) - A classical theorem in the topology of a manifold asserts any connected closed two-dimensional manifold is a classical surface (Anatoly Fomenko 311) - The reader will see that for any classical surface X; w11 ŒX D w2 ŒX\. (Anatoly Fomenko 311) - Theorem\. If a closed manifold is a boundary of a compact manifold, then all its Stiefel–Whitney numbers are zeroes\. (Anatoly Fomenko 311) - EXERCISE 30\. Prove that if n C 1 is not a power of 2, then neither RPn nor CPn is a boundary of a compact manifold\. (Anatoly Fomenko 312) - The most striking fact, however, is that this necessary condition is also sufficient for a closed manifold to be a boundary of a compact manifold (Anatoly Fomenko 312) - \(Corollary: Every closed three-dimensional manifold is the boundary of some compact four-dimensional manifold; this is a classical theorem of Rokhlin\.\) (Anatoly Fomenko 312) - If X is a closed oriented manifold of dimension 4m, then the value of the class p j 1 \.X/ : : : pjr \.X/; j1 C C jr D m on the fundamental homology class of X is called a Pontryagin number and is denoted as pj1 :::jr ŒX\. (Anatoly Fomenko 312) - There also is a Thom theorem which asserts that if all the Pontryagin numbers of a closed orientable manifold are zeroes \(for example, if its dimension is not divisible by 4\), then a union of several copies of X \(taken all with the same orientation\) is a boundary of some compact manifold (Anatoly Fomenko 312) - Moreover, every set of integers fpj1 :::jr j j 1 C C jr D mg becomes, after a multiplication of all the numbers in the set by the same positive integer, the set of Pontryagin numbers of some closed oriented manifold of dimension 4m\. (Anatoly Fomenko 312) - F A useful corollary of the Thom theorem \(and the fact that if Y D X1 X2 is the disjoint union of two closed oriented 4m-dimensional manifolds, then p j 1 :::jr ŒY D pj1 :::jr ŒX1 C pj1 :::jr ŒX2 (Anatoly Fomenko 313) - EXERCISE 31\. Suppose that for every closed oriented n-dimensional manifold there is assigned an integer \.X/ with the following properties: tie F \(1\) If X is a boundary of a compact oriented manifold, then \.X/ D 0; \(2\) \.X1 X2 / D \.X1 / C \.X2 /\. Prove that X \.X/ D aj1 :::jr pj1 :::jr ŒX; j 1 CCj r Dn=4 where aj1 :::jr are some rational numbers not depending on X\. In particular, \.X/ D 0 if n is not divisible by 4\. (Anatoly Fomenko 313) - Denote by \.X/ the signature of the intersection index form in the 2mdimensional homology of a 4m-dimensional closed oriented manifold X\. The theorem in Sect\. 17\.10 shows that satisfies condition \(1\); condition \(2\) for the signature is obvious\. Hence, the signature is a rational linear combination of Pontryagin number (Anatoly Fomenko 313) - D \.1 x2 /2mC1 (Anatoly Fomenko 313) - where x 2 H 2 \.CP2m / D Z (Anatoly Fomenko 313) - \.CP < 2m 2m / D i 0; if i > (Anatoly Fomenko 314) - ! (Anatoly Fomenko 314) - In particular, the Pontryagin number p1 ŒX of every closed orientable fourdimensional manifold X is divisible by (Anatoly Fomenko 314) - \(Hence, 7p2 ŒX p11 ŒX is divisible by 45, and if the first Pontryagin class of a closed orientable eight-dimensional manifold is zero, then its signature is divisible by 7\. (Anatoly Fomenko 314) - \) The formulas \(\), \(\) form the beginning of an infinite chain of formulas relating the signature to the Pontryagin numbers\. The work of explicitly writing these formulas was done in the 1950s by F\. Hirzebruch\. He calculated the Pontryagin numbers of manifolds of the form CP2m1 CP2mk \(which, essentially, we have done\) and, using the fact that the signatures of all these manifolds are equal to 1, he found the coefficients of the Pontryagin numbers in the formulas for signatures\. (Anatoly Fomenko 314) - As we know, the Euler class of a manifold can be expressed through the Betti numbers of this manifold\. (Anatoly Fomenko 315) - d\. It turns out that although the Stiefel–Whitney classe (Anatoly Fomenko 315) - still they are homotopy invariant; (Anatoly Fomenko 315) - For Pontryagin classes, however, the homotopy invariance fails \(the only homotopy invariant nonzero polynomial in Pontryagin classes is the signature\)\. (Anatoly Fomenko 315) - (Anatoly Fomenko 315) - \. In the 1960s, S\. Novikov proved the difficult theorem of topological invariance of rational Pontryagin classes \(a homeomorphism between two smooth closed orientable manifolds takes Pontryagin classes into Pontryagin classes modulo elements of finite order; these elements of finite order may be nonzero—there are examples\)\. (Anatoly Fomenko 315) - A decade before that, V\. Rokhlin, A\. Schwarz, and R\. Thom proved this statement for homeomorphisms, establishing a correspondence between some smooth triangulations of two smooth manifolds \(see Rokhlin and Schwarz [72], Thom [85]\)\. This result leads naturally to the problem of “combinatorial calculation of Pontryagin classes,” that is, their calculation via triangulation \(compare to Exercise 29\)\. At present, this problem has been solved only for the first Pontryagin class (Anatoly Fomenko 315) - Definition 2\. A filtration of C is a family of subgroups ps if p < q, then Fp C Fq C (Anatoly Fomenko 316) - iltered group C r \.Fr C=Fr 1 C/\. (Anatoly Fomenko 319) - n this way, we (Anatoly Fomenko 319) - If an Abelian group C possesses a differential d and a filtration fFp Cg, then we (Anatoly Fomenko 319) - Lobtain a filtration fFp Hg of H\. If C has a differential d and a grading C D r2Z Cr , then we usually assume that d is homogeneous of some degree u 2 Z (Anatoly Fomenko 319) - Z, which means that for all r; d\.Cr / CrCu \. (Anatoly Fomenko 319) - Notice that the homology group of a differential graded group with homogeneous differential \(of some degree u\) has a natural grading (Anatoly Fomenko 319) - L If C has a filtration, fFp Cg, and a grading, g, are called compatible if for every p; Fp C L LC D r D r \.Fp C \ (Anatoly Fomenko 319) - L r Cr , C \ Cr /\. (Anatoly Fomenko 319) - Here is the simplest \(?\) example\. Let C be a free Abelian group with two generators: a and b \(so C D Za ˚ Zb\)\. Consider the filtration 0 D F 1 C F0 C F1 C D C with F0 C D Z\.a C b/ and the grading C D C0 ˚ C1 with C0 D Za; C1 D Zb (Anatoly Fomenko 319) - For a better understanding, we can notice that the filtration and the grading are compatible if every Fp C is generated by “homogeneous elements,” that is, by elements belonging to the groups Cr \. (Anatoly Fomenko 320) - Since d\.Fp C/ Fp C, the differential d induces a differential dp0 W Fp C=Fp 1 C ! Fp C=Fp 1 C (Anatoly Fomenko 320) - d the direct sum of all dp0 becomes a homogeneous differential of degree 0, d0 W Gr C ! Gr C\. 0 Question: Are H\.Gr C; d 0 / and Gr H\.C; d/ the same? Answer: not, in general (Anatoly Fomenko 320) - his shows that the group H\.Gr C/ should be bigger than Gr H\.C/\. (Anatoly Fomenko 320) - This is what the spectral sequence exists for: a gradual, “monotonic” transition from H\.Gr C/ to Gr H\.C/\. (Anatoly Fomenko 320) - m (Anatoly Fomenko 323) - Remark 2\. Spectral sequences were first introduced in 1945 by J\. Leray in the context of the sheaf theory \(see Leray [53]\)\. (Anatoly Fomenko 323) - The significance of spectral sequences for algebraic topology was demonstrated in 1951 by J\.-P\. Serre in his doctoral dissertation [75]\. With the appearance of homological algebra \(the term was used as the title of the famous book by Cartan and Eilenberg [29]\), spectral sequences became the main technical tool in this area\. (Anatoly Fomenko 323) - Let X be a topological space, and let " ˛1 ˛2 ˛3 " ˛1 ˛2 ˛3 0 G G0 G1 G2 : : : be an exact sequence of Abelian groups \(we assume it infinite to the right, but it desirable—that Gn D 0 for n big enough\) (Anatoly Fomenko 329) - Exercise 7 shows that the spectral sequence II E connects homology groups with coefficients in G with homology with coefficients in Gp \. In this sense, it is similar to the coefficient homology sequence\. Actually, if Gp D 0 for all p 2, this spectral sequence is algebraically equivalent to the coefficient homology sequence (Anatoly Fomenko 329) - Let g be a Lie algebra over the field C, and let M be a g-module \(that is a complex vector space endowed by a linear map W g ! End M such that Œg; h D \.g/ ı \.h/ \.h/ ı \.g/; (Anatoly Fomenko 330) - The \(co\)homology of the complex fCn \.gI M/; dg is called the cohomology of the Lie algebra g with the coefficients in M and is denoted as H n \.gI M/ (Anatoly Fomenko 330) - with the differential; thus, there arises a spectral sequence with the limit term Gr H n \.gI M/; this is the Hochschild–Serre spectral sequence\. (Anatoly Fomenko 330) - EXERCISE 10\. \(1\) Prove that the spaces F p CpCq \.gI M/ form a filtration compatible (Anatoly Fomenko 330) - 21\.3 A New Understanding of the Homology Sequence of a Pair (Anatoly Fomenko 334) - Let \.X:A/ be a topological pair\. It can be regarded as a “two-term filtration,” (Anatoly Fomenko 334) - which is the same as the homological sequence of the pair \.X; A/\. (Anatoly Fomenko 335) - Let D \.E; B; F; p/ be a locally trivial fibration with a CW base B with skeletons B p \. Consider a filtration fFp Eg of the space E with Fp E D p 1 \.Bp / (Anatoly Fomenko 335) - Let sW I ! B be a path joining points b0 ; b1 2 B\. As explained in Sect\. 9\.6 \(for Serre fibrations, but locally trivial fibrations are Serre fibrations\), this path determines a \(homotopically uniquely defined\) homotopy equivalence p Š \.b 0 / ! p Š 1 \.b1 /, and hence isomorphisms Hn \.p 1 \.b0 /I G/ Hn \.p 1 \.b1 /I G/\. This isomorphism may depend on the path s \(although it stays the same if the path s is replaced by a homotopic path\)\. Actually, the fibration is called homologically simple if this isomorphism does not depend on the path for any n and G (Anatoly Fomenko 336) - For example, if the base B is simply connected, then the fibration is homologically simple\. (Anatoly Fomenko 336) - Corollary\. \.E/ D \.B/ \.F/\. (Anatoly Fomenko 337) - \. \(1\) A usual way to prove that a spectral sequence of a fibration \(and, actually, any spectral sequence\) is degenerate is to show 2 2 that every element of Epq is represented by a genuine cycle of E, not just by a chain whose boundary has a filtration not exceeding p 2\. (Anatoly Fomenko 338) - \(2\) Exercise 2 shows that of fibered spaces with a given base and fiber, the direct product has “the biggest homology” (Anatoly Fomenko 338) - Let X be a topological space\. A local system of groups \(D an ensemble of groups D a locally trivial sheaf\) over X is a function which assigns to every point x 2 X a group G x and to every path sW I ! X joining x0 with x1 an isomorphism s W Gx0 ! Gx1 which depends only on the homotopy class of the path s and possesses the property ss0 D s0 ı s (Anatoly Fomenko 339) - Choose a base point x0 in the base X of a local system fGx ; s g\. Then every loop with the beginning at x0 determines an automorphism of the group Gx0 , and in this way there arises a group action of the group 1 \.X; x0 / in Gx0 \. (Anatoly Fomenko 339) - A singular n-dimensional ns chain of X with coefficients in G is defined as a \(finite\) linear combination P i gi fi , P where fi W n ! X is a singular simplex and gi 2 Gfi \.cn / \. The group of all such chains is denoted as Cn \.XI G/, and the boundary operator @ D @n W Cn \.XI G/ ! Cn 1 \.XI G/ is defined by the formula n X @\.gf / D \.1/i f ısn;i \.g/i f : iD0 The homology arising is denoted as Hn \.XI G/; the cohomology H n \.XI G/ is defined in a similar way (Anatoly Fomenko 340) - EXERCISE 8\. Let X D RPn ; G be an Abelian group and let TW G ! G be an automorphism with T 2 D id\. Let the generator of the group 1 \.RPn / act in G as T\. Denote by G the local system arising\. Prove that the homomorphism @W Cr \.RPn I G/ ! Cr 1 \.RPn I G/; 0 < r n k k G G acts like id CT for r even and like id T for r odd\. Compute the homology H r \.RPn I G/ in the general case and in the case G D Z; T D id\. (Anatoly Fomenko 341) - C: Main Theorem for Nonsimple Fibrations Theorem\. E 2 pq D Hp \.BI fHq \.p 1 \.x//g/; where fHp \.p 1 //g is the local system described in Example \.2/ following the definition of a local system \(see the beginning of Sect\. 22\.2\.A\) (Anatoly Fomenko 341) - A: Homology of the Special Unitary Group SU\.n/ (Anatoly Fomenko 342) - Since we know the \(integral\) homology of the base and the fiber, we can display a 2 full diagram of the E2 -term of this spectral sequence \(Fig\. 81\)\. (Anatoly Fomenko 342) - Theorem\. Let X be a topological space \(with a base point\), and let the space X be \.n 1/-connected, that is, 0 \.X/ D 1 \.X/ D : : : D n 1 \.X/ D 0: Then H r \.X/ Š Hr 1 \.X/ for r 2n 2; and a similar isomorphism holds for homology and cohomology with arbitrary coefficients\. (Anatoly Fomenko 344) - We will prove in the next lecture that this isomorphism and this epimorphism are actually induced by a certain continuous map, namely, by the map \.X/W †X ! X acting by the formula X \.s; t/ D s\.t/ \(where s 2 X and t 2 I\): The isomorphism and the epimorphism can be described as † 1 \.X / H † \.X / r 1 \.X/ ! Hr \.†X/ ! Hr \.X/: (Anatoly Fomenko 346) - In Sect\. 10\.1, we proved Freudenthal’s theorem, which states that the suspension homomorphism r \.Sn / ! rC1 \.SnC1 is an isomorphism for r 2n 2 and an epimorphism for r D 2n 1\. I (Anatoly Fomenko 346) - Theorem\. Let X be an \.n 1/-connected CW complex\. Then the suspension homomorphism †W r \.X/ ! rC1 \.†X/ is an isomorphism if r 2n 2 and is an epimorphism if r D 2n 1\. (Anatoly Fomenko 346) - Remark 2\. The last results may serve as one more illustration of the Eckmann–Hilton duality described in Lecture 4\. The operations and † are dual to each other\. The spaces X and †X have equal \(co\)homology groups \(with a dimension shift by 1\); their homotopy groups are the same in “stable dimensions,” that is, in dimensions less than twice the connectivity of X\. On the other hand, X and X have the same homotopy groups \(again, with a dimension shift\), while their \(co\)homology groups are the same in stable dimensions\. (Anatoly Fomenko 347) - Mark one obvious but important property of homomorphisms of spectral sequences: If for some r, the homomorphism Er ! 0Er belonging to a homomorphism between spectral sequences is an isomorphism, then so are all homomorphisms Es ! 0Es with s > r \(including s D 1\)\. Moreover, if two homomorphisms between two spectral sequences coincide on Er for some r, then they coincide on Es for all s > r\. (Anatoly Fomenko 348) - m, the spectral sequence of a fibration does not d (Anatoly Fomenko 349) - Corollary\. Starting from the E2 -term, the spectral sequence of a fibration does not depend on the CW structure of the base\. (Anatoly Fomenko 349) - There arise a monomorphism and an epimorphism E 1 2 p0 ! Ep0 and 2 1 E0q ! E0q : (Anatoly Fomenko 350) - Proposition\. These homomorphisms coincide with the homology homomorphisms induced by the inclusion F ! E and the projection E ! B\. (Anatoly Fomenko 350) - Consider differentials d m m0 0 m W Em0 m ! E0;m 1 and d m0;m 1 W Em0;m 1 ! Emm0 (Anatoly Fomenko 351) - m As we noticed before, Em0 and Em0; m0;m 1 m 1 are subgroups of Hm \.EI G/ and H \.EI G/, while m m0 E0;m 1 and Em are quotients of groups Hm 1 \.FI G/ and H m \.EI G/\. Hence, our differentials have the form (Anatoly Fomenko 351) - If A; B are Abelian groups, then a homomorphism of a subgroup of A into the quotient of B is called a partial multivalued homomorphism of A into B\. For partial multivalued homomorphisms A ! B we will sometimes use the h h notation A Ü B\. (Anatoly Fomenko 352) - \. Notice that partial multivalued homomorphisms always have f “inverses”: (Anatoly Fomenko 352) - These homomorphisms are called, respectively, homological and cohomological transgression \(see Fig\. 88\)\. Elements of the domain of transgression are called transgressive \. As far as we know, this term is used only in the cohomology case\. Theorem\. Homological and cohomological transgressions coincide, respectively, with the following compositions: (Anatoly Fomenko 352) - A: Gysin’s Sequence (Anatoly Fomenko 353) - Let \.E; B; Sn; p/ be a homologically simple fibration with a spherical fiber \(the condition of homological simplicity is equivalent to the condition of orientability: (Anatoly Fomenko 353) - p d ` p d p d ` p d : : : ! HmC1 \.B/ ! Hm n \.B/ ! Hm \.E/ ! Hm \.B/ ! : : : which is called the \(homological\) Gysin sequence\. (Anatoly Fomenko 354) - These five-term fragments may be merged into one infinite exact sequence, (Anatoly Fomenko 354) - B: Wang’s Sequence Let D \.E; Sn ; F; p/ be a fibration with a spherical base\. \(If n 2, this fibration is automatically homologically simple; if n D 1, then we need to assume that the fibration is homologically simple, but in this case, the construction presented here requires some clarification\.\) The E2 -term of the homological spectral sequence consists of two identical columns, zeroth and nth; each contains homology of (Anatoly Fomenko 354) - Precisely as in Gysin’s case, we get a short exact sequence 0 ! Coker n dn;m nC1 ;m nC1 ! Hm \.E/ ! Ker n dn;m n ;m n !0 and then develop it into a long exact sequence r : : : ! r ! Hm nC1 d \.F/ ! d ! Hm i \.F/ ! i ! Hm r \.E/ ! r d ! Hm n \.F/ ! :::: This is the homological Wang sequence (Anatoly Fomenko 354) - Fig\. 91 The spectral sequence which implies the Serre exact sequence (Anatoly Fomenko 356) - to the homotopy sequence of the same fibration, but unlike the homotopy sequence, it is finite \(exists only in the “stable” dimensions\)\. (Anatoly Fomenko 356) - This exact sequence is called the Serre exact sequence \. It has a strong resemblance to the homotopy sequence of the same fibration, but unlike the homotopy sequence, (Anatoly Fomenko 356) - formed by homotopy and Serre sequences and Hurewicz homomorphisms, is commutative\. (Anatoly Fomenko 357) - EXERCISE 5\. Prove that for n 1 m 2n 2, the diagram (Anatoly Fomenko 357) - Proposition\. Transgression Hm \.XI G/ Ü Hm 1 \.XI G/ is, for every m, a partial multivalued homomorphism inverse to the homomorphism † \.X / H † \.X / m 1 \.XI G/ ! Hm \.†XI G/ ! Hm \.XI G/: (Anatoly Fomenko 357) - map \.X; A/ ! \.X=A; pt/\. Now, we can state that the factorization theorem for homotopy groups holds in “stable dimensions”; that is, the following holds\. Proposition\. Suppose that for a CW pair \.X; A/, the homotopy groups r \.X/; r \.A/ are trivial for r < n\. Then the homomorphism q \.X; A/ ! q \.X=A/ is an isomorphism for q < 2n 2 and is an epimorphism for q D 2n 2\. (Anatoly Fomenko 358) - Hence, our map †F ! X=A induces, in stable dimensions, a homology isomorphism, and, according to Whitehead’s theorem, it also induces, in the same dimensions, a homotopy group isomorphism (Anatoly Fomenko 360) - In H n \.FI /, there is the fundamental class c\.F/ (Anatoly Fomenko 360) - or cF is the characteristic class of the path fibration X X EX ! X: (Anatoly Fomenko 360) - EXERCISE 6\. Prove that the image of cF with respect to the \(cohomological\) transgression equals the characteristic class C of the fibration \. (Anatoly Fomenko 360) - [Hint: We need to prove that the fundamental class and the characteristic class have the same image under the homomorphism (Anatoly Fomenko 360) - These images are equal, actually, to the first [\.n C 1/-dimensional] obstruction to extending the map idW F ! F to a map E ! F\.] (Anatoly Fomenko 360) - 0 0 The multiplication is bihomogeneous; that is, if ˛ 2 Erpq and ˇ 2 Erp q , then ˛ˇ 2 ErpCp 0 0 ;qCq \. \(2\) The differentials satisfy the product rule: If ˛ 2 Erpq ; ˇ 2 Erp 0 0 q , then d rpCp;qCq 0 \.˛ˇ/ D \.drpq ˛/ˇ C \.1/pCq ˛\.drp 0 0 q ˇ/: (Anatoly Fomenko 361) - The multiplication in E2 coincides with the multiplication in the cohomology of B with the coefficients in the cohomology ring of F\. (Anatoly Fomenko 361) - called multiplicative if be a filtration of the Abelian group A\. This filtration is L \.F p A/\.F q A/ F pCq A for all p and q\. The group Gr A D \.F p A=F pC1 A/ adjoint to A with respect to the multiplicative filtration as above has a natural structure of a graded ring: If ˛ 2 F p A=F pC1 A and ˇ 2 F q A=F qC1 A and a 2 F p A; b 2 F q A are representative of ˛ and ˇ, then ab 2 F pCq A represents some element of F pCq A=F pCqC1 A, and we take this element for ˛ˇ (Anatoly Fomenko 361) - The multiplication in Gr A is far less rich than the multiplication in A\. For example, if the multiplication in Gr A is trivial \(the product of any two elements is zero\), then for A this means only that \.F p A/\.F q A/ F pCqC1 A (Anatoly Fomenko 361) - On the contrary, any statement showing a nontriviality of the multiplication in Gr A implies, as a rule, a similar statement for A\. (Anatoly Fomenko 361) - EXERCISE 1\. Prove that if Gr A has no zero divisors, then A has no zero divis (Anatoly Fomenko 363) - divisors, then A has no zero divisors\. (Anatoly Fomenko 363) - Proposition\. Let A be a ring or an algebra over some field possessing a finite \.F nC1 A D 0/ multiplicative filtration, and let fxi g be a system of \(multiplicative\) generators of Gr A with xi 2 F pi A=F pi C1 A\. Further, lete xi e x i be a representative of xi in F pi A A\. Then fe xi g is a system of generators of A\. (Anatoly Fomenko 363) - Note that property \(5\) shows that the multiplication in the E2 -term is associative pq p0 q0 pq p0 q0 and skew-commutative [the latter means that if ˛ 2 E2 and ˇ 2 E2 , then ˇ˛ D \.1/\.pCq/\.p 0 0 Cq / ˛ˇ]\. (Anatoly Fomenko 365) - Theorem\. There is a multiplicative isomorphism H \.SU\.n/I Z/ Š H \.S3 S5 : : : S2n 1 I Z/: Remark 1\. A more common way to describe the preceding result is to say that H \.SU\.n/I Z/ is an exterior algebra \(over Z\) with n 1 generators of dimensions 3; 5; : : : ; 2n 1 (Anatoly Fomenko 366) - EXERCISE 3\. Prove that there is a multiplicative isomorphism H \.CV\.n; k/I Z/ Š H \.S2\.n k/C1 S2\.n k/C3 : : : S2n 1 I Z/ (Anatoly Fomenko 367) - A: Symplectic Groups EXERCISE 4\. Prove that there is a multiplicative isomorphism H \.Sp\.n/I Z/ Š H \.S3 S7 : : : S4n 1 I Z/: (Anatoly Fomenko 367) - Fig\. 93 n 2 S 1 Cohomological spectral sequence of the fibration V\.n; 2/ ! Sn (Anatoly Fomenko 368) - Theorem\. If K D Q, R, or C, then 8 3 8 < H H \.SO\.n/I K/ D H 3 \.S S7 : : : S4m 1 I K/; if n D 2 \.\.S3 S7 : : : S4m 5 /S2m 1 I K/; < if n D 2m: (Anatoly Fomenko 368) - The map SO\.n/ ! V\.n; 2/, which assigns to an orthogonal matrix the 2-frame formed by its first two rows, is a fibration with the fiber SO\.n 2/ [one can notice that V\.n; 2/ D SO\.n/=SO\.n 2/; (Anatoly Fomenko 368) - Lemma\. If n is odd, then H \.V\.n; 2/I K/ D H \.S2n 3 I K/: (Anatoly Fomenko 368) - Theorem\. There is an additive isomorphism H \.SO\.n/I Z2 / Š H \.S1 S2 S3 : : : Sn 1 I Z2 /: (Anatoly Fomenko 369) - Moreover, there exist elements xn;i D xi 2 H i \.SO\.n/I Z2 /; i D 1; 2; : : : ; n 1; with the following properties: \(1\) \(2\) \(3\) The monomials xi1 xi2 : : : xis ; 1 i1 < i2 < : : : < is n 1; form an additive basis in H \.SO\.n/I Z2 /: The cohomology homomorphism induced by the inclusion map SO\.n 1/ ! SO\.n/ takes xn;i with i < n 1 into xn 1;i : x n;n 1 is the image of the generator of the group H n 1 \.Sn 1 I Z2 / with respect to the homomorphism induced by the projection SO\.n/ ! Sn 1 \. (Anatoly Fomenko 370) - Remark 1\. The theorem shows that H \.SO\.2m C 1/I Z2 / 6Š H \.Sp\.2m C 1/I Z2 / although, according to Sect\. 24\.4\.A, H \.SO\.2m C 1/I K/ Š H \.Sp\.2m C 1/I K/ for K D Q; R; C\. (Anatoly Fomenko 371) - The ring H \.SO\.n/I Z 2 / is generated by yi 2 H 2i 1 \.SO\.n/I Z2 /; i D 1; 2; : : : ; Œn=2; k with the defining system of relations \(besides the commutativity relations\) y2i i D 0, where 2ki is such a power of 2 that n i 2ki < 2n \(see Kac [50]\)\. The reader can find some further information on the cohomology of Lie groups in the article by Fuchs [39]\. (Anatoly Fomenko 371) - Theorem\. Z; if m 0 is divisible by n 1; H m \.Sn I Z/ D 0; if m is not divisible by n 1: 1/ Moreover, there exist generators xk 2 H k\.n \.Sn I Z/ such that xk x` D ˛k;` xkC` , where (Anatoly Fomenko 371) - Let us now construct a nontrivial fibration with the base and the fiber homeomorphic to S2 \. (Anatoly Fomenko 373) - We get a four-dimensional manifold which is known to us as the connected sum E D CP2 #CP2 of two copies of CP2 (Anatoly Fomenko 373) - We see that the cohomology ring of E is not isomorphic to that of S2 S2 \(z is a square\), (Anatoly Fomenko 374) - Remark\. If we replace in our construction C by R, then X will become a Möbius band, and E will turn into a Klein bottle\. So our E can be regarded as a “complex Klein bottle\.” Also, we can replace C by H\. This will give us an interesting nontrivial S 4 S4 fibration E ! S4 \. (Anatoly Fomenko 374) - EXERCISE 4\. Let the homology of a connected closed orientable manifold X be known\. Find the homology of the manifold of nonzero tangent vectors to X\. (Anatoly Fomenko 374) - EXERCISE 5\. Find the \(already known to you\) cohomology ring of CPn using the S 1 S1 spectral sequence of the Hopf fibration S2nC1 ! CPn \. (Anatoly Fomenko 374) - EXERCISE 6\. Find the ring of rational cohomology of SU\.n/=SO\.n/\. (Anatoly Fomenko 374) - EXERCISE 7\. Find the ring of rational cohomology of SO\.2n/=SU\.n/, at least for some small n \(the first interesting case is n D 3\)\. (Anatoly Fomenko 374) - Q: Have you ever had the experience where you found a problem to be impossible to solve, and then after putting it aside for some time, an idea suddenly occurred leading to the solution? A: Yes, of course this happens quite often\. For instance, when I was working on homotopy groups \(1950\), I convinced myself that, for a given space X, there should exist a fiber space E, with base X, which is contractible; such a space would indeed allow me \(using Leray’s methods\) to do lots of computations on homotopy groups and Eilenberg–MacLane cohomology\. But how to find it? It took me several weeks \(a very long time, at the age I was then1 \. \. \. \) to realize that the space of “paths” on X had all the necessary properties—if only I dared call it a “fiber space,” which I did\. This was the starting point of the loop space method in algebraic topology; many results followed quickly\. (Anatoly Fomenko 375) - We begin with a quotation from an interview which Jean-Pierre Serre gave in Singapore in February 1985 \(published in Mathematical Intelligencer, 1986, 8, 8–13\)\. (Anatoly Fomenko 375) - uppose that we want to calculate the homotopy groups of some space X\. Consider the sequence of spaces, X; X; X; X; : : :\. Imagine that there is a way of calculating the homology of the loop space Y, provided that the homology of Y is known\. For this, we can use the spectral sequence of the Y Y path fibration EY ! Y \. But the first nontrivial homology group of k X is, by Hurewicz’s theorem, the same as its first nontrivial homotopy group, say, r \.k X/, which is, in turn, rCk \.X/\. (Anatoly Fomenko 375) - We will not directly use this method here\. We prefer another method, also belonging to Serre: the method of killing spaces\. \(One can say that, roughly speaking, these two Serre’s methods are closely related to each other and lead to similar results\.\) (Anatoly Fomenko 375) - If n \.X/ D is the first nontrivial homotopy group of X, then XjnC1 is related to X by two fibrations: Xj nC1 K\.;n 1/ XjnC1 X ! K\.; n/; K\.;n 1/ XjnC1 ! X: (Anatoly Fomenko 376) - The first of them was constructed in Sect\. 11\.9: Its projection belongs to the homotopy class of maps corresponding to the fundamental class FX 2 H n \.XI /\. The projection of the second fibration is homotopic to the inclusion XjnC1 ! X of the fiber of the first fibration; in other words, it is the fibration over X induced by K\.;n 1/ K\.;n 1/ the path fibration EK\.; n/ !K\.; n/ with respect to the projection of the first fibration\. In still other words, both fibrations may be described as parts of the following commutative diagram of four fibrations: (Anatoly Fomenko 376) - We usually will use the second of these fibrations\. In principle, if we know the cohomology of K\.; n 1/, then we can at least try to find the cohomology of Xj nC1 , and hence the first nontrivial homotopy group of XjnC1 , which is the second nontrivial homotopy group of X\. And so on\. (Anatoly Fomenko 376) - 25\.2 First Application: A Computation of nC1 \.Sn / (Anatoly Fomenko 376) - At the moment, we know almost nothing of the cohomology of the Eilenberg– MacLane spaces\. Still we know that K\.Z; 2/ D CP1 , and this makes it possible to find H \.S3 j4 I Z/\. (Anatoly Fomenko 376) - and hence 4 \.S3 j4 / D Z2 \) 4 \.S3 / D Z2 \) nC1 \.Sn / D Z2 for n 3\. The reader who was able to get through the exercises in Sect\. 10\.5 already knows this fact\. But even for that reader the easiness of the proof may be strong evidence of a big advantage of the method of spectral sequences\. At the same time, we can see a great importance of the “Eilenberg–MacLane cohomology” H \.K\.; n/I / (Anatoly Fomenko 377) - The computation of the Eilenberg–MacLane cohomology turned out to be hard work\. Still, the problem was fully solved \(for finitely generated Abelian \) in the 1950s, mostly by the French topologists A\. Borel, H\. Cartan, and J\.-P\. Serre\. The easiest part of this work was the computation of the rational cohomology ring H \.K\.; n/I Q/\. T (Anatoly Fomenko 378) - Theorem\. If is a finitely generated \(finite\) Abelian group, then for every finitely generated Abelian group G and every n > 0 and q > 0, the group H q \.K\.; n/I G/ is finitely generated \(finite\)\. (Anatoly Fomenko 378) - For n D 1, the statement is true: We know the spaces K\.Z; 1/ D S1 ; K\.Z2 ; 1/ D RP 1 ; K\.Zm ; 1/ D L1 m \(the infinite-dimensional lens space\) for m > 2 and their cohomology: (Anatoly Fomenko 378) - We say that a class C of Abelian groups is given if every Abelian group belongs or does not belong to C, and \(1\) isomorphic groups belong or do not belong to C simultaneously; \(2\) if a group belongs to C, then all its subgroups and all its quotients belong to C; \(3\) if a subgroup H of a group G and the quotient G=H belong to C, then G belongs to C\. (Anatoly Fomenko 379) - Examples: finitely generated groups; finite groups; periodic groups; finite p-groups\. (Anatoly Fomenko 379) - EXERCISE 1\. Prove that if C is a class of finitely generated Abelian groups, 2 C, and G is a finitely generated Abelian group, then H q \.K\.; n/I G/ 2 C for all n > 0; q > 0\. (Anatoly Fomenko 379) - Theorem\. Let X be a simply connected topological space such that all homology groups Hq \.X/; q > 0 are finitely generated \(finite\)\. Then all the homotopy groups q \.X/ are finitely generated \(finite\)\. (Anatoly Fomenko 380) - Remark\. The requirement of X being simply connected is not unnecessary: For example, the homology groups of S1 \_ S2 are all finitely generated, while the group 2 \.S1 \_ S2 / is not finitely generated (Anatoly Fomenko 380) - F EXERCISE 2\. Let E ! B be a homologically simple fibration\. Prove that if the homology groups of positive dimensions of two of the three spaces E; B; and F are finitely generated \(finite\), then the same is true for the homology groups of the third space\. (Anatoly Fomenko 380) - Corollary\. The homotopy groups m \.S3 / are finite for all m 4\. (Anatoly Fomenko 380) - Proof\. It was shown in Sect\. 25\.2 that all the groups Hm \.S3 j4 /; m > 0, are finite\. Hence, by the theorem, the groups m \.S3 j4 / are all finite, and m \.S3 j4 / D m \.S3 / for all m 4\. (Anatoly Fomenko 380) - EXERCISE 3\. Prove that if all homology groups of some positive dimension of a simply connected \(or homotopically simple\) space belong to a class C of finitely generated Abelian groups, then all the homotopy groups of this space also belong to the class C\. (Anatoly Fomenko 380) - EXERCISE 4\. Prove the following “Hurewicz C-theorem”: If for some simply connected space X, all homotopy groups m \.X/ with m < n belong to a class C \(as in Exercise 3\), then the groups Hm \.X/ with m < n also belong to C and the Hurewicz homomorphism n \.X/ ! Hn \.X/ is a C-isomorphism \(that is, its kernel and cokernel belong to C\)\. Prove also the inverse statement where homology and homotopy groups are swapped\. (Anatoly Fomenko 382) - Remark\. It follows from the Hurewicz C-theorem that for every prime p the order of the group m \.S3 / with 4 m < 2p is not divisible by p (Anatoly Fomenko 382) - p\. To prove that, we apply the Hurewicz C-theorem to the space S 3 j4 and the class C of finite Abelian groups of the order not divisible by p\. (Anatoly Fomenko 382) - Let be a finitely generated Abelian group\. Then Š Z ˚ : : : ˚ Z ˚ , where is a finite Abelian group\. Accordingly, K\.; n/ D K\.Z; n/ : : : K\.Z; n/ K\.; n/, and, by Künneth’s formula, H \.K\.; n/I Q/ D H \.K\.Z; n/I Q/ ˝ : : : ˝ H \.K\.Z; n/I Q/ ˝ H \.K\.; n/I Q/: (Anatoly Fomenko 382) - By the theorem in Sect\. 26\.1\.A, H \.K\.; n/; Q/ D H \.ptI Q/ \(if the integral homology groups of a topological spaces are finite, then its rational cohomology is trivial\)\. Hence, we can remove the last factor in the formula for H \.K\.; n/I Q/ and all we need to compute is H \.K\.Z; n/I Q/\. Theorem\. ƒ Q \.x/; dim x D n; if n is odd; H \.K\.Z; n/I Q/ D QŒx; dim x D n; if n is even: (Anatoly Fomenko 382) - Let us explain the notation\. If K is a field, then ƒK \.x1 ; : : : ; xm / denotes the exterior algebra with generators x1 ; : : : ; xm , that is, the algebra with these generators and relations xi xj D xj xi and x2i D 0 (Anatoly Fomenko 382) - \. The dimension of this algebra is 2m , and an \(additive\) basis formed by monomials xi1 : : : xis with 1 i1 < : : : < is m\. (Anatoly Fomenko 382) - Corollary\. If rank D r, then ƒQ \.x1 ; : : : ; xr /; dim xi D n; if n is odd; H \.K\.; n/I Q/ D QŒx 1 ; : : : ; xr ; dim xi D n; if n is even: (Anatoly Fomenko 384) - 26\.3 Ranks of the Homotopy Groups of Spheres Theorem\. n 1; if q D n; or n is even and q D 2n 1; n rank q \.S / D 0 in all other cases: (Anatoly Fomenko 384) - Remark\. We already know that n \.Sn / D Z \(Sect\. 10\.2\) and that 4n 1 \.S2n / contains an element of an infinite order: the Whitehead square of the generator of 2n \.S 2n / \(see Sect\. 16\.5\)\. Now we are going to show that 4n 1 \.S2n / D Z ˚ a finite group; (Anatoly Fomenko 385) - and that all the other groups q \.Sn / with q ¤ n are finite\. (Anatoly Fomenko 385) - We say that the rational cohomology of a space X forms a free skew-commutative algebra with generators xs 2 H ms \.XI Q/ if these generators are not tied by any relations besides the relations of the skew commutativity: xs xt D \.1/ms mt xt xs \. In other words, H \.XI Q/ D ƒQ \.odd dimensional xs / ˝ QŒeven dimensional xs : (Anatoly Fomenko 386) - Actually, it is true that the rational cohomology of an arbitrary H-space, in particular, of an arbitrary loop space, forms a free skew-commutative algebra (Anatoly Fomenko 386) - Theorem \(Cartan–Serre\)\. Let X be a simply connected space with finitely generated homology groups\. Suppose that the rational cohomology of X is a free skew-commutative algebra, H \.XI Q/ D ƒQ \.x1 ; : : : ; xm / ˝ QŒy1 ; : : : ; y` ; (Anatoly Fomenko 386) - where the xs are odd-dimensional rational cohomology classes of X and the yt are even-dimensional rational cohomology classes of X\. Then the rank of q \.X/ equals the number of q-dimensional elements among x1 ; : : : ; xm ; y1 ; : : : ; y` \. (Anatoly Fomenko 386) - In other words, for a simply connected space X with a free skew-commutative rational cohomology algebra, there is a dimension-preserving bijection between free additive generators of homotopy groups and free multiplicative generators of a rational cohomology algebra\. (Anatoly Fomenko 386) - The Cartan–Serre theorem implies some statements already known to us \(for example, the theorem of ranks of homotopy groups of odd-dimensional spheres\) and also some new statements, such as the following one\. Corollary\. Z ˚ a finite group; if q D 3; 5; 7; : : : ; 2n C 1; q \.SU\.n// D a finite group for all other qI Z ˚ a finite group; if q D 3; 7; 11; : : : ; 4m 1; q \.SO\.2m C 1// D a finite group for all other qI Z ˚ a finite group; if q D 3; 7; 11; : : : ; 4m 5; 2m 1; q \.SO\.2m// D a finite group for all other qI Z ˚ a finite group; if q D 3; 7; 11; : : : ; 4n 1; q \.Sp\.n// D a finite group for all other q: (Anatoly Fomenko 387) - Remark\. The exactness of the homotopy sequences of the fibrations SO\.n 1/ SO\.n/ ! Sn 1 SU\.n 1/ ; SU\.n/ ! SU\.n 1/ ! S2n 1 Sp\.n 1/ ; Sp\.n/ ! Sp\.n 1/ ! S4n 1 ; and the triviality of the homotopy groups q \.Sm / with q < m imply the isomorphisms q \.SO\.n 1// Š q \.SO\.n// for q < n 2; q \.SU\.n 1// Š q \.SU\.n// for q < 2n 2; q \.Sp\.n 1// Š q \.Sp\.n// for q < 4n 2 (Anatoly Fomenko 387) - Thus, for n large, the groups q \.SO\.n//; q \.SU\.n//; and q \.Sp\.n// do not depend on n; these stable groups are denoted as q \.SO/; q \.SU/; and q these groups of groups of infinite matrices, SO \.Sp/S[actually, D n SO\.n/; ese stable groups are oups S are homotopy groups S SU D n SU\.n/; Sp D n Sp\.n/]\. Unlike the homotopy groups q \.SO\.n//; S q \.SU\.n//; and q \.Sp\.n// \(which are known only partially\), the stable homotopy groups are fully known: 8 Z for odd q; Z for odd q; 8 < Z; if q 1 mod 4; Z for odd q; q \.U/ D q \.SO/ D Z2 ; if q 0; 1 mod 8; 0 for even qI < q \.SO/ D 0 for all other qI (Anatoly Fomenko 388) - q \.Sp/ D qC4 \.SO/ \(compare with the last corollary\)\. This fact was proved in the late 1950s by Raoul Bott, who applied methods of variational calculus in the large \(this result is broadly known under the name of the Bott periodicity \)\. A presentation of Bott’s proof is contained in the last chapter of Milnor’s book [57]\. (Anatoly Fomenko 388) - We have already noticed that the theorem on the ranks of homotopy groups of an odd-dimensional sphere is a corollary of the Cartan–Serre theorem\. However, the case of an even-dimensional sphere is not covered by Cartan–Serre, since the cohomology ring of an even-dimensional sphere has an undesirable relation: The square of an even-dimensional generator is equal to 0 (Anatoly Fomenko 388) - A reader may create an impression that as soon as we know the rational cohomology ring of, say, a simply connected CW complex, we can find the ranks of its homotopy groups\. This impression is wrong, however, as the following example shows\. Take the bouquet of two two-dimensional spheres, S 2 \_ S2 (Anatoly Fomenko 388) - EXERCISE 5\. Using methods of this lecture, prove that rank 4 \.S2 \_ S2 / D 2 \(the linearly independent elements of this group are Œ Œs1 ; s2 ; s1 and Œ Œs1 ; s2 ; s2 ; where s 1 and s2 are the generators of the groups 2 of the two spheres and the square brackets denote the Whitehead product; Sect\. 10\.5\)\. (Anatoly Fomenko 388) - A question arises: Is it possible to express in terms of rational cohomology information about a \(say, simply connected, CW\) space sufficient for determining the ranks of homotopy groups? The answer is yes; however, for this purpose, we need to consider, in addition to the classical cohomological multiplication, a sequence of “higher multiplications,” the Massey products \( (Anatoly Fomenko 389) - e (Anatoly Fomenko 389) - Let a 2 ˛; b 2 ˇ; c 2 be cocycles (Anatoly Fomenko 389) - Then there are cochains e; f such that ıe D ab and ıf D bc\. Consider the cochain ec\.1/p af \. It is a cocycle: ı\.ec \.1/p af D \.ab/c \.1/p \.1/p a\.bc/ D 0; and its cohomology class is called the Massey product of ˛; ˇ; and is denoted as h˛; ˇ; i\. (Anatoly Fomenko 389) - product is a partial multivalued operation\. (Anatoly Fomenko 389) - Thus, the Massey (Anatoly Fomenko 389) - EXERCISE 6\. Compute the Massey products in the cohomology of the space \.S 2 \_ S2 / [ D5 considered above\. (Anatoly Fomenko 389) - EXERCISE 7\. Compute the Massey products in the cohomology of the complement to the Borromean rings \(compare Exercise 52 in Lecture 17\)\. (Anatoly Fomenko 389) - It is possible to extend this construction to still “higher” Massey products\. For example, for a quadruple of cohomology classes ˛; ˇ; ; ı such that ˛ˇD0; ˇ D0; ı D 0; h˛; ˇ; i 3 0; hˇ; ; ıi 3 0, one can construct a cohomology class h˛; ˇ; ; ıi with a still bigger indeterminacy than triple Massey product (Anatoly Fomenko 389) - A general algebraic description of Massey products is contained in the article by Fuchs and Weldon [41]\. (Anatoly Fomenko 389) - It turns out that the rational cohomology of a simply connected space, given with the multiplicative structure and the whole infinite sequence of Massey products, determines the ranks of homotopy groups \(with Whitehead products and a sequence of Whitehead–Massey products in homotopy groups tensored with Q\)\. (Anatoly Fomenko 389) - A different approach \(related to, but not based on Massey products\) to rational homotopy types was developed by Dennis Sullivan as the minimal model theory\. The main ingredient of this theory consists in assigning to a space a rational cochain complex furnished with an associative skew-commutative multiplication whose cohomology is the rational cohomology of the space \(in Sullivan’s original construction it is the complex of piecewise rational differential forms\) (Anatoly Fomenko 389) - A space is called formal if there exists a multiplicative homomorphism from the cohomology (Anatoly Fomenko 389) - ring into a minimal model which assigns to every cohomology class a cocycle representing this class (Anatoly Fomenko 390) - For formal spaces, a rational cohomology algebra determines ranks of homotopy groups \(by a procedure similar to that of this lecture\)\. Examples of formal spaces include loop spaces, spheres \(both oddand even-dimensional\), Kähler manifolds, and symmetric spaces\. See the article by Deligne, Griffiths, Morgan, and Sullivan [33]; there is also a famous book by Félix, Halperin, and Thomas [36] (Anatoly Fomenko 390) - Theorem\. If p0 is a prime different from p, then H \.K\.Zp0 s I Zp / Š H \.ptI Zp /: (Anatoly Fomenko 390) - Corollary\. If is a finite Abelian group whose order is not divisible by p, then every H q \.K\.; n/I Z/; q > 0 is a finite Abelian group whose order is not divisible by p\. (Anatoly Fomenko 390) - 27\.2 A Partial Computation of H \.K\.Zp ; n/I Z/ (Anatoly Fomenko 390) - S1 Fig\. 104 The spectral sequence of the fibration L1 p ! CP1 The cohomology of the space K\.Zp ; 1/ D L 1 p is known to (Anatoly Fomenko 391) - he space K\.Zp ; 1/ D L 1 p is known to us: Zp for an even q > 0; H q \.K\.Zp ; 1/I Z/ D 0 for an odd q: (Anatoly Fomenko 391) - It is easy to see that, in the positive dimensions, there is a multiplicative isomorphism H \.K\.Zp ; 1/I Z/ Š Zp Œx; x 2 H 2 \.K\.Zp ; 1/I Z/: (Anatoly Fomenko 391) - Theorem\. For an even n 2, the ring H \.K\.Zp ; n/I Z/ in dimensions n C 4p 2 is generated additively by the following elements of order p: nC1 ; nC2p 1 ; nC2p ; nC4p 3 ; nC4p 2 ; (Anatoly Fomenko 394) - nd also nC1 nC2p 1 and nC1 nC2p if their dimensions belong to our range\. For an odd n 3, this ring, in the same dimensions, is additively generated by the following elements of order p: knC1 \.k 1/I knC1 nC2p 1 \.k 0/I knC1 nC2p \.k 0/I nC4p 3 ; nC4p 2 : (Anatoly Fomenko 394) - 27\.3 A Partial Computation of the Cohomology K\.Z; n/ mod p (Anatoly Fomenko 395) - Theorem\. In positive dimensions n C 4p 3, the ring H \.K\.Z; n/I Zp / is isomorphic to the ring H \.K\.Zp ; n 1/I Z/\. (Anatoly Fomenko 395) - Indeed, there is a multiplicative isomorphism H (Anatoly Fomenko 395) - Remark\. We have to warn the reader that the multiplicative arguments alone do not provide a full computation of the rings H \.K\.Zp ; n/I Z/ and H \.K\.Z; n/I Zp /\. Rather soon, there arises a situation when there are several different possibilities for a differential not contradicting the product rule (Anatoly Fomenko 395) - In particular, for n 3, the rings H H all dimension (Anatoly Fomenko 395) - ements of the order p 2 in H \.K\.Z; n/I Z/ first appear in dimension n C 2p2 1\. We leave the proof of this fact to the reader \(who is recommended to begin by considering the integral K\.Z;2/ K\.Z;2/ cohomological spectral sequence of the fibration ! K\.Z; 3/ \)\. (Anatoly Fomenko 395) - 27\.4 A Partial Computation of the p-Component of the Homotopy Groups of Spheres Theorem\. If n 3, then nC2p 3 \.Sn / Zp ; nC4p 5 \.Sn / 4p 5 n n ı nCq \.S / \.Zp ˚ Zp / qD0 Zp , and the quotient does not contain elements of order p\. (Anatoly Fomenko 395) - We cannot say more, since we do not know the action of the differential labeled by the question mark\. Actually, this differential is not trivial, and this is sufficient for the proof of the theorem\. But the proof of this requires using Steenrod powers, which will appear in Chap\. 4 (Anatoly Fomenko 397) - Corollary\. The order of the stable homotopy group kS \(see Sect\. 10\.4\) is not divisible by p for 1 k < 2p 3 and 2p 3 < k < 4p 5 and is divisible by p but not by p2 for k D 2p 3 and 4p 5\. (Anatoly Fomenko 398) - Proposition\. All nontrivial p-components of kS with k 2p\.p1/1 are contained in the following formula: 8 < Zp ; if k D 2i\.p 1/ 1; p-component of kS D Zp ; if k D 2p\.p 1/ 2; < Zp2 ; if k D 2p\.p 1/ 1: (Anatoly Fomenko 398) - The proof of this result, as well as of some further results, is contained in the classical book by Toda [86]\. We can also refer the reader to the later book by Mosher and Tangora [63]\. (Anatoly Fomenko 398) - Let n; q be two integers and let ; G be two Abelian groups\. We say that a cohomology operation ' of type \.n; q; ; G/ is given if for every CW complex X a map 'X W H n \.XI / ! H q \.XI G/ is given and is natural with respect to X, in the sense that the diagram 'X 'X H n \.XI / ! H q \.XI G/ XI / ! H q \.X x x x x ? ? ? ? ? ? 'Y 'Y H n \.YI / ! H q \.YI G/ is commutative for every continuous map f W X ! Y\. (Anatoly Fomenko 399) - Notice that the map 'X is not assumed to be a homomorphism of the group H n \.XI / into the group H q \.XI G/\. (Anatoly Fomenko 399) - Since G is an Abelian group, the set of cohomology operations of the type \.n; q; ; G/ is an Abelian group\. We denote this group by O\.n; q; ; G/\. (Anatoly Fomenko 399) - 28\.2 Classification Theorem\. There is a \(canonical\) isomorphism O\.n; q; ; G/ Š H q \.K\.; n/I G/: (Anatoly Fomenko 399) - Proof of Theorem\. Recall that there is a natural bijection H n \.XI /$ \.X; K\.; n// between n-dimensional cohomology of X with coefficients in and homotopy classes of continuous maps X ! K\.; n/ (Anatoly Fomenko 400) - The construction of this bijection uses a remarkable cohomology class F 2 H n \.K\.; n/I /, which has several equivalent definitions; for example, the universal coefficients formula isomorphism H n \.K\.; n/I / D Hom\.Hn \.K\.; n//; / D Hom\.; / connects F with id \. Namely, for an ˛ 2 H n \.XI /, there exists a homotopically unique continuous map h˛ W X ! K\.; n/ such that h ˛ \.F / D ˛\. Our bijection is ˛ $ Œh˛ \. (Anatoly Fomenko 400) - Let us assign to a cohomology operation ' 2 O\.n; q; ; G/ the cohomology class 'K\.;n/ \.F / 2 H q \.K\.; n/I G/\. (Anatoly Fomenko 400) - \. The formula ' 7! 'K\.;n/ \.F / determines a map of O\.n; q; ; G/ into H q \.K\.; n/I G/, which is, obviously, a group homomorphism\. We will show that it is actually an isomorphism\. (Anatoly Fomenko 400) - Corollary\. A nonzero cohomology operation does not lower the dimension [that is, if 0 ¤ ' 2 O\.n; q; ; G/, then q n\.] (Anatoly Fomenko 400) - Here is an example of a cohomology operation which is not a homomorphism\. Let be a ring, and let n be an even number\. The raising to a square H n \.XI / ! H 2n \.XI / is a cohomology operation which, in general, is not a homomorphism \(it certainly is a homomorphism if D Z2 \)\. (Anatoly Fomenko 400) - We know the cohomology H \.K\.; n/I Q/ for all finitely generated groups \. (Anatoly Fomenko 401) - In particular, if is finite, then H \.K\.; n/I Q/ D 0, so there are no nontrivial cohomology operations from cohomology with finite coefficients into rational cohomology\. (Anatoly Fomenko 401) - \. If n is odd, then H \.K\.Z; n/I Q/ D H \.Sn I Q/, so every nontrivial cohomology operation from odd-dimensional integral cohomology into rational cohomology preserves dimension and is a fixed rational number times the homomorphism induced by the inclusion Z ! Q\. If n is even, there also appear operations assigning to an integral cohomology class its powers \(rationalized\)\. (Anatoly Fomenko 401) - Consider now cohomology operations raising the dimension by one\. Since K\.Z; n/ has no \.n C 1/-dimensional cells, there are no nontrivial cohomology operations from integral cohomology into any other cohomology raising dimension by one\. T (Anatoly Fomenko 401) - The space K\.Zp ; n/ has precisely one \.n C 1/-dimensional cell \(obtained by attaching DnC1 to Sn by a spheroid Sn ! Sn of degree p\)\. Thus, H nC1 \.K\.Zp ; n/I Zp / is, at most, Zp , so there is, up to a multiplication by a constant, at most one cohomology operation of the type \.n; nC1; Zp ; Zp /\. But we know such an operation: the Bockstein homomorphism \(see Sect\. 15\.3\)\. Thus, the Bockstein homomorphism \(up to a multiplication by a constant\) is the only cohomology operation in mod p cohomology raising dimension by one\. (Anatoly Fomenko 401) - EXERCISE 1\. Prove that any cohomology operation of type \.n; n C 1; C; A/ \(where C and A are Abelian groups\) is the connecting homomorphism in the coefficient exact sequence corresponding to some short exact sequence 0 ! A ! B ! C ! 0\. (Anatoly Fomenko 401) - 28\.4 Stable Cohomology Operations A: Definition A stable cohomology operation of type \.r; ; G/ is a sequence of cohomology operations 'n 2 O\.n; n C r; ; G/; n D 1; 2; 3; : : : ; such that for every X and every n the diagram H n \(X; π\) ⏐ ⏐ ⏐ ⏐Σ H n+1 \(ΣX; π\) \(ϕ n \) X −−−−−→ \(ϕ n+1 \) ΣX −−−−−→ H n+r \(X; G\) ⏐ ⏐ ⏐ ⏐Σ H n+r+1 \(ΣX; G\) \(where † is the suspension isomorphism\) is commutative\. (Anatoly Fomenko 401) - The set \(Abelian group\) of all stable cohomology operations of type \.r; ; G/ is denoted as Stab O\.r; ; G/\. For example, the different versions of the Bockstein homomorphism \(Sect\. 15\.3\) are stable cohomology operations of types \.1; Zp ; Z/ and \.1; Zp ; Zp /\. (Anatoly Fomenko 403) - Theorem\. A stable cohomology operation is compatible with a cohomology sequence of a CW pair; that is, for every CW pair \.X; A/ and every stable cohomology operation ' D f'n g 2 Stab O\.r; ; G/ the diagram i ∗ δ∗ j ∗ i∗ δ∗ j∗ H n \(X; π\) H n \(A; π\) H n+1 \(X, A; π\) H n+1 \(X; π\) \.\.\. \.\.\. \.\.\.\.\.\.\.\. \.\.\.\.\. \.\.\. \.\.\. \.\.\. \.\.\. \.\.\. \.\.\. \.\.\. \. \. \. \.\. \.\.\. \.\.\. \.\.\. \. \.\.\. \.\.\. \(ϕn \)X \(ϕn \)A \.\.\. \.\.\. \(ϕn+1 \)\(X,A\) \(ϕn+1 \)X i ∗ δ∗ j ∗ i∗ δ∗ j∗ n+r \(X; G\) H n+r \(A; G\) H n+r+1 \(X, A; G\) H n+r+1 H n+r \(X; G\) H n+r \(A; G\) H n+r+1 \(X, A; G\) H n+r+1 \(X; G\) is commutative\. (Anatoly Fomenko 403) - Let us point out an important corollary of the last theorem\. Let \.E; B; F; p/ be a fibration with a simply connected base, and let be a stable cohomology operation 0q of the type \.r; ; G/\. Suppose that the cohomology class ˛ 2 H q \.FI / D E2 is transgressive (Anatoly Fomenko 403) - e, that is, d2 ˛ D d3 ˛ D D dq ˛ D 0\. (Anatoly Fomenko 403) - Then the class q \.˛/ 2 H qCr \.FI G/ D E 0;qCr 2 is also transgressiv (Anatoly Fomenko 403) - Moreover, if \.˛/ D dqC1 ˛ (Anatoly Fomenko 403) - contains ˇ 2 H qC1 \.BI /, then \.'q \.˛// contains 'qC1 \.ˇ/ 2 H qCrC1 \.BI G/\. (Anatoly Fomenko 403) - qC1;0 2 E2 (Anatoly Fomenko 403) - \(Less precisely, but more sonorously, this can be expressed by the words cohomology operations commute with transgression\.\) (Anatoly Fomenko 404) - All this follows from the presentation of transgression as a composition ı \.p / 1 H ı q \.FI / ! H qC1 \.E; FI / Ü H qC1 \.B; ptI / D H qC1 \.BI / stab (Anatoly Fomenko 404) - A stable cohomology operation of the type \.r; ; G/ is a sequence 'n 2 O\.n; r C n; ; G/ D H rCn \.K\.; n/I G/: (Anatoly Fomenko 404) - Thus, in the language of algebra, Stab O\.r; ; G/ D lim \.H rCn \.K\.; n/I G/; fn /; (Anatoly Fomenko 404) - the inverse \(projective\) limit of the sequence f 4 f3 f4 f3 : : : ! H rC3 \.K\.; 3/I G/ ! H rC2 \.K\.; 2/I G/ f 2 f2 ! H rC1 \.K\.; 1/I G/ (Anatoly Fomenko 404) - [recall that the inverse limit lim \.Gn ; fn / of a sequence f 4 f3 f2 f4 f3 f2 : : : ! G3 ! G2 ! G1 of groups and homomorphisms is the group of sequences fgn 2 Gn g such that f n \.gn / D gn 1 for all n] (Anatoly Fomenko 404) - The condition of commuting with † means that fn \.'n / D 'n 1 , where fn is the composition i n H rCn \.K\.; n/I G/ ! n ! H rCn \.†K\.; n 1/I G/ † 1 † ! H rCn 1 \.K\.; n 1/I G/; (Anatoly Fomenko 404) - We know that if n > r C 1, then the homomorphism fn W H rCn \.K\.; n/I G/ ! H rCn 1 \.K\.; n 1/I G/ (Anatoly Fomenko 404) - s an isomorphism \(it is inverse to the transgression in the spectral sequence of K\.;n 1/ K\.;n 1/ the fibration ! K\.; n/ \)\. (Anatoly Fomenko 405) - We will say that a cohomology group \(or a cohomology class\) of K\.; n/ has a stable dimension if its dimension < 2n; otherwise, we say that it has a nonstable dimension\. (Anatoly Fomenko 405) - E: The Algebra of Stable Operations \(the Steenrod Algebra\) (Anatoly Fomenko 405) - The multiplicative structure in H \.K\.; n/I G/ \(where G is a ring\) does not determine any multiplication for stable operations \(a product of two cohomology classes of stable dimensions never has a stable dimension (Anatoly Fomenko 405) - on\)\. However, the composition L product turns r 0 Stab O\.r; G; G/ into a graded ring, whether G is a ring or not\. This ring of stable operations is a unitary associative ring, in general, not commutative\. Notice that the cohomology H \.XI G/ becomes a \(graded\) module over this ring, and all induced homomorphisms f \.YI G/ ! H \.XI G/ become module homomorphis (Anatoly Fomenko 405) - If G is a field, then the ring of stable cohomology operations becomes an algebra\. If G D Zp , then this algebra is called the Steenrod algebra and is denoted as Ap (Anatoly Fomenko 405) - 29\.1 An Introduction We begin with a construction of some important elements of the Steenrod algebra A 2 which are called Steenrod squares\. Steenrod squares Sqi are stable cohomology operations which are additive homomorphisms Sq i W H n \.XI Z2 / ! H nCi \.XI Z2 /: (Anatoly Fomenko 405) - They are defined for all i 0 and possess the following properties \(in addition to the properties required by the definition of stable cohomology operations\): \(1\) 8 Sq 8 < 0; if i > dim ˛; i ˛ D ˛ 2 ; if i D dim ˛; < ˛; if i D 0: (Anatoly Fomenko 405) - \(2\) The following Cartan’s multiplication formula holds: Sq X i \.˛ ˇ/ D Sqp ˛ Sqq ˇ: (Anatoly Fomenko 407) - Add to this that since Sq1 is not zero \(Sq1 ˛ D ˛ 2 for dim ˛ D 1\), it must be the Bockstein homomorphism \(see Sect\. 28\.3\)\. (Anatoly Fomenko 407) - Denote the fundamental class in H n \.K\.Z2 ; n/I Z2 / by en (Anatoly Fomenko 407) - To finish the proof, notice that for any space X, the diagonal map W †X ! †X †X is homotopic to a map taking †X into †X \_†X: The homotopy ht W †X ! †X †X is defined by the formula ht \.x/ D \.'t \.x/; t \.x//, where 't ; t W †X ! †X are two homotopies of the identity map \.'0 D 0 D id†X / such that '1 \.C1 X/ D x 0 ; 1 \.C2 X/ D x0 , where C1 X; C2 X are two cones composing †X and x0 2 †X is an \(arbitrarily chosen\) base point (Anatoly Fomenko 408) - On the other hand, the cross-product of any two cohomology classes of X of positive dimensions has zero restriction to †X \_ †X \(at least in the case when X is a CW complex\)\. Hence, in the cohomology of †X, the cup-product of any two classes of positive dimensions is zero\. (Anatoly Fomenko 408) - The Steenrod algebra A D A2 is the algebra of all stable cohomology operations over the field Z2 where the multiplication is defined as the composition\. (Anatoly Fomenko 411) - It turns out that the algebra A is multiplicatively generated by Steenrod squares Sq i ; that is, every stable cohomology operation in Z2 -cohomology is a linear combination of iterations of Steenrod squares\. (Anatoly Fomenko 411) - However, the Steenrod squares do not form a free system of multiplicative generators: There are relations between them\. In particular, an additive basis of the algebra A is not formed by all iterations of Steenrod squares, but only by iterations Sq I D Sqi1 Sqi2 : : : Sqik for which the sequence I D fi1 ; i2 ; : : : ; ik g satisfies the conditions i 1 2i2 ; i2 2i3 ; : : : ; ik 1 2ik (Anatoly Fomenko 411) - \(such sequences are usually called admissible sequences, and the corresponding iterations of Steenrod squares are called admissible iterations \)\. (Anatoly Fomenko 411) - The multiplicative structure of A is determined by the so-called Adem relations \(see Adem [11]\): If a < 2b, then a b ! X bc1 a b Sq Sq D SqaCb c Sqc c a 2c c (Anatoly Fomenko 411) - Theorem\. H \.K\.Z2 ; n/I Z2 / is a polynomial algebra with the generators SqI en for all admissible sequences I D fi1 ; : : : ; ik g with exc I < n\. (Anatoly Fomenko 412) - Theorem\. The Steenrod algebra A2 is additively generated by admissible iterations Sq I \(without restrictions on excesses\)\. (Anatoly Fomenko 416) - Thus, all stable cohomology operations in the cohomology with coefficients in Z2 are sums of iterations of Steenrod squares (Anatoly Fomenko 416) - \. Moreover, we see that there must exist many relations between these iterations, since an arbitrary iteration is equal to a sum of admissible iterations\. (Anatoly Fomenko 416) - Suppose that we are given a certain \(noncommutative\) polynomial in Steenrod squares, P\.Sq 1 ; Sq2 ; : : : /\. How can we prove that it is equal to zero? Theoretically, we need to prove the equality P\.Sq 1 ; Sq2 ; : : : /x D 0 for every cohomology class x 2 H n \.XI Z2 / of every, say, CW complex X\. Visibly, there arise two difficulties, and each of them seems to be a dead end\. First, how do we observe all cohomology classes of all CW complexes? Second, even for an individual x, how do we compute Sq I x for an iteration SqI of Steenrod squares? (Anatoly Fomenko 417) - The second difficulty does not exist in the case when x is a product of onedimensional cohomology classes: Then SqI x can easily be found with Cartan’s formula\. And it turns out that we actually do not need anything else\. Here is an explanation\. (Anatoly Fomenko 417) - RP1 N generators xi 2 H 1 \.PI Z2 / coming from the ith factor Z2 Œx1 ; : : : ; xN with the RP1 \. Let u D uN D x 1 : : : xN 2 H N \.PI Z2 /\. Theorem\. Let ' D P\.Sq1 ; Sq2 ; : : : / D P Ij j Sq be a polynomial in Steenrod P Ij squares, and let N jIj j for all j\. If '\.uN / D 0, then ' D 0 [that is, '\.x/ D 0 for any x 2 H \.XI Z2 / and any X]\. (Anatoly Fomenko 417) - EXERCISE 1\. Prove that if q N, then Sqq \.u/ D u eq \.x1 ; : : : ; xN /, where eq is the qth elementary symmetric polynomial [if q > N, then, certainly, Sqq \.u/ D 0]\. (Anatoly Fomenko 417) - q then the map W Aq D A2 ! H NCq \.PI Z2 /, \.'/ D '\.u/, is a monomorphism\. Let Bq D \.Aq / H NCq \.PI Z2 / D Z 2 Œx1 ; : : : ; xN NCq (Anatoly Fomenko 417) - Let us call a polynomial from Z2 Œx1 ; : : : ; xN special if every monomial in this polynomial has the form x k1 k2 kN 21 x22 : : : x2N (Anatoly Fomenko 417) - Lemma\. Bq is the same as the set of symmetric special polynomials of degree NCq\. (Anatoly Fomenko 418) - The previous theorem paves a road to finding relations in the Steenrod algebra\. Here is the first example: Obviously, Sqn \.u/ D Symm\.x21 : : : xn2 xnC1 : : : xN / and Sq 1 Sqn \.u/ D \.n C 1/ Symm\.x21 : : : x2nC1 xn2 : : : xN /: (Anatoly Fomenko 419) - Thus, Sq nC1 ; if n is even; Sq 1 Sqn D 0; if n is odd: Or, Sq 2 Sq2 \.u/ D Sq2 Symm\.x21 x22 x3 : : : xN / D Symm\.x41 x22 x3 : : : xn / C 6 Symm\.x21 x22 x23 x24 x5 : : : xN /; Sq 3 Sq1 \.u/ D Sq3 Symm\.x21 x2 : : : xN / D Symm\.x41 x22 x3 : : : xn / C 4 Symm\.x21 x22 x23 x24 x5 : : : xN /; (Anatoly Fomenko 420) - which shows that Sq2 Sq2 D Sq3 Sq1 \. (Anatoly Fomenko 420) - We already mentioned in Sect\. 30\.1 that a complete system of relations between the Steenrod squares is formed by the so-called Adem’s relation: If a < 2b, then Œa=2 a b bc1 ! X bc1 a b Sq Sq D SqaCb c Sqc : a 2c (Anatoly Fomenko 420) - To prove this relation, we need only to check that the leftand right-hand sides take equal values on u 2 H N \.PI Z2 / (Anatoly Fomenko 420) - \. Both values can be calculated by means of Cartan’s formu (Anatoly Fomenko 420) - EXERCISE 4 \(Bullet–MacDonald\)\. as one identity, Prove that the Adem relations may be presented P\.s 2 C st/P\.t2 / D P\.t2 C st/P\.s2 /; where P\.u/ D Sqi ui \. (Anatoly Fomenko 421) - EXERCISE 2\. Prove that the Steenrod squares Sq1 ; Sq2 ; Sq4 ; Sq8 ; Sq16 ; : : : form a system of generators of the algebra A2 , and this system is minimal\. In other words, prove that Sqn can be expressed as a polynomial in Sqi with 0 < i < n if and only if n is not a power of 2\. (Anatoly Fomenko 421) - 30\.6 Computing O\.n; q; Z; Z2 / The computation of the cohomology of K\.Z; n/ modulo 2 was done by J\.-P\. Serre simultaneously with the computation of the cohomology of K\.Z2 ; n/ modulo 2\. No wonder: The two computations are essentially the same \(induction with respect to n based on the Borel theorem\)\. (Anatoly Fomenko 421) - based on the Borel theorem\)\. Here is the final result\. Theorem 1\. If n 2, then the ring H \.K\.Z; n/I Z2 / is the ring of polynomials \(with coefficients in Z2 \) of generators SqI en where en 2 H n \.K\.Z; n/I Z2 / is the generator and I D \.i1 ; i2 ; : : : ; ik / is an admissible sequence with exc I < n and ik > 1 [the last inequality is the only difference between the results for K\.Z; n/ and K\.Z2 ; n/]\. (Anatoly Fomenko 421) - yie L Theorem 2\. The vector space q Stab O\.q; Z; Z2 / has a basis consisting of all operations SqI where I D \.i1 ; i2 ; : : : ; ik / is an admissible sequence with ik > 1\. (Anatoly Fomenko 421) - Let p be a prime number\. For p > 2, as well as for p D 2, the only \(up to a factor\) cohomology operation in the cohomology modulo p raising the dimension by 1 is the Bockstein homomorphism ˇ\. However, while for p D 2 the operation ˇ D Sq 1 is just one of the Steenrod squares, for p > 2 this operation plays a very special role\. The operations similar to other Steenrod squares also exist\. Namely, there exists a unique stable cohomology operation Pip 2 Stab O\.2i\.p 1/; Zp ; Zp / such that Pip x D xp for x 2 H 2i \.XI Zp /\. This operation is called the \(pth\) Steenrod power \. (Anatoly Fomenko 422) - It turns out that the Steenrod algebra Ap is multiplicatively generated by the operations ˇp D ˇ and P ip \. (Anatoly Fomenko 422) - A sequence I is called admissible if i 1 pi2 ; i2 pi3 ; : : : ; ik 1 pik : For an admissible I, we refer to StI as to an admissible iteration\. There i (Anatoly Fomenko 422) - is a theorem: Admissible iterations StI form an additive basis in the Zp -algebra Ap \. (Anatoly Fomenko 422) - The relations between the iterations StI are generated by the Adem relations \(see [12]\) (Anatoly Fomenko 422) - We will need some new notations\. Let K be a field\. Denote as ƒ\.m; K/ the graded K-algebra with the basis f1; xg with deg x D m; x2 D 0 \(this algebra is called the exterior algebra with the generator x\)\. Furthermore, denote as P\.m; K/ the K-algebra with the basis f1; x D x\.1/ ; x\.2/ ; : : : g with h \.k/ k defined by the formula x\.k/ x\.`/ D ! deg x D km and the multiplication C ` \.kC`/ x \(this algebra is called the algebra k of modified polynomials \)\. Of these algebras, we will form graded tensor products, which are defined as the usual tensor products with the multiplication acting by the formula \.a ˝ b/ \.c ˝ d/ D \.1/deg b deg c ac ˝ b (Anatoly Fomenko 423) - Fix a prime p and a group … D Z or Zps and set up a definition of a sequence of numbers satisfying the condition \.Cp /\. Let us be given a sequence of integers I D \.i1 ; i2 ; : : : ; ik /\. We say that I satisfies the condition \.Cp / if \(1\) i1 pi2 ; i2 pi3 ; : : : ; ik 2 pik 1 ; ik 1 2\.p 1/; \(2\) ik D 0 if … D Z; \(3\) ik D 0 or 1 if … D Zps ; \(4\) i` 0 or 1 mod 2\.p 1/ for 1 ` k\. Theorem \(Cartan [28]\)\. For n 1 and a prime p > 2, the algebra H \.K\.…; n/I Zp / \(where … D Z or Zps \) is isomorphic to the tensor product of exterior algebras ƒ\.m; Zp / with generators of odd degrees and algebras of usual polynomials with generators of even degrees\. For n 2 and p D 2, the algebra H \.K\.…; n/I Z2 / \(where … D Z or Z2s \) is isomorphic to the tensor product of algebras of usual polynomials\. In both cases, the number of generators of degree n C q is equal to the number of sequences I, with jIj D q satisfying the condition \.Cp / and the condition pi < \.p 1/\.n C q/\. (Anatoly Fomenko 423) - Remark\. This theorem implies all the preceding classification results (Anatoly Fomenko 424) - It turns out that the homology algebra of K\.…; n/ can also be fully described [the multiplication in the homology of K\.…; n/ is induced by the structure of an H-space: K\.…; n/ D K\.…; n C 1/]\. (Anatoly Fomenko 424) - Theorem \(Cartan [28]\)\. For n 1 and a prime p > 2, the algebra H \.K\.…; n/I Zp / \(where … D Z or Zps \) is isomorphic to the tensor product of exterior algebras ƒ\.m; Zp / with generators of odd degrees and algebras of modified polynomials with generators of even degrees\. For n 2 and p D 2, the algebra H \.K\.…; n/I Z2 / \(where … D Z or Z2s \) is isomorphic to the tensor product of algebras of modified polynomials\. In both cases, the number of generators of degree nCq is equal to the number of sequences I with jIj D q satisfying the condition \.Cp /\. (Anatoly Fomenko 424) - Theorem of a Choice of a Basis \(Cartan\)\. Let … D Z and en 2 H n \.K\.Z; n/I Zp / be the generator\. Then, for the generators of the exterior algebras and algebras of usual polynomials in the theorem concerning H \.K\.Z; n/I Zp /, one can take the classes StIp \.en / for all I satisfying the conditions \.Cp / and pi1 < \.p 1/\.n C jIj/\. (Anatoly Fomenko 424) - The general homotopy direction of this book forces us to consider Steenrod squares mainly as a tool for calculating homotopy groups\. Indeed, Steenrod squares are very useful for this: In the next section, we will perform the calculation of the groups nC2 \.Sn /, and in the next chapter we will calculate the stable groups nCq \.Sn / for q 13\. (Anatoly Fomenko 424) - 3\. One should not forget, however, that the homotopy calculations are, in some sense, a side effect of Steenrod’s theo (Anatoly Fomenko 424) - od’s theory\. (Anatoly Fomenko 424) - We already see that H nC2 \.Sn jnC2 I Z2 / D Z2 ; and hence, nC2 \.Sn / D HnC2 \.Sn jnC2 / D \.Z or Z2s / ˚ \.a finite group of odd order/: But the group nC2 \.Sn / is finite \(see Sect\. 26\.3\) and does not contain elements of an odd order \(Sect\. 27\.4\)\. Thus, nC2 \.Sn / D Z2s : (Anatoly Fomenko 426) - Actually, the fact that Sq 1 w ¤ 0, that is, bnC2 \.w/ ¤ 0, already implies that s D 1\. (Anatoly Fomenko 426) - But we will prove it using spectral sequences\. (Anatoly Fomenko 426) - EXERCISE 1\. Compute nC3 \.Sn /\. [Hint: Consider the “underkilling space” E1 with K\.Z 2 ;nC2/ K\.Z2 ;nC2/ a fibration E1 ! Sn jnC3 \. Show that HnC3 \.E1 I Z2 / is again Z2 \. The same K\.Z 2 ;nC2/ K\.Z2 ;nC2/ will be true for the next “underkilling space,” E2 ; E2 ! E1 , and only the third underkilling space, E3 , manages to kill the 2-component of nC3 \.Sn /\. Hence, the 2-component of nC3 \.Sn / is Z8 \. Furthermore, we know from Sect\. 27\.4 that the group nC3 \.Sn / has a 3-component Z3 and has no p-components with p > 3\. Thus, nC3 \.Sn / D Z24 \. \(If you do this exercise, you will give an algebraic proof of the main result of one of the most difficult topological works of the pre-French epoch, that is, the work by Rokhlin [71] in which the group nC3 \.Sn / was calculated by a geometric method\.\)] (Anatoly Fomenko 427) - f we assign to a real vector bundle the cohomology class Sqk wm \./ of its base, then we will get a new characteristic class of real vector bundles with values in Z2 cohomology\. But there is no such thing as “new characteristic classes”: We proved in Sect\. 19\.4\.D that every characteristic class of real vector bundles with values in Z2 cohomology is a polynomial in the Stiefel–Whitney classes\. What is this polynomial in our case? It turns out that the following formula holds: k k mCjk1 ! X mCjk1 k Sq wm D wk j wmCj : j (Anatoly Fomenko 427) - First, we check that the leftand right-hand sides of the equality take equal values on the fibration with the base RPN RPN , where is the \(onedimensional\) Hopf bundle over RPN and N and the number of factors are both sufficiently large\. (Anatoly Fomenko 427) - calculations for products of real projective spaces provide a key tool in the theories of both Stiefel–Whitney classes and Steenrod squares\.\) (Anatoly Fomenko 428) - This shows that if w1 D 0, that is, if the vector bundle is orientable, then Sq1 wm D 1 w mC1 for every even m (Anatoly Fomenko 428) - m\. But Sq1 is the Bockstein homomorphism (Anatoly Fomenko 428) - Let be a real vector bundle of dimension n with the base B and the total space E\. Fix Euclidean structures in all fibers of \(such that the square-of-length function is continuous on the total space of \) and denote by D\./ and S\./ the spaces of unit ball and unit sphere bundles associated with \. The quotient space T\./ D D\./=S\./ is called the Thom space of (Anatoly Fomenko 428) - EXERCISE 2\. Prove that if the bundle is trivial, then T\./ D †n \.B t pt/ \(where † denotes the base point version of suspension; see Sect\. 2\.6\)\. (Anatoly Fomenko 428) - EXERCISE 3\. Prove the equality T\. / D T\./#T\./\. (Anatoly Fomenko 428) - EXERCISE 5\. Give a geometric description of the Thom space of the tautological vector bundle over the Grassmannian\. (Anatoly Fomenko 428) - Thus, for an oriented vector bundle with a CW base, there arise Thom isomorphisms Š tW H Š Š Š (Anatoly Fomenko 429) - where G is an arbitrary coefficient group\. Moreover, if G D Z2 , then the assumption of orientability for is not needed\. (Anatoly Fomenko 429) - group\. Moreover, if G D Z2 , then the assumption of orientability for is not needed\. Thom isomorphisms possess many naturality properties, of which we mention the commutative diagrams (Anatoly Fomenko 429) - which arise for an arbitrary continuous map f W B 0 ! B \(between CW complexes\); T\.f /W T\.f / ! T\./ is the naturally arising map\. (Anatoly Fomenko 430) - Proposition\. Let be an oriented vector bundle with a CW base B, let G be a ring, and let ˛1 2 H q1 \.BI G/; ˛2 2 H q2 \.BI G/\. Then t\.˛1 ^ ˛2 / D t\.˛1 / ^ ˛2 : (Anatoly Fomenko 430) - \(Hint: For a compact B, the vector bundle can be embedded into a vector bundle with a closed oriented smooth base B (Anatoly Fomenko 430) - isomorphism in homology up to an arbitrarily high dimension (Anatoly Fomenko 431) - Corollary\. For an arbitrary ˛ 2 H q \.BI G/, t\.˛/ D t\.1/ ^ ˛: The cohomology class t\.1/ 2 H n \.T\./I G/ is called the Thom class of the bundle \. (Anatoly Fomenko 431) - Remark\. We can generalize the proposition to the case when ˛1 2 H q1 \.BI G1 /; ˛2 2 H q2 and the cup-product ˛1 ^ ˛2 2 H q1 Cq2 \.BI G/ is taken with respect to some pairing G1 G2 ! G (Anatoly Fomenko 431) - Let us do some final remark for the case when is the normal bundle of a smooth submanifold Y of a smooth manifold X\. EXERCISE 8\. Prove that in this case T\./ D X=\.X U/, where U is a tubular neighborhood of Y\. (Anatoly Fomenko 431) - EXERCISE 10\. Let X be an m-dimensional closed orientable manifold embedded in S n , and let U be a tubular neighborhood of X in Sn \. Prove that the diagram (Anatoly Fomenko 431) - \(where t denotes the Thom isomorphism associated with the normal bundle of X in S n \) is commutative\. (Anatoly Fomenko 431) - C: An Sq-definition of the Stiefel–Whitney Classes Theorem\. For an arbitrary vector bundle with a CW base, w m \./ D t 1 Sqm t\.1/ \(the equality holds in Z2 -cohomology\)\. (Anatoly Fomenko 432) - EXERCISE 11\. Prove this theorem\. [Hints: A standard proof of this result \(which can be found, for example, in the book by Milnor and Stasheff [60]\) consists in systematically checking for the classes t 1 Sqm t\.1/ the axioms of Stiefel–Whitney classes \(mentioned in the beginning of Sect\. 19\.5\)\. Another approach is based on the splitting principle\. Since both sides of the equality in the theorem are characteristic classes of real vector bundles with values in the Z2 -cohomology, it is sufficient to prove the equality for products of Hopf bundles\. It is not hard to do this \(using the results of Exercises 3 and 4\)\.] (Anatoly Fomenko 432) - Corollary\. If is the normal bundle of a submanifold Y of X, then i Š \.wm \.// D Sqm iŠ \.1/ D Sqm \.DX 1 ŒY/ \(where i is the inclusion map Y ! X\)\. (Anatoly Fomenko 432) - Theorem \(Wu\)\. Let X be a closed manifold\. Then, for every ˛ 2 H \.XI Z 2 /, h\.Sq/ 1 w\.X/; D\.˛/i D hSq ˛; ŒXi: This “Wu formula” completely determines the Stiefel–Whitney classes of \(the tangent bundle of\) a closed smooth manifold\. Its standard proof can be found in the books by Milnor and Stasheff [60] \(Sect\. 11\) or Spanier [79] \(Sect\. 10 of Chap\. 6\)\. (Anatoly Fomenko 432) - The tangent bundle of X is the same as the normal bundle of \.X/ in X X, where W X ! X X is the diagonal map (Anatoly Fomenko 432) - Corollary 1\. The Stiefel–Whitney classes of a smooth manifold are its homotopy invariants\. \(This result was mentioned in Sect\. 19\.6\.E\.\) (Anatoly Fomenko 433) - Corollary 2 \(The Stiefel Theorem\)\. Every closed orientable three-dimensional manifold is parallelizable\. Proof\. To prove that a closed orientable three-dimensional manifold X is parallelizable, it is sufficient to construct two linearly independent \(at every point\) vector V\.3;2/ V\.3;2/ fields on X, that is, to construct a section of the fibration E ! X associated with the tangent bundle of X (Anatoly Fomenko 433) - EXERCISE 13\. Prove that if X is an n-dimensional manifold, then wm \.X/ D 0 for m > n ˛\.n/ where ˛\.n/ is the number of ones in the binary representation of n and w m \.X/ is the mth Stiefel–Whitney class of the normal bundle of X in an Euclidean space\. \(This result belongs to W\. S\. Massey\. The proof is quite involved but does not use anything not known to the reader\.\) (Anatoly Fomenko 433) - FYI\. The results in Exercises 13 and 14, together with Exercise 26 in Sect\. 19\.6, gave rise to a known Massey conjecture that every closed \(actually, not necessarily closed\) n-dimensional smooth manifold can be immersed into R2n ˛\.n/ \(and, as follows from Exercise 14, the dimension 2n ˛\.n/ cannot be reduced in this statement\)\. The Massey conjecture was proved by Ralph Cohen [31]\. (Anatoly Fomenko 434) - In the early 1930s, H\. Hopf gave a homotopy classification of continuous maps of an n-dimensional CW complex X into the ndimensional sphere\. It turned out that these homotopy classes bijectively correspond to cohomology classes in H n \.XI Z/1 \(see Sect\. 18\.4\); namely, the homotopy class of a map f W X ! Sn is bijectively characterized by the cohomology class f \.s/ 2 H n \.XI Z/, where s 2 H n \.Sn I Z/ is the canonical generator\. (Anatoly Fomenko 434) - Let X be a CW complex\. We pointed out before that the diagonal map W X ! X X is not cellular (Anatoly Fomenko 434) - Steenrod extended this classification to maps X ! Sn , where dim X D n C 1\. (Anatoly Fomenko 434) - But one should not think that the goal of the groundbreaking work of Steenrod was studying any cohomology operations\. The work \(Steenrod [81]\) was devoted to an old homotopy problem\. In the early 1930s, H\. Hopf gave a homotopy classification of continuous maps o (Anatoly Fomenko 434) - Remark 1\. According to Steenrod’s construction, the reason for the existence of Steenrod squares lies in the noncommutativity of multiplication of cellular \(or classical\) cochains (Anatoly Fomenko 435) - As we mentioned briefly at the end of Lecture 26, a \(skew-\) commutative ring of cochains can be constructed when the coefficients are taken in Q\. Rema (Anatoly Fomenko 435) - \. Actually, if the coefficients of a cohomology theory lie in Z or Z2 , then this multiplication cannot be commutative \(otherwise, Steenrod squares could not have existed\)\. A (Anatoly Fomenko 435) - 31\.4 Nonexistence of Spheroids with an Odd Hopf Invariant Theorem\. If n is not a power of 2, the group 2n 1 \.Sn / does not contain elements whose Hopf invariant is odd\. \(See Remark 5 in Sect\. 16\.5\.\) (Anatoly Fomenko 435) - Remark\. This theorem was proved by Hopf in the 1930s\. However, the proof was complicated and did not seem convincing to topologists of that time; even some \(false!\) counterexamples to it were published by some prominent mathematician (Anatoly Fomenko 435) - Proof of Theorem\. Let ˛ 2 2n 1 \.Sn / be an element with an odd Hopf invariant, and let Y D Sn [f D2n , where f W S2n 1 ! Sn is a spheroid of the class ˛\. Then q Z2 ; if q D 0; n; 2n; q H \.YI Z2 / D 0 for all other q; and, by definition of the Hopf invariant, the squaring operation H n \.YI Z2 / ! H 2n \.YI Z2 / is nontrivial\. But this operation is the same as Sqn , and if n is not a power of 2, then Sqn may be presented as a polynomial of Sqi with 0 < i < n \(see Exercise 2 in Sect\. 30\.5\)\. But since H q \.YI Z2 / D 0 for n < q < 2n, any such polynomial is zero on H n \.YI Z2 /\. This contradiction proves the theorem\. (Anatoly Fomenko 436) - Remark\. As we mentioned in Lecture 16, there are no elements with odd Hopf invariant in 2n 1 \.Sn / with n > 8\. (Anatoly Fomenko 436) - \. One of the possible proofs of this fact consists in a construction, which shows that the operations Sq 16 ; Sq32 ; : : : , indecomposable within the class of usual \(“primary”\) cohomology operations, are decomposable within the class of \(“secondary”\) cohomology operation\. (Anatoly Fomenko 436) - Another \(remarkably simple\) proof using K-theory will be presented in Chap\. 6 (Anatoly Fomenko 436) - Let p and q be relatively prime integers, 1 < q < p; p > 2\. Consider the 3 2 the 2i 2piq z t 1 e ransf p ;z or e 2 T of the sphere S C acting by the formula T\.z mation p \. Obviously, T generates a free action of the group Zp in 1 ; z2 / D S3 \. The quotient S3 =Zp is denoted by L\.p; q/ and is called the \(three-dimensional\) lens space\. (Anatoly Fomenko 436) - The question is, for which p; q; p0 ; q0 is the lens space L\.p; q/ homeomorphic, or homotopy equivalent, to the lens space L\.p 0 ; q0 /? (Anatoly Fomenko 436) - Theorem\. The lens spaces L\.p; q/ and L\.p; q0 / are homotopy equivalent if and only if q0 k2 q mod p for some integer k\. (Anatoly Fomenko 436) - The lens space has a natural cell decomposition which is obtained from the following Zp -invariant cell decomposition of S3 with p cells in every dimension 0; 1; 2; 3\. The circle S 1 D f\.z; 0/ j jzj D 1g S3 is divided by the p 2ki points e0k D \.e p ; 0/; k D 0; 1; : : : ; p 1, into the union of p arcs; we denote the \(open\) arc joining e0k with e0kC1 \(throughout this section, we regard subscripts as 0k with e0kC1 2ki residues modulo p\) as e1k \. Next, we put e2k D f\.z1 ; z2 / 2 S3 j z2 2 R>0 e p g and take for e3k the domain between e2k and e2kC1 \. We get the desired cell decomposition of S3 \. \(Some people say that this decomposition reminds them of an orange; if the reader finds this comparison helpful, we appreciate it\.\) The transformation T maps the cells e0k ; e1k ; e2k ; e3k homeomorphically onto e0kC1 ; e1kC1 ; e2kCq ; e3kCq \. Thus, our cell decomposition of S3 gives rise to a cell decomposition of L\.p; q/, with one cell in every dimension, e0 ; e1 ; e2 ; e3 \. All the cells in S3 have natural orientations preserved by T; thus, the cells in L\.p; q/ are also naturally oriented\. (Anatoly Fomenko 437) - For example, the lens spaces L\.5; 1/; L\.5; 2/ are not homotopy equivalent \(since 2 6 ˙k2 mod 5\)\. (Anatoly Fomenko 438) - For example, there is no orientation reversing map \(that is, simply no map of degree 1\) of L\.3; 1/ \(this fact was proved by H\. Kneser long before the theory discussed in this section was created\)\. (Anatoly Fomenko 438) - \. Moreover, elementary number theory states that if p is prime and p 3 mod 4, then every q not divisible by p is congruent either to a square or to a negative square \(for example, 1; 2; 3; 4; 5; 6 12 ; 32 ; 22 ; 22 ; 33 ; 12 mod 7\), (Anatoly Fomenko 438) - Homotopy equivalent lens spaces are not necessarily homeomorphic\. For example, the lens spaces L\.7; 1/; L\.7; 2/ are homotopy equivalent, but not homeomorphic\. This fact can be established with the help of the so-called Reidemeister–Franz torsion\. The reader can find details in the book by Dubrovin, Fomenko, and Novikov [35] \(Sect\. 11 of Chap\. 1, Problems 5–8\), or in the book by de Rham, Maumary, and Kervaire [34]\. (Anatoly Fomenko 438) - If we know the cohomology of the space X, we can find, relatively easily, the “stable part” of the cohomology of the first killing space of X, and then the same for the second, the third, and so on killing spaces (Anatoly Fomenko 439) - s\. Hurewicz’s theorem gives us, every time, the corresponding homotopy group\. This method \(the Serre method\) does not permit us, however, to find the homotopy groups without dealing with other difficulties\. (Anatoly Fomenko 439) - As we demonstrated in the previous lecture, the information about the action of stable cohomology operations may be used for computing stable homotopy groups\. (Anatoly Fomenko 439) - For example, let us imagine that the first nontrivial homotopy group of X has dimension N and equals Z2 \. Consider the spectral sequence of the fibration K\.Z 2 ;N 1/ Xj K\.Z2 ;N 1/ N ! X \. (Anatoly Fomenko 439) - n the upper row, there is the cohomology K\.Z2 ; N 1/ modulo 2, which coincides, in the stable dimensions, with the Steenrod algebra A2 (Anatoly Fomenko 441) - Thus, the cohomology of XjN is fully known to us, but the action of the cohomology operations in this cohomology is known to us only partially (Anatoly Fomenko 441) - Thus, we have no full information on the action of cohomology operations in the cohomology of XjN , and hence we cannot find the cohomology of the next killing space\. (Anatoly Fomenko 441) - The modern topology has no way to overcome this difficulty; the homotopy groups of spheres have not been calculated yet\. (Anatoly Fomenko 441) - t\. But there is a possibility to “expose the difficulty in a pure form,” namely, to collect all the computations related to stable homotopy groups into one spectral sequence\. The initial term of this spectral sequence will be known, and all the difficulties will be concentrated in the computations of the differential\. This is what the Adams spectral sequence is\. (Anatoly Fomenko 441) - Certainly, the role of the Adams spectral sequence is not restricted to exposing difficulties: It also allows us to resolve part of them\. Namely, the Adams spectral sequence may be furnished with an amount of additional structures \(for example, the multiplicative structure\) which do not show themselves at the level of usual killing spaces and which provide rich information on the action of the differentials\. (Anatoly Fomenko 442) - The Serre method of computing homotopy groups consists in killing the cohomology groups ordered by their dimension: First, we kill the nth group, then pass to the \.nC1/st, and so on\. The Adams method also consists in killing cohomology groups, but in a different order\. Take a space X\. For example, let it be \.N 1/-connected, and we want to find the p-components of its homotopy groups from the dimension N to the dimension N C n\. At the first step we kill all the cohomology of X modulo p in these dimensions\. (Anatoly Fomenko 442) - \. One can do this in the following way\. Every additive generator of H NCq \.XI Zp / determines a map X ! K\.Zp ; N C q/\. (Anatoly Fomenko 442) - C q/\. All these generators determine a map of X into the product …i K\.Zp ; N C qi /\. There arises (Anatoly Fomenko 442) - N C qi /\. There arises a fibration The spectral sequence of this fibration looks as follows\. In the upper row, there is a “free Ap -module”; that is, the action of cohomology operations in this row, at least in the dimensions not very much exceeding N, is free: There are no relations which do not follow from the relations in the Steenrod algebra Ap itself\. (Anatoly Fomenko 442) - Fig\. 116 The spectral sequence of the Adams killing (Anatoly Fomenko 442) - There arises a sequence of Adams killing spaces, X\.1/; X\.2/; X\.3/; : : :, with a chain of fibrations : : : X\.3/ ! X\.2/ ! X\.1/ ! X whose fibers are products of K\.Zp ; n/s\. (Anatoly Fomenko 443) - Notice that the killing of all elements of Zp -cohomology of X, as well as all subsequent killing, may be made in a more economical way\. For example, if we kill the element and, say, Pip ¤ 0, we do not need to kill Pip separately: It will be killed automatically simultaneously with \. Speaking more algebraically, we should not kill all elements of some basis of the vector space over Zp , but rather all elements of some system of generators of an Ap -module (Anatoly Fomenko 443) - other words, consider an Ap -module H \.XI Zp /\. Then there exists a free Ap -module F0 with an Ap -epimorphism F0 ! H \.XI Zp /\. [It is easy to understand: F 0 is a free module whose free generators bijectively correspond to the chosen generators of the module H \.XI Zp /\.] The kernel of this epimorphism is again an Ap -module, but not necessarily free\. We construct a free Ap -module F1 and an Ap -epimorphism of it onto our kernel, in other words, an Ap -homomorphism F1 ! F0 whose image is the kernel of the epimorphism F0 ! H \.XI Zp /\. Repeating this construction, we get an exact sequence : : : ! F3 ! F2 ! F1 ! F0 ! H \.XI Zp / ! 0 of Ap -modules and Ap -homomorphisms in which all the modules Fi are free\. Such a sequence is called a free resolution of the module H \.XI Zp /\. (Anatoly Fomenko 443) - odule\. In other w (Anatoly Fomenko 443) - Let us return to geometry\. Our process is, in some sense, converging; that is, the cohomology of the spaces X\.k/ becomes less and less and, again in some sense, completely vanishes at the limit\. However, the Serre killing process was directly related \(via Hurewicz’s theorem\) to the homotopy groups of the space\. Namely, we always killed the cohomology of the smallest dimension, and every killing corresponded to some element of the homotopy group\. It is not so, however, for the Adams killing\. (Anatoly Fomenko 443) - Thus, the Adams method contains more killings than the Serre method\. (Anatoly Fomenko 443) - count the number of generators killed by the Adams method in some dimension N C q, we will get an upper estimate for the p-compone (Anatoly Fomenko 444) - ponent (Anatoly Fomenko 444) - The limit term of the Adams spectral sequence will be adjoint to the p-components of the stable homotopy groups of X \. (Anatoly Fomenko 444) - Let A be an associative algebra \(with the unit\) over a field K, and let it be Z-graded: L AD q Aq D : : : ˚ A 1 ˚ A0 ˚ A1 ˚ : : : ; Ar As ArCs : (Anatoly Fomenko 444) - A left module le L over A \(or an A-module\) is a vector space T over K which is Z-graded, T D q Tq , and is furnished with a bilinear map A T ! T; \.a; t/ 7! at such that \(1\) If a 2 Aq ; t 2 Tr , then at 2 TqCr \. \(2\) b\.at/ D \.ba/t for a; b 2 A; t 2 T\. (Anatoly Fomenko 444) - A subset U of a A-module T is called a system of generators if every t 2 T can be presented as a finite sum a1 u1 C : : : C an un with a1 ; : : : ; an 2 A; u1 ; : : : ; un 2 U\. A system of generators is called homogeneous if every u 2 U belongs to some Tq ; is called homog S in other words, U T ı where T ı D q Tq (Anatoly Fomenko 445) - If a presentation as above \(with all ui S different\) is unique for every \(D for some\) t 2 T, then the system of generators U is called free, or a basis\. A module which possesses a free system of generators is called free (Anatoly Fomenko 445) - S An obvious construction extends an arbitrary graded set U D q Uq to a free A-module T D FU U with a basis U\. (Anatoly Fomenko 445) - A homomorphism \(or module homomorphism or A-homomorphism \) of an A-module T into an A-module T 0 is a linear map f W T ! T 0 such that f \.at/ D af \.t/ for all a 2 A; t 2 T\. A homomorphism f W T ! T 0 is called homogeneous 0 of degree d if f \.Tq / TqCd ; a homogeneous homomorphism of degree 0 is called simply a homogeneous homomorphism\. Kernels and images of homogeneous A-homomorphisms \(of any degree\) are \(in the obvious sense\) A-modules\. (Anatoly Fomenko 445) - Obviously, for every A-module T there is an exact sequence 0 ! KT ! FT ı ! T ! 0; where is the extension of the inclusion map T ı ! T\. (Anatoly Fomenko 445) - In particular, every A-module is an image of a \(homogeneous\) epimorphism of a free module (Anatoly Fomenko 445) - An A-module P is called projective if every diagram of the form M \.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\. \.\.\.\.\.\. \.\.\.\.\.\.\.\.\.\. N\. \.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\. \.\.\.\.\.\. \.\.\.\.\.\.\.\.\. 0 \.\.\.\.\.\.\.\. \.\.\.\.\.\. \.\.\.\.\.\. \.\.\.\.\.\. \.\.\. \.\.\.\.\.\. \.\.\. \.\.\.\.\.\. \.\.\. \. \.\.\.\.\.\. \.\.\. \. \.\.\. \. \.\.\.\.\.\.\. \.\.\. \. P with exact row can be extended to a commutative diagram \.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\. N\. \.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\. M\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\. N\. \.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\.\. 0 \.\.\.\.\. \.\.\.\. \.\.\.\.\. \.\.\.\.\.\.\.\. \.\. \. \.\.\. \. \.\.\.\. \.\. \.\. \. \.\. \.\. \. \.\.\. \. \.\.\.\. \.\.\.\. \.\.\. \.\. \. \.\. \.\. \. \.\.\. \. \.\.\. \.\. \.\. \.\.\. \.\. \.\.\.\.\.\. \. \.\.\. \. \.\.\.\.\.\. \.\.\.\. \.\.\.\.\. \.\.\.\. \. \.\.\.\.\. \.\.\.\. \. \.\.\.\.\. \.\.\.\. \. \. \.\.\.\.\.\. \. \.\. P\. (Anatoly Fomenko 445) - In other words, any homomorphism of P into a quotient module M=R can be “lifted” to M, that is, factored as P ! M ! M=R where the second arrow is the projection of a module onto the quotient module\. (Anatoly Fomenko 447) - Proposition\. An A-module P is projective if and only if it is a direct summand of some free module\. (Anatoly Fomenko 447) - Proof\. First, let us prove that every free module is projective (Anatoly Fomenko 447) - Let T be an arbitrary A-module\. Then there exists an exact sequence : : : ! Pq ! Pq 1 ! : : : ! P1 ! P0 ! T; where Pq are projective modules\. Such a sequence is called a projective resolution of T\. If all the modules Pq are free, then the sequence is called a free resolution\. (Anatoly Fomenko 448) - Here is a construction of a free resolution\. For every module T we have an exact sequence 0 ! KT ! FT ! T ! 0 with the module FT being free\. Put T1 D KT T2 D K T1 ; T3 D KT2 ; : : : : We have exact sequences 0 ! T1 ! FT ! T ! 0; 0 ! T2 ! FT1 ! T1 ! 0; 0 ! T3 ! FT2 ! T2 ! 0; ::::::::::::::::::::: of which we can compose one long exact sequence: (Anatoly Fomenko 448) - the following proposition which is sometimes called the fundamental lemma of homological algebra\. (Anatoly Fomenko 448) - The degree of nonexactness of this sequence is measured by the homology of this complex, which is denoted as TorAn \.M; N/: (Anatoly Fomenko 450) - We get a sequence : : : ! P2 ˝A N ! P1 ˝A N ! P0 ˝A N \.! 0 ! 0 ! : : :/; which, in general, is not exact but still is a complex\. Th (Anatoly Fomenko 450) - 33\.4 Tor and Ext \(Compare with Sect\. 15\.4\.\) Let M be a right A-module and let N be a left A-module\. Fix a projective resolution : : : ! P2 ! P1 ! P0 ! M ! 0 of M and apply to the portion : : : ! P2 ! P1 ! P0 of this resolution the functor ˝ A N 1 with values in graded vector spaces over K\. W (Anatoly Fomenko 450) - It is well known that the two-variable functor HomA \.M; N/ is covariant with respect to N and contravariant with respect to M\. (Anatoly Fomenko 452) - EXERCISE 10\. An alternative way of defining the Ext operation is to use injective resolutions of the module N\. (Anatoly Fomenko 452) - Let X be a topological space\. Our goal is to find its stable homotopy groups qS \.X/ D NCq \.†N X/, where N q \(according to the generalized Freudenthal theorem, Sect\. 23\.3\.C, this group does not depend on N\)\. The main case: X D S 0 \(the twopoint space\); then qS \.X/ D NCq \.SN / D qS \. (Anatoly Fomenko 454) - Since A acts in H H e , we can consider He \.X/ as an A-modul nonnegative \(that is, all components of negative degrees are zero\)\. (Anatoly Fomenko 454) - The modules Bi are not cohomology modules for any topological space, because nonzero cohomology modules are never free (Anatoly Fomenko 454) - e; its grading is Let us construct for this A-module a free resolution (Anatoly Fomenko 454) - [for example, in the cohomology modules there are always relations of the form Pip \.x/ D 0 for i > \.p 1/ dim x]\. The cohomology module with the smallest possible amount of relations is H We want to “approximate” the modules Bi with modules of this kind\. (Anatoly Fomenko 454) - e want to “approximate” the modules Bi with modules of this kind\. Let N be a large number\. The A-module H the grading is shifted by N\. (Anatoly Fomenko 454) - Next, we want to replace the chain of maps between X\.i/ by a filtration\. For this, we have to turn all these maps into embeddings\. Construct the cylinders of all these maps and attach them to each other as shown in Fig\. 119 (Anatoly Fomenko 456) - 34\.3 The Adams Theorem Theorem\. Let X be a CW complex with finite skeletons, and let p be a prime number\. Then there exists a spectral sequence fErs;t ; drs;t W Ers;t ! ErsCr;tCr 1 j r 1; t s 0g, with the following properties\. \(1\) \(2\) There is a canonical isomorphism E 2s;t Š Exts;t A \.H H \.X/I Zp /: Here Zp is regarded as an A-module with the trivial action of L q degree of the generator being 0\. There is a canonical isomorphism 0 Aq with the E s;t rC1 D Ker drs;t = Im drs r;t rC1 : s;t s;t s;t \(3\) For r > s, Im s;t T drs r;t rC1 D 0; thus, ErC1 D Ker drs;t Ers;t ; let E1 s;t D s;t r>s Er \. Claim: For t > s, there exists a chain of subgroups : : : BsC1;tC1 Bs;t : : : B1;t sC1 B0;t s tS s \.X/ such that Bs;t =BsC1;tC1 D E1 s;t \. s;t T \(4\) t sDm B is the subgroup of mS \.X/ consisting of all elements whose order is s;t finite and is not divisible by p\. (Anatoly Fomenko 460) - Notice that the adjointness of the limit term of the spectral sequence to the stable homotopy group of X is substantially different from the adjointness in the Leray theory\. For every m 0, there are, in general, infinitely many groups s;t E1 with 0 s; t s D m, as shown in the picture\. (Anatoly Fomenko 461) - The groups in the marked squares are all finite—actually, they are all finite sums of groups Zp ; however, the group to which the sum is adjoint may have elements of infinite order\. A typical example: The group Zp ˚ Zp ˚ Zp ˚ Zp ˚ : : : is adjoint to Z with respect to the filtration : : : p4 Z p3 Z p2 Z pZ Z\. (Anatoly Fomenko 461) - As we know from Sect\. 9\.8, the projection of a Serre fibration establishes an isomorphism between the relative homotopy groups of the total space modulo a fiber and the homotopy groups of the base (Anatoly Fomenko 461) - 34\.5 A Digression: A Remark on the Resolutions Let us describe the most convenient free resolution : : : ! B3 ! B2 ! B1 ! B0 ! H of the A-module H (Anatoly Fomenko 465) - Thus, for our resolution, the complex fHomA \.Bk ; Zp /g has a trivial differential, and, consequently, Ext kA \.H H (Anatoly Fomenko 465) - \.X/, choose generators in this kernel in the same way, get a B1 , and so on\. With a resolution constructed in this way, we will have all the homomorphisms Hom A \.Bk ; Zp / ! HomA \.BkC1 ; Zp / trivia (Anatoly Fomenko 465) - Every finite Abelian group is decomposed into the sum of two groups: “p-component” and “non-p-component\.” We will denote these groups as compp G and compp G, respectively\. Thus, G D compp G ˚ comp p G, the order of compp G is a power of p, and the order of compp G is not divisible by p (Anatoly Fomenko 465) - Lemma 1\. The group compp NCq \.X\.s// does not depend on s\. (Anatoly Fomenko 465) - e \.X 0 / ! H e \.X/\. The fundamental lemma of homological algebra \(the theorem of Sect\. 33\.3\) provides a \(homotopically unique\) homomorphism between the projective resolutions, that is, a commutative diagram (Anatoly Fomenko 468) - Corollary\. The Adams spectral sequence of X, beginning from the E2 -term, depends only on the stable homotopy type of X\. (Anatoly Fomenko 470) - In the general case, the homotopy groups NCq \.X\.s// are not finite, but they are finitely generated \(since X was assumed to be a CW complex with finite skeletons\)\. (Anatoly Fomenko 470) - The decomposition G D compp G ˚ compp G \(see Sect\. 34\.6\) in the case when G is finitely generated, but not necessarily finite, should be replaced by the decomposition G D Compp G ˚ compp G, where compp G is the same as before, the subgroup of G consisting of 0 and all elements of finite order prime to p, and Compp G Š \.G= comp p G/\. Thus, Compp G is the sum of a group of the form Z ˚ : : : ˚ Z and a finite group whose order is a power of p\. (Anatoly Fomenko 470) - If the group NCq \.†N X/ is infinite, then we cannot expect that Compp NCq \.X\.s// will be zero for sufficiently large s; actually, all the groups NCq \.X\.s/ have the same ran (Anatoly Fomenko 470) - The composition map hW †X ! †X may also be described as the smash product X#S 1 ! X#S1 of the identity map X ! X and a map S 1 ! S1 of degree pk \. (Anatoly Fomenko 470) - Fig\. 124 A pk -fold map †X ! †X (Anatoly Fomenko 471) - 0 Lemma\. The stable homotopy groups of the space X 0 are finite\. \(They are actually p-groups\.\) (Anatoly Fomenko 471) - The multiplicative structure in the cohomological Leray spectral sequence appeared because the cohomology groups of which this spectral sequence is made possess a multiplicative structure\. The groups of which the Adams spectral sequence is made are homotopy groups which do not possess any multiplicative structure\. Well, there certainly is the Whitehead product, but it is useless for the Adams spectral sequence since it is zero in the stable dimensions (Anatoly Fomenko 473) - Still, there is a way to introduce a multiplication in the Adams spectral sequence of a space X, at least in the case when X is a sphere; but this case is very important and interesting\. (Anatoly Fomenko 473) - Let ˛ 2 kS \.S0 /; ˇ 2 `S \.S0 /\. We can regard ˛ as an element of NCk \.SN / and ˇ as an element of NCkC` \.SNCk / \(where N k; `\)\. Spheroids aW SNCk ! S N ; bW SNCkC` ! SNCk representing ˛ and ˇ can be composed to create a spheroid a ı bW SNCkC` ! SN , which, in turn, belongs to some class in NCkC` \.SN / D S kC` C` \.S0 /\. We denote this element as ˛ ı ˇ and call it the composition product of ˛ and ˇ (Anatoly Fomenko 473) - Proposition 2\. The composition product is skew-commutative, that is, ˇ ı ˛ D \.1/k` ˛ ı ˇ\. (Anatoly Fomenko 475) - Proposition 3\. The composition product is distributive: (Anatoly Fomenko 475) - Since the associativity of product is obvious, we can say that 0 the direct sum S \.S 0 / D Lthe Sco q q \.S 0 position / acquires the structure of graded associative skew-commutative ring\. (Anatoly Fomenko 475) - Remark\. As we noticed in the proof of Proposition 3, the composition products exist not only in stable homotopy groups, and not only of spheres\. However, even for a homotopy group of spheres, the composition product does not have any good algebraic properties (Anatoly Fomenko 475) - Certainly, it is associative: (Anatoly Fomenko 475) - However, no kind of commutativity can exist: If ˛ 2 m \.Sn / and ˇ 2 ` \.Sm /, then ˛ ı ˇ cannot be equal to ˙ˇ ı ˛ simply because the latter is not define (Anatoly Fomenko 475) - If 2 2 \.S2 / is the class of the identity spheroid and ˛ 2 3 \.S2 / is the Hopf class, then \.2/ ı ˛ is not 2˛ \(which would have followed from the left distributivity\), but rather 4˛\. (Anatoly Fomenko 475) - Here is the simplest example: If 2 2 \.S2 / is the class of the identity spheroid and (Anatoly Fomenko 475) - L An associative unitary graded algebra A D m Am over a field K is called a Hopf algebra if the following axioms hold\. \(1\) \(2\) \(3\) \(4\) A m D 0 for negative m, and A0 D K\. There is defined a diagonal map or comultiplication W A ! A ˝K A which is a homomorphism between graded algebras \(the multiplication in A ˝K A is defined by the rule \.˛ ˝ ˇ/ \.˛ 0 ˝ ˇ 0 / D \.1/deg ˇ deg ˛ tiplication in 0 ˛˛ 0 ˝ ˇˇ 0 \)\. If deg a D d > 0, then \.a/ D a ˝ 1 C : : : C \.1/d 1 ˝ a, where : : : denotes a sum of tensor products of elements of positive degrees\. The comultiplication is coassociative ; that is, the diagram is commutative\. (Anatoly Fomenko 476) - A remarkable property of the definition of a Hopf algebra is its symmetry with respect to the multiplication W A ˝K A ! A \.\.a ˝ b/ D ab/ and comultiplication W A ! A ˝K A\. This symmetry displays itself in the following fact\. E 2\. Prove that if A is a Hopf algebra, then the graded dual space A D E 2\. Prove that if A is a Hopf algebra, then the graded dual space A D m ERCISE m m \.A / is a Hopf algebra with respect to the multiplication W A ˝K A ! A LXERCISE and the comultiplication W A ! A ˝K A \. (Anatoly Fomenko 476) - Example 1\. Cohomology \(as well as homology\) of an H-space X with coefficients in a field is a Hopf algebra: The multiplication and comultiplication are induced by the diagonal map X ! X X and the product map X X ! X \(plus Künneth’s formula\)\. (Anatoly Fomenko 476) - Example 2\. The Steenrod algebra is a Hopf algebra: The comultiplication is defined by the formula X \.ˇ/ D ˇ ˝ 1 1 ˝ ˇ; \.Psp / D Psp ˝ Ptp sCtDi P s \(in particular, if p D 2, then \.Sqi / D sCtDi Sq ˝ Sqt \)\. (Anatoly Fomenko 477) - If A is a Hopf algebra and B; C are A-modules, then the tensor product B ˝K C also has a natural structure of an A-module\. Indeed, B ˝K C is naturally an \.A ˝K A/module, and the algebra homomorphism W A ! A ˝K A turns it into an A-module\. (Anatoly Fomenko 477) - Remark\. The Hopf algebra structure on the Steenrod algebra is compatible with Künneth’s formula\. Namely, if X and Y are arbitrary spaces, then H H \.XI Zp / ˝ H (Anatoly Fomenko 477) - Proposition\. For any Hopf algebra A, the A-module A ˝K A is free\. (Anatoly Fomenko 477) - Corollary\. If B and C are free A-modules, then B ˝K C is also a free A-module\. (Anatoly Fomenko 477) - Let A be a Hopf algebra, and let M 0 ; M 00 ; N 0 ; N 00 be A-modules\. We will define a multiplication \(pairing\) Exts 0 A 0 0 ;t \.M 0 ; N 0 / ˝K Ext 0 sA 00 00 ;t \.M 00 ; N 00 / ! ExtA 0 s 0 00 ;t0 Ct00 Cs \.M 0 ˝K M 00 ; N 0 ˝K N 00 /: (Anatoly Fomenko 478) - EXERCISE 3\. If M 0 D M 00 D N 0 D N 00 D K \(with the natural structure of A-modules\),Lthen the constr structure in s;t Exts;t A \.K; K/\. (Anatoly Fomenko 478) - Definition\. The spaces Ext s;t A \.K; K/ are called cohomology spaces of A and are denoted by H s;t \.A/\. (Anatoly Fomenko 478) - Thus, we have defined the \(bigraded\) cohomology algebra of a Hopf algebra\. (Anatoly Fomenko 478) - EXERCISE 4\. This is an associative skew-commutative algebra\. (Anatoly Fomenko 478) - 35\.3 The Multiplicative Structure in the Adams Spectral Sequence Theorem \(Adams\)\. If X D S 0 , then in the Adams spectral sequence, there arises a multiplication Ers;t ˝ E rs 0 0 ;t ! ErsCs 0 0 with the following properties\. (Anatoly Fomenko 479) - We begin with a computation of the E2 -term of the Adams spectral sequence for the case of 2-components of stable homotopy groups of spheres, that is, the cohomology of the Steenrod algebra mod 2\. (Anatoly Fomenko 482) - But even without this, we see now that the orders of the 2-components of the stable groups nCk \.Sn / with k D 1; 2; 3; 4; 5; 6; 7 are 2, 2, 8, 1, 1, 2, 16 or 8 (Anatoly Fomenko 488) - We see that the problem of the computation of the stable homotopy groups of spheres falls into two parts: the computation of the cohomology of the Steenrod algebra and the computation of the differentials of the Adams spectral sequence\. (Anatoly Fomenko 491) - The first part is reduced to a purely mechanical work \(or to a computer program\) and can be done up to any degree\. In the book Stable Homotopy Theory of Adams \(see Adams [2]\) the result of this computation is presented for t s 17\. (Anatoly Fomenko 491) - s;t Hence, E1 D E2s;t for t s 13, and we can derive the following statement concerning the first 13 stable homotopy groups of spheres\. The orders of 2-components of the groups rS \.S 0 / for r D 1; 2; : : : ; 13 are as follows: 2; 2; 8; 1; 1; 2; 16; 4; 8; 2; 8; 1; 1: (Anatoly Fomenko 492) - 2 E2s;t ; hr0 u ¤ 0 and u represents some 2 tS s s \.S0 /, then 2r ¤ 0 in 2 tS s s \.S0 /\. In particular, we see from Figs\. 128 and 129 that 3S \.S 0 /; 7S \.S 0 /; and 11 S \.S 0 / are cyclic groups of orders 8, 16, and 8\. (Anatoly Fomenko 492) - 8\. This gives us full information about the groups rS \.S0 / with r 13, except the cases r D 8; 9\. (Anatoly Fomenko 492) - The group 8S \.S0 / has order 4\. It has a subgroup Z2 (Anatoly Fomenko 492) - the quotient of 8S \.S 0 / over this subgroup is also a group Z2 (Anatoly Fomenko 492) - and the quotie (Anatoly Fomenko 492) - thus, 2v D 0 and 8S \.S0 / Š Z2 ˚ Z2 \. (Anatoly Fomenko 493) - Finally, the group 9S \.S0 / has order 8 (Anatoly Fomenko 493) - subgroup isomorphic to Z2 ˚Z2 ; (Anatoly Fomenko 493) - The quotient of 9S \.S0 / over this subgroup is Z (Anatoly Fomenko 493) - Z 2 , which leaves us with two possibilities for 9S \.S0 /: Z 2 ˚ Z2 ˚ Z2 or Z4 ˚ Z2 (Anatoly Fomenko 493) - Thus, 9S \.S0 / Š Z2 ˚ Z2 ˚ Z2 , (Anatoly Fomenko 493) - ˚ Z2 , and we now know the 2-components of all groups rS \.S0 / with r 13: Z 2 ; Z2 ; Z8 ; 0; 0; Z2 ; Z16 ; Z2 ˚ Z2 ; Z2 ˚ Z2 ˚ Z2 ; Z2 ; Z8 ; 0; 0: (Anatoly Fomenko 493) - In addition to all this, we remark that the computation of 2-components of 14th and further stable homotopy groups of spheres encounters a difficulty: The grading arguments no longer guarantee the triviality of differentials (Anatoly Fomenko 493) - Within our dimension range, the only necessity for considering the Adams spectral sequence arises for p D 3\. Indeed, we already know \(see Sect\. 27\.4\) that the first S p-component in the stable homotopy groups of spheres is Zp 2p 3 0 p 3 \.S / and the S second is Zp 4p 5 0 p 5 \.S /; thus, besides S Z5 7 0 \.S / and S Z7 11 \.S 0 /, the groups S n \.S 0 / with n 13 do not have p-components with p > 3\. (Anatoly Fomenko 493) - The 3-components are as follows: 0; 0; Z3 ; 0; 0; 0; Z3 ; 0; 0; Z3 ; Z9 ; 0; Z3 (Anatoly Fomenko 495) - The composition multiplication in these groups is trivial\. Thus, we have n \.Sn / nC1 \.Sn / nC2 \.Sn / nC3 \.Sn / nC4 \.Sn / nC5 \.Sn / nC6 \.Sn / nC7 \.Sn / nC8 \.Sn / nC9 \.Sn / nC10 \.Sn / nC11 \.Sn / nC12 \.Sn / nC13 \.Sn / DZ D Z2 D Z2 D Z24 D0 D0 D Z2 D Z240 D Z2 ˚ Z2 D Z2 ˚ Z2 ˚ D Z6 D Z504 D0 D Z3 for n 1; for n 3; for n 4; for n 5; for n 6; for n 7; for n 8; for n 9; for n 10; Z 2 for n 11; for n 12; for n 13; for n 14; for n 15: (Anatoly Fomenko 495) - Here each group Exts;t which occurs in the E2 -term can be effectively computed; the process is purely algebraic\. However, no such effective method is given for computing the differentials dr in the spectral sequence or to determine the group extension by which p S \.X; Y/ is built up from the E1 -term; these are topological problems\. (Anatoly Fomenko 495) - In fact, when a chance has arisen to show that such a differential d r is nonzero, it has been regarded as an interesting problem, and duly solved\. However, the difficulty of actually computing groups Exts;t A \.L; M/ has remained the greatest obstacle to the method\. (Anatoly Fomenko 496) - As we can see from this quotation from Adams’ article \(Adams [8]\), its author did not think that the existence of the algorithm for computing the cohomology is equivalent to its computing (Anatoly Fomenko 496) - existence of the algorithm for computing the cohomology is equivalent to its computing\. Adams devoted several works to this subject\. Here we will formulate his most important results\. Theorem of the First Three Rows \(Adams [1]\)\. (Anatoly Fomenko 496) - Triviality Theorem \(Adams [7]\)\. (Anatoly Fomenko 496) - Periodicity Theorem \(Adams [7]\)\. F (Anatoly Fomenko 496) - We conclude this lecture with an example of an Adams spectral sequence with a nontrivial differential\. (Anatoly Fomenko 497) - Let X D K\.Z4 ; n/, where n is large enough\. (Anatoly Fomenko 497) - Thus, we have constructed a secondary operation '\. It is a family of partial multivalued homomorphisms H N \.XI Zp / Ü H NCn 1 \.XI Zp / \(partial because they are defined only on the intersection of the kernels of the operations ˛i ; multivalued because they are defined modulo the images of the operations ˇi \)\. (Anatoly Fomenko 500) - EXERCISE 2\. Formulate and prove the properties of the secondary operations, including their naturality and stability\. (Anatoly Fomenko 500) - The best-known example of a secondary cohomology operation is the so-called second Bockstein homomorphism\. (Anatoly Fomenko 500) - m\. It corresponds to the relation ˇ 2 D 0 and thus is defined on the kernel and takes values modulo the image of the usual, “first” Bockstein homomorphism\. (Anatoly Fomenko 500) - If one does not restrict this theory to stable operations, and considers arbitrary coefficient groups, then it becomes true that the cohomology with the action of all this higher cohomology operations fully determines the homotopy type of a simply connected space\. (Anatoly Fomenko 500) - On the other hand, elements of the stable homotopy groups of spheres arise from the Adams spectral sequence\. What is the relation between \(primary and higher\) cohomology operations and stable homotopy groups of spheres? (Anatoly Fomenko 501) - We already know that the homotopy groups of a space \(even of a CW complex\) do not fully determine its homotopy type\. There are two exceptions: when all the homotopy groups are trivial, and when there is only one nontrivial homotopy group\. (Anatoly Fomenko 502) - Suppose that a CW complex X has trivial homotopy groups of all dimensions except n1 and n2 , and n1 \.X/ D …1 ; n2 \.X/ D …2 \. Then there (Anatoly Fomenko 502) - This shows that a space of the form S N [DNCq with a nontrivial action of Sqq exists only for q D 1; 2; 4; 8\. We explained in Sect\. 31\.4 that this is equivalent to the fact that a group 4n 1 \.S 2 n/ contains an element with odd Hopf invariant only for n D 1; 2, and 4\. Let us mention a more classical \(and actually equivalent\) approach to the problem of the odd Hopf invariant \(known as the Frobenius conjecture\) (Anatoly Fomenko 502) - The cohomology classes ui are the fundamental classes, and means the transgressions\. The transgression images f 1 ; f2 ; f3 ; : : : (Anatoly Fomenko 503) - These cohomology operations are called Postnikov factors\. (Anatoly Fomenko 503) - arises a \(homotopically unique\) map X ! K\.…1 ; n1 / which may be considered a fibration, and the fiber is K\.…2 ; n2 /\. In the cohomology spectral sequence of this fibration with coefficients in …2 , the fundamental class u2 2 H n2 \.K\.…2 ; n2 /I …2 / D 20;n 2 E 20;n2 is transgressive and is taken by the transgression into some element f1 of E 2n2 C1;0 D H n2 C1 \.K\.…1 ; n1 /I …2 /\. This f1 , on one hand, fully determines the fibration, and hence \(the homotopy type of\) X, and on the other hand it can be regarded as a \(primary\) cohomology operation H n1 \.I …1 / ! H n2 C1 \.I …2 /\. Thus, the homotopy type of a space X with two nontrivial homotopy groups is determined by these groups plus one primary cohomology operation, which is called the \(first\) Postnikov factor of X\. (Anatoly Fomenko 503) - Thus, according to the Postnikov theory, the homotopy type of a simply connected CW complex is determined by the homotopy groups plus a sequence of cohomology operations, a primary one, a secondary one, and so on, such that every operation from this sequence in defined on the kernel of the previous one\. (Anatoly Fomenko 504) - EXERCISE 8\. Prove that the first Postnikov factor of Sn \(with n 3\) is Sq2 \. Try to describe the second Postnikov factor\. (Anatoly Fomenko 504) - K-theory emerged as an independent part of topology in the late 1950s, when the limits of the possibilities of the methods based on spectral sequences and cohomology operations \(and studied in Chaps\. III–V\) became visible (Anatoly Fomenko 505) - It consisted of replacing cohomology as the basic homotopy invariant by an entirely new object, the so-called K-functor\. (Anatoly Fomenko 505) - The first works in K-theory were published in 1959, and as early as 1963 most of its results were completed \(including all the results described below, with the exception of the proof of the Adams conjecture\)\. And it is remarkable that all works in K-theory of this period belong to four authors: Frank Adams, Michael Atiyah, Raoul Bott, and Friedrich Hirzebruch\. There is no name of Alexander Grothendieck on this list, for he has no works in topology; but according to common opinion, it was Grothendieck who had first conceived of the key ideas of K-theory that found their applications not only in topology\. The topological applications of K-theory obtained during its “heroic period” were really impressive\. Let us mention only a simple proof of the nonexistence of division algebras in dimensions different from 1, 2, 4, 8; the precise computation of the maximal number of linearly independent vector fields on the sphere of an arbitrary dimension; the computation of the order of the image of the “J-homomorphism” n \.SO/ ! nS ; theorems of nonembeddability and nonimmersibility of various manifolds into Euclidean spaces; as well as various (Anatoly Fomenko 505) - Let X be a finite CW complex\. Usually we assume that X has a base point, x0 \. Denote by F\.X/ the set of equivalence classes ` If X is connected, then F\.X/ D n of 0 Fn complex vector bundles with the base X\. \.X/, where Fn \.X/ is the set of classes of n-dimensional vector bundles with the base X\. If X is not connected, then we do not (Anatoly Fomenko 506) - integrality theorems for rational linear combinations of characteristic classes\. The solution of the problem of the index of an elliptic operator found in 1963 by Atiyah and Singer would have been totally impossible without K-theory\. (Anatoly Fomenko 506) - for the usual cohomology \(for example, with the coefficients in Z\) the following Whitehead theorem holds: If X and Y are simply connected CW complexes, and if a continuous map f W X ! Y induces an isomorphism f W H \.YI Z/ ! H \.XI Z/, then f is a homotopy equivalence \(see Sect\. 14\.5\)\. K-theory also assigns to a \(finite\) CW complex a graded group that depends on it in a functorial way, but there is no analog of Whitehead’s theorem: It is quite possible that f W K \.Y/ ! K \.X/ is an isomorphism, but f is not a homotopy equivalence\. Some extraordinary theories were conceived \(or, better to say, recollected\) that have all the advantages of K-theory and are free of this flaw\. The best known of these theories is the cobordism theory, which exists in a variety of versions \(the same is true for K-theory, by the way\)\. For cobordisms, the analog of Whitehead’s theorem is valid; thus, from the point of view of homotopy theory, cobordism theory is not worse than cohomology\. The topologists were able to construct for cobordisms an analog of the Adams spectral sequence, the so-called Adams–Novikov spectral sequence, which was repeatedly applied to homotopy calculations\. A theory of cobordism-valued characteristic classes proved to be very (Anatoly Fomenko 506) - useful\. At last the assertion that cobordisms are better than K-theories had a precise statement: The \(complex\) K-theory may be embedded into the \(complex\) cobordism theory as a direct summand\. We pay a tribute to cobordism theories by devoting a separate lecture to them \(Lecture 44\)\. (Anatoly Fomenko 506) - \. (Anatoly Fomenko 507) - Notice that the dimension of a virtual vector bundle may be negative\. (Anatoly Fomenko 507) - Definition\. K\.X/ D G\.F\.X//\. The elements of the group K\.X/ are sometimes called virtual vector bundles over X\. A continuous map f W X ! Y defines, in the obvious way, the “induced homomorphism” f W K\.Y/ ! K\.X/\. Homotopic continuous maps always induce equal homomorphism; thus, K\.X/ is a homotopy invariant of X\. (Anatoly Fomenko 507) - EXERCISE 4\. This map is not necessarily an injection: Give an example\. (Anatoly Fomenko 507) - Denote by G\.F/ the set of equivalence classes of formal differences in F, and define an addition operation in G\.F/ by the formula fa bg C fc dg D f\.a C c/ \.b C d/g\. (Anatoly Fomenko 507) - This is the Grothendieck group of the semigroup F\. The formula a 7! fa 0g defines a map F ! G\.F/\. (Anatoly Fomenko 507) - There is a general operation in algebra which converts a commutative semigroup with zero into a commutative group \(the so-called Grothendieck group\)\. Namely, let F be a commutative semigroup with zero\. Consider the set of all formal differences a b; a 2 F; b 2 F and introduce the following equivalence relation in this set: \.a b/ \.c d/ if there exists some e 2 F such that a C d C e D b C c C e\. Obviously, this relation is reflexive and symmetric\. (Anatoly Fomenko 507) - require that the dimensions of the fibers over different components be the same; in this case we define the dimension of the bundle as the dimension of the fiber over the base point, if there is one\. There are two binary operations in F\.X/: direct sum and tensor product\. With respect to addition \(direct summation\), F\.X/ is a commutative semigroup with zero; (Anatoly Fomenko 507) - Definition\. K\.X/ D KerŒdimW K\.X/ ! Z\. (Anatoly Fomenko 508) - Obviously, K\.X/ D K\.X/ ˚ Z always\. e Theorem\. K\.X/ is the set of classes of stably equivalent vector bundles over X\. (Anatoly Fomenko 508) - Proposition 1\. \(i\) For any virtual vector bundle ˛ 2 K\.X/ there exist a usual vector bundle a and an integer N such that ˛ D fa Ng: \(Recall that in the theory of vector bundles N denotes the standard trivial vector bundle of dimension N; X CN ! X\.\) \(ii\) Virtual vector bundles ˛ D fa Ng and ˇ D fb Mg are equal if and only if dim a D dim b and the vector bundles a and b are stably equivalent \(see the definition in Sect\. 19\.1\)\. (Anatoly Fomenko 508) - Lemma\. For any vector bundle a over \(a finite CW complex\) X there exists a vector bundle a over X such that the sum a ˚ a is trivial\. Proof of Lemma\. According to Sect\. 19\.4, a D f , where is the tautological bundle over an appropriate complex Grassmannian and f is a continuous map of X into this Grassmannian\. Hence, we need to prove the lemma only in the case when a itself is the tautological bundle over the Grassmannian CG\.NI n/\. (Anatoly Fomenko 508) - EXERCISE 7\. Prove the following stronger form of the homotopy lemma: The spaces BU and U are homotopy equivalent, and the spaces BU\.n/ and U\.n/ are homotopy equivalent\. EXERCISE 8\. A further strengthening of the homotopy lemma: The spaces above are homotopy equivalent as H-spaces\. APPLICATION\. K\.S K\.S \.U/ D Z\. / D Z\. Notice that the group K\.S over S 2 D CP 1 e 2 / is gen \. [In other words, K\.S erated by the stable class of the Hopf bundle e 2 / is generated by the class of the virtual b 1\.] Indeed, the Hopf bundle is not stably trivial since its first Chern class undle c1 \./ (Anatoly Fomenko 509) - Homotopy Lemma\. For every r, there is an isomorphism r \.U/ D rC1 \.BU/\. The same is true if BU and U are replaced by BU\.n/ D CG\.1; n/ and U\.n/\. (Anatoly Fomenko 509) - BU D CG\.1; 1/ D lim lim CG\.N; n/; U D lim U\.n/: lim lim CG\.N; n/; U D lim n!1 N!1 n!1 (Anatoly Fomenko 509) - Corollary\. K\.X/ D \.X; BU/\. (Anatoly Fomenko 509) - Example\. K q \.pt/ D K K q 0 \.S // D K\.S q \.BU/: (Anatoly Fomenko 510) - For q 0, put K q \.X; A/ D K\.† K\.† q \.X=A//; e q q K \.X/ D K \.X; ;/; K (Anatoly Fomenko 510) - K 0 \.X/ D K\.X=;/ K\.X=;/ D K\.X (Anatoly Fomenko 510) - does not vanish \(s (Anatoly Fomenko 510) - D: K-Sequences of Pairs Lemma\. The functor K \.X; A/, the sequence K\.X=A/ ! K\.X/ of homomorphisms induced by the inclusion A ! X and the projection X ! X=A is exact\. (Anatoly Fomenko 511) - In K-theory, the groups K q \.X; A/ with q > 0 are, certainly, also studied, but just to define them we need the so-called Bott periodicity, (Anatoly Fomenko 511) - E: Attempts to Generalize K to the Case of Infinite CW Complexes (Anatoly Fomenko 512) - There is no satisfactory generalization of K-theory to the case of infinite CW complex (Anatoly Fomenko 512) - VERSION 1\. K\.X/ is the Grothendieck group of the semigroup of the equivalence classes of vector bundles\. (Anatoly Fomenko 512) - Remark\. The geometric construction, which we used for the definition of the K-sequences of pairs and triples, may be continued\. As a result we will get a sequence of continuous maps A ! X ! X=A Ü †A ! †X ! †\.X=A/ Ü †2 A ! : : : ; where dotted arrows are defined up to a homotopy\. This sequence is called the Puppe sequence\. The Puppe sequence is homotopy exact in the sense that any threeterm fragment of this sequence is homotopy equivalent to a fragment of the form B ! Y ! Y=B\. (Anatoly Fomenko 512) - Definition\. A continuous map of a CW complex X into some topological space is called a phantom map if it is not homotopic to a constant, but its restriction to any finite CW subspace of X is homotopic to a constant\. (Anatoly Fomenko 513) - \(1\) If all the oddnumbered Betti numbers of X are 0, then k\.X/ D K\.X/\. \(2\) Let X D K\.Z; 3/\. numbered Betti numbers of X are 0, then k\.X/ D K\.X/\. \(2\) Let X D K\.Z; 3/\. Then K\.X/ Z is the completion of the group Z e k\.X/ D b D 0, while e Z=Z, where b Z is the subgroup of the with respect to all finite index subgroups\. \(In other words, b infinite product Z2 Z3 Z4 : : : consisting of such sequences \.a2 ; a3 ; a4 ; : : : / that 2 ; a3 ; a4 ; : : : / that Z by the formula aqr aq mod q for any integers q; r; the group Z is em n 7! \.res2 n; res3 n; res4 n; : : : /; in other words, Z bedded into b Z is the group of stabilizing b Z\.\) (Anatoly Fomenko 513) - The biggest advantage of K-theory is its relatively easy computability: actually, the methods of computing groups K\.X/, developed in Lecture 39 ahead, if applied to an infinite CW complex X, yield precisely K\.X/\. However, there are no exact sequences of pairs and triples in this theory: (Anatoly Fomenko 513) - k\.X/ ! K\.X/ is an epimorphism\. ERCISE 9\. Prove that this map e e What is the kernel of this map? This kernel consists of classes of maps X ! BU which are homotopic to a constant on any finite CW subspace Y X, but still not homotopic to a constant on the whole X\. Is such an odious thing possible? It turns out that yes! (Anatoly Fomenko 513) - VERSION 2\. K\.X/ K\.X/ D \.X; BU/; K\.X/ D K\.X/ K\.X/ ˚ Z\. e e VERSION 3\. K\.X/ D lim K\.Y/, where the limit is taken with respect to all finite m CW subcomplexes of X\. (Anatoly Fomenko 513) - Theorem \(Bott\)\. For any finite CW complex X, the mapping K\.X/ ˚ K\.X/ ! K\.X S2 /; \.˛1 ; ˛2 / 7! \.˛1 ˝ 1/ C \.˛2 ˝ / is an isomorphism\. (Anatoly Fomenko 514) - We use this isomorphism to define K q \.X; A/ for all q: By definition, 2N K q \.X; A/ D K q \.X; A/; N > max\.0; q/ (Anatoly Fomenko 515) - Thus, we obtain Corollary\. There is an isomorphism that is natural with respect to X: K\.X/ Š K\.† In the case X D Sq , this implies [since K\.S Corollary of corollary\. For any q, q \.BU/ Š q 2 \.BU/; and hence q \.U/ Š q 2 \.U/\. It was this theorem that was originally proved by Bott and which is known as \(unitary\) Bott periodicity\. Since 1 \.U/ D Z and 2 \.U/ D 2 \.SU/ ˚ 2 \.S1 / D 2 \.SU/ ˚ 2 \.S1 / D 0, Z for i odd; i \.U/ D 0 for i even; and Z for i even; i \.BU/ D K\.S K\.S 0 f or i odd: (Anatoly Fomenko 515) - ch\. 1 ˚ 2 / D ch 1 C ch 2 ; ch\.1 ˝ 2 / D ch1 ch2 : The first property lets us extend ch to the K-theory: chW K\.X/ ! H even \.XI Q/I the second property makes this K-theory’s character multiplicative\. Furthermore, this definition obviously gives rise to a definition of the character H even \.X; AI Q/ for q even; H eve chW K q \.X; A/ ! H odd H odd \.X; AI Q/ for q oddI (Anatoly Fomenko 520) - This H looks like K in some respects: It has exact sequences of pairs and triples, but also it is 2-periodic: Hq D HqC2 \. The character ch becomes then a sequence of homomorphisms chW K q \.X; A/ ! Hq \.X; A/; (Anatoly Fomenko 520) - Let chQ D ch ˝QW K q \.X; A/ ˝ Q ! Hq \.X; A/ ˝ Q D Hq \.X; A/: Theorem\. The map chQ is an isomorphism for any Q and any \.finite CW\) pair \.X; A/\. In particular, K\.X/ ˝ Q Š H even \.XI Q/; and the rank of the group K\.X/ is equal to the sum of the even Betti numbers of X\. (Anatoly Fomenko 520) - Let chQ D ch ˝QW K q \.X; A/ ˝ Q ! Hq \.X; A/ ˝ Q D Hq \.X; A/: Theorem\. The map chQ is an isomorphism for any Q and any \.finite CW\) pair \.X; A/\. In particular, K\.X/ ˝ Q Š H even \.XI Q/; and the rank of the group K\.X/ is equal to the sum of the even Betti numbers of X\. (Anatoly Fomenko 520) - HOMOTOPY AXIOM\. If f g, then f D g \. EXACTNESS AXIOM\. The sequences of pairs are exact\. \(Corollary: The sequences of triples are exact\.\) FACTORIZATION AXIOM\. The projection \.X; A/ ! \.X=A; pt/ induces for all q def isomorphisms Hq \.X; A/ Š Hq \.X=A; pt/ \. D H H DIMENSION AXIOM\. Hq \.pt/ D 0 for q ¤ 0\. (Anatoly Fomenko 522) - These axioms are called Eilenberg–Steenrod axioms\. If we add to these axioms the statement that H0 \.pt/ D Z, then the following uniqueness theorem will hold: The theory fHq ; @ ; fq g satisfying all the axioms above is unique and coincides with the theory of usual \(singular\) homology\. (Anatoly Fomenko 522) - Actually, this theorem was proven in Lecture 13, where we calculated the homology of CW complexes \(and CW pairs\) using only the properties of homology listed above (Anatoly Fomenko 522) - EXERCISE 15\. Prove that hq \.X1 t X2 / D hq \.X1 / ˚ hq \.X2 / for any extraordinary homology theory h \(and that the same is true for extraordinary cohomology\)\. (Anatoly Fomenko 522) - EXERCISE 15\. Prove that hq \.X1 t X2 / D hq \.X1 / ˚ hq \.X2 / for any extraordinary homology theory h \(and that the same is true for extraordinary cohomology\)\. (Anatoly Fomenko 522) - Actually, this theorem was proven in Lecture 13, where we calculated the homology of CW complexes \(and CW pairs\) using only the properties of homology listed above (Anatoly Fomenko 522) - These axioms are called Eilenberg–Steenrod axioms\. If we add to these axioms the statement that H0 \.pt/ D Z, then the following uniqueness theorem will hold: The theory fHq ; @ ; fq g satisfying all the axioms above is unique and coincides with the theory of usual \(singular\) homology\. (Anatoly Fomenko 522) - Theorem\. Let 'W h ! k be a homomorphism of the theory h into the theory k such that for any q '\.pt;;/ W hq \.pt/ ! kq \.pt/ is an isomorphism\. Then '\.X;A/W hq \.X; A/ ! kq \.X; A/ is an isomorphism for any q and \.X; A/; in particular, in this case h Š k\. (Anatoly Fomenko 523) - Theorem\. Let 'W h ! k be a homomorphism of the theory h into the theory k such that for any q '\.pt;;/ W hq \.pt/ ! kq \.pt/ is an isomorphism\. Then '\.X;A/W hq \.X; A/ ! kq \.X; A/ is an isomorphism for any q and \.X; A/; in particular, in this case h Š k\. (Anatoly Fomenko 523) - Remark\. If h q \.pt/ Š kq \.pt/, but the isomorphism is not induced by any map h ! k, then the theories h and k may be nonisomorphic\. For example, K q q \.pt/ Š HZ \.pt/; q where HZ \.X; A/ D H even \.odd/ \.X; AI Z/, but, as we will prove in Lecture 39, K\.RPn / 6Š H even \.RPn I Z/: (Anatoly Fomenko 524) - The role similar to that of BU is played in the real K-theory by the space BSO D GC \.1; 1/ [in another version of it by the space BO D G\.1; 1/], and the role of the Bott 2-periodicity is played by the real Bott 8-periodicity i \.SO/ Š iC8 \.SO/\. (Anatoly Fomenko 524) - Definition\. A spectrum, or an -spectrum, is a sequence of CW complexes Wi and continuous maps fi W Wi ! WiC1 \.i 1/\. (Anatoly Fomenko 524) - Notice that for any topological spaces A; B with base points there exists a natural one-to-one correspondence between continuous maps A ! B and †A ! B (Anatoly Fomenko 524) - In particular, \.A; B/ D \.†A; B/\. (Anatoly Fomenko 524) - Definition\. Let \.X; A/ be a CW pair, and let W D fWi ; fi g be an arbitrary spectrum\. The homology and cohomology groups of the pair \.X; A/ with the coefficients in W are defined by the formulas \(in which q 2 Z\) h q \.X; AI W/ D lim lim N \.†N \.X=A/; WNCq /; hq \.X; AI W/ D lim lim N NCq \.WN #\.X=A//: (Anatoly Fomenko 524) - Definition\. Let \.X; A/ be a CW pair, and let W D fWi ; fi g be an arbitrary spectrum\. The homology and cohomology groups of the pair \.X; A/ with the coefficients in W are defined by the formulas \(in which q 2 Z\) h q \.X; AI W/ D lim lim N \.†N \.X=A/; WNCq /; hq \.X; AI W/ D lim lim N NCq \.WN #\.X=A//: (Anatoly Fomenko 524) - In particular, \.A; B/ D \.†A; B/\. (Anatoly Fomenko 524) - Notice that for any topological spaces A; B with base points there exists a natural one-to-one correspondence between continuous maps A ! B and †A ! B (Anatoly Fomenko 524) - Definition\. A spectrum, or an -spectrum, is a sequence of CW complexes Wi and continuous maps fi W Wi ! WiC1 \.i 1/\. (Anatoly Fomenko 524) - The role similar to that of BU is played in the real K-theory by the space BSO D GC \.1; 1/ [in another version of it by the space BO D G\.1; 1/], and the role of the Bott 2-periodicity is played by the real Bott 8-periodicity i \.SO/ Š iC8 \.SO/\. (Anatoly Fomenko 524) - Remark\. If h q \.pt/ Š kq \.pt/, but the isomorphism is not induced by any map h ! k, then the theories h and k may be nonisomorphic\. For example, K q q \.pt/ Š HZ \.pt/; q where HZ \.X; A/ D H even \.odd/ \.X; AI Z/, but, as we will prove in Lecture 39, K\.RPn / 6Š H even \.RPn I Z/: (Anatoly Fomenko 524) - Examples\. The usual homology and cohomology \(with coefficients in G\) correspond to the Eilenberg–MacLane spectrum in which Wi is K\.G; i/ and fi is the standard homotopy equivalence K\.G; i/ ! K\.G; i C 1/ (Anatoly Fomenko 525) - The complex K-theory corresponds to the periodic -spectrum U; BU; U; BU; : : : with the homotopy equivalences U ! BU and BU ! U \(the first equivalence was announced in Exercise 7; the second is one of the forms of Bott periodicity\)\. The homology in this spectrum, the so-called K-homology, is also very important (Anatoly Fomenko 525) - One more example: The spherical spectrum S in which Wi D Si and the map S i ! SiC1 correspond to the standard homeomorphism †Si ! SiC1 \. Cohomology groups are stable homotopy groups qS qS \.X; A/ and the so-called stable cohomotopy groups q S S \.X; A/ D lim lim N \.†NCq \.X=A/; SN /: (Anatoly Fomenko 525) - Finally, remark that an arbitrary theory of extraordinary homology or cohomology is obtained by the procedure described above from some -spectrum\. This can easily be deduced from the so-called Brown representability theorem (Anatoly Fomenko 525) - which can be found, for example, in Spanier [79]\. (Anatoly Fomenko 525) - which can be found, for example, in Spanier [79]\. (Anatoly Fomenko 525) - Finally, remark that an arbitrary theory of extraordinary homology or cohomology is obtained by the procedure described above from some -spectrum\. This can easily be deduced from the so-called Brown representability theorem (Anatoly Fomenko 525) - One more example: The spherical spectrum S in which Wi D Si and the map S i ! SiC1 correspond to the standard homeomorphism †Si ! SiC1 \. Cohomology groups are stable homotopy groups qS qS \.X; A/ and the so-called stable cohomotopy groups q S S \.X; A/ D lim lim N \.†NCq \.X=A/; SN /: (Anatoly Fomenko 525) - The complex K-theory corresponds to the periodic -spectrum U; BU; U; BU; : : : with the homotopy equivalences U ! BU and BU ! U \(the first equivalence was announced in Exercise 7; the second is one of the forms of Bott periodicity\)\. The homology in this spectrum, the so-called K-homology, is also very important (Anatoly Fomenko 525) - Examples\. The usual homology and cohomology \(with coefficients in G\) correspond to the Eilenberg–MacLane spectrum in which Wi is K\.G; i/ and fi is the standard homotopy equivalence K\.G; i/ ! K\.G; i C 1/ (Anatoly Fomenko 525) - Hence, we get some isomorphisms K 0 1 \.X/ ˝ Q Š H \.XI Q/; K \.X/ ˝ Q Š H odd \.XI Q/\. even \(8\) These isomorphisms coincide with chQ \(see Sect\. 38\.4\)\. (Anatoly Fomenko 528) - Hence, we get some isomorphisms K 0 1 \.X/ ˝ Q Š H \.XI Q/; K \.X/ ˝ Q Š H odd \.XI Q/\. even \(8\) These isomorphisms coincide with chQ \(see Sect\. 38\.4\)\. (Anatoly Fomenko 528) - The Atiyah–Hirzebruch spectral sequence [rather its \.Erp;q rp ; drp;q rp /-version of Sect\. 39\.1\.B] exists for any extraordinary homology or cohomology theory; it has the form H \.XI h \.pt// \) h \.X/ (Anatoly Fomenko 530) - A further generalization of this spectral sequence provides a “spectral sequence of a fibration \.E; B; F; p/”: H \.BI h \.F// \) h \.E/ (Anatoly Fomenko 530) - 39\.2 Examples of Calculations A: K \.CPn / (Anatoly Fomenko 530) - 39\.2 Examples of Calculations A: K \.CPn / (Anatoly Fomenko 530) - A further generalization of this spectral sequence provides a “spectral sequence of a fibration \.E; B; F; p/”: H \.BI h \.F// \) h \.E/ (Anatoly Fomenko 530) - The Atiyah–Hirzebruch spectral sequence [rather its \.Erp;q rp ; drp;q rp /-version of Sect\. 39\.1\.B] exists for any extraordinary homology or cohomology theory; it has the form H \.XI h \.pt// \) h \.X/ (Anatoly Fomenko 530) - Theorem\. Multiplicatively, K 0 \.CPn / D ZŒ =\. nC1 /, where D 1 \. is the Hopf bundle\)\. (Anatoly Fomenko 531) - Theorem\. Multiplicatively, K 0 \.CPn / D ZŒ =\. nC1 /, where D 1 \. is the Hopf bundle\)\. (Anatoly Fomenko 531) - B: K \.RPn / (Anatoly Fomenko 532) - K 1 n \.RP / D 0 for n even; Z for n oddI (Anatoly Fomenko 532) - Theorem\. K e K\.RP n / is a cyclic group of order 2Œn=2 \. (Anatoly Fomenko 532) - Theorem\. K e K\.RP n / is a cyclic group of order 2Œn=2 \. (Anatoly Fomenko 532) - K 1 n \.RP / D 0 for n even; Z for n oddI (Anatoly Fomenko 532) - B: K \.RPn / (Anatoly Fomenko 532) - e \.RP / is cyclic\. Corollary of Proof\. Any complex vector bundle over RP2m or RP2mC1 is stably equivalent to the vector bundle kCR , and the vector bundles kCR and lCR are stably equivalent if and only if k l mod 2m \. (Anatoly Fomenko 533) - Other computations of K-functors in particular, for Grassmann manifolds and flag manifolds, are contained in Chapter IV of Karoubi’s book [51] \(see Sect\. 38\.3\.C\)\. (Anatoly Fomenko 533) - H nC3 \.K\.Z; n/I Z/ D Z2 \.n 3/: (Anatoly Fomenko 533) - Thus, there exists a unique nontrivial stable cohomology operation which maps integral cohomology into integral cohomology and raises dimensions by 3\. This operation acts as (Anatoly Fomenko 533) - orem, that K K e0 \.RP2m / 2 m is spanned by ˇ; ˇ ; : : : ; ˇ , where ˇ D f \. But D 1\. It is easy to see that f D CR , (Anatoly Fomenko 533) - Thus, there exists a unique nontrivial stable cohomology operation which maps integral cohomology into integral cohomology and raises dimensions by 3\. This operation acts as (Anatoly Fomenko 533) - H nC3 \.K\.Z; n/I Z/ D Z2 \.n 3/: (Anatoly Fomenko 533) - Other computations of K-functors in particular, for Grassmann manifolds and flag manifolds, are contained in Chapter IV of Karoubi’s book [51] \(see Sect\. 38\.3\.C\)\. (Anatoly Fomenko 533) - e \.RP / is cyclic\. Corollary of Proof\. Any complex vector bundle over RP2m or RP2mC1 is stably equivalent to the vector bundle kCR , and the vector bundles kCR and lCR are stably equivalent if and only if k l mod 2m \. (Anatoly Fomenko 533) - After we reduce it mod 2, this operation becomes Sq Sq D Sq , and it is natural to denote it by Sq Sq (Anatoly Fomenko 534) - homology Theories H n \.XI Z/ 2 2 ! H n \.XI Z2 / Sq 2 Sq2 ! H nC2 \.XI Z2 / ˇ ˇ ! H nC3 \.XI Z/: After we reduce it mod 2, this operation becomes Sq Sq D Sq , and it is natural to denote it (Anatoly Fomenko 534) - Corollary of Proof\. There exist simply connected finite CW complexes X; Y and a continuous map f W X ! Y which is not a homotopy equivalence, but which induces an isomorphism f W K \.Y/ ! K \.X/\. \( (Anatoly Fomenko 535) - Notice that there exists only partial information about the subsequent differentials\. It is known that for any prime p the differential dr with r < 2p 1 has order not divisible by p, while the differential d2p 1 has order not divisible by p \. See the details in the article by Buchstaber [24]\. (Anatoly Fomenko 535) - A cohomology operation in K-theory \(and in any extraordinary cohomology theory as well\) is defined precisely in the same way as the usual cohomology operation\. Namely, it is a class of homomorphisms X W K\.X/ ! K\.X/ (Anatoly Fomenko 535) - defined for all finite CW complexes X and satisfying the condition of the commutativity of diagrams K\.X/ ˛ ˛X ! K\.X/ \. ? ? ? ? ? \. ? ? ? ? ? y K\.Y/ ˛ y ˛Y ! K\.Y/ for all continuous maps f W Y ! X (Anatoly Fomenko 535) - An approach to the classification of such operations is suggested by the general idea of Sect\. 28\.2, which reduces the classification problem to the problem of the calculation of the K-functor for BU \(with simultaneous overcoming of the difficulties related to BU being an infinite CW complex; see Sect\. 38\.2\.E\)\. All these problems have been long solved; the results related to the complex case are presented in Sect\. 4\.7 of Karoubi’s book \(see Sect\. 38\.3\.C\), while the real case is studied in the article by Anderson [13]\. (Anatoly Fomenko 535) - An approach to the classification of such operations is suggested by the general idea of Sect\. 28\.2, which reduces the classification problem to the problem of the calculation of the K-functor for BU \(with simultaneous overcoming of the difficulties related to BU being an infinite CW complex; see Sect\. 38\.2\.E\)\. All these problems have been long solved; the results related to the complex case are presented in Sect\. 4\.7 of Karoubi’s book \(see Sect\. 38\.3\.C\), while the real case is studied in the article by Anderson [13]\. (Anatoly Fomenko 535) - defined for all finite CW complexes X and satisfying the condition of the commutativity of diagrams K\.X/ ˛ ˛X ! K\.X/ \. ? ? ? ? ? \. ? ? ? ? ? y K\.Y/ ˛ y ˛Y ! K\.Y/ for all continuous maps f W Y ! X (Anatoly Fomenko 535) - A cohomology operation in K-theory \(and in any extraordinary cohomology theory as well\) is defined precisely in the same way as the usual cohomology operation\. Namely, it is a class of homomorphisms X W K\.X/ ! K\.X/ (Anatoly Fomenko 535) - Notice that there exists only partial information about the subsequent differentials\. It is known that for any prime p the differential dr with r < 2p 1 has order not divisible by p, while the differential d2p 1 has order not divisible by p \. See the details in the article by Buchstaber [24]\. (Anatoly Fomenko 535) - s (Anatoly Fomenko 536) - As to the complex case, we restrict ourselves to considering the most important specific cohomology operations in the K-theory\. The readers of Karoubi’s book know that these operations exhaust the whole variety of the cohomology operation in the complex K-theory—in the same sense in which the Steenrod squares and Steenrod powers exhaust the entire variety of ordinary cohomology operations\. (Anatoly Fomenko 537) - A cohomology operation in K-theory should assign to a virtual vector bundle with the base X another virtual vector bundle with the same base X (Anatoly Fomenko 537) - One can associate with a vector bundle a vector bundle ˝ , or ƒ2 , or S2 , or one can take any complex linear representation \(of some dimension N\) of the group GL\.n; C/ [or U\.n/] and assign, with the help of this representation, some N-dimensional complex vector bundle to a given n-dimensional complex vector bundle\. The common flaw of all these constructions is the lack of additivity: (Anatoly Fomenko 537) - Therefore, these constructions cannot be extended to the K-theory\. (Anatoly Fomenko 537) - : \.1 ˚ 2 / ˝ \.1 ˚ 2 / ¤ \.1 ˝ 1 / ˚ \.2 ˝ 2 /, (Anatoly Fomenko 537) - mentioned above\) to compose from these operations an additive combination\. : : be the elementary symmetric polynomials [that is, Let e1 ; e2 ; e3 ek \.x1 ; : : : ; xm / D : : 1i1 <