## Highlights - the Whitehead theorem which states that \.fW X ! Y is a homotopy equivalence if and only if f is an isomorphism on homotopy groups—cf\. Sect\. 9\.2 for a proof\) is true (Philip Griffiths, John Morgan 16) - if X ¤ X\.n/ for any n \(thus X is infinite dimensional\), then X has the weak topology with respect to the X\.n/ ’s meaning that U X is an open set if and only if U \ X\.n/ is open for all n\. (Philip Griffiths, John Morgan 17) - Also, a map fW X ! Y from a CW complex to a space is continuous if and only if its restriction to each skeleton X\.n/ is continuous (Philip Griffiths, John Morgan 17) - \. Any simplicial complex K has the natural structure of a CW complex\. The n-cells of this CW structure are exactly the n-simplices\. Conversely, if X is a CW complex, then there is a simplicial complex K and a homotopy equivalence from K to X (Philip Griffiths, John Morgan 18) - A CW pair \.X; A/ is a pair of spaces A X such that X is obtained from A by attaching cells (Philip Griffiths, John Morgan 18) - \.Of course, as we have already remarked, any simplicial complex is a CW complex\. The converse is true up to homotopy\. (Philip Griffiths, John Morgan 20) - It is frequently easier to work with an inclusion rather than an arbitrary map\. This is always possible up to homotopy equivalence\. Theorem 2\.3\. Given fW X ! Y, there is a space Mf , the mapping cylinder of f, inclusions jW X ! Mf and iW Y ! Mf , and a map W Mf ! Y (Philip Griffiths, John Morgan 20) - f\. Thus, we may replace Y by a homotopy equivalent space in which X is included\.\) (Philip Griffiths, John Morgan 21) - Thus, we may consider any map as an inclusion without leaving the category of CW complexes (Philip Griffiths, John Morgan 21) - \. If W E ! B has the h\. l\. p\. \(D homotopy lifting property\), then it is said to be a fibration\. For any b 2 B, the fiber Fb D l \(b\) is the preimage of the point\. In a fibration, any two fibers are homotopy equivalent provided that the base is path connected (Philip Griffiths, John Morgan 21) - Homotopy Exact Sequence of a Fibration\. Here is the statement (Philip Griffiths, John Morgan 22) - boundary is (Philip Griffiths, John Morgan 23) - Proposition 2\.5\. Let x0 2 X be given and let P\.X; x0 / denote the space of paths in X beginning at x0 \. Then W P\.X; x0 / ! X is a fibration (Philip Griffiths, John Morgan 23) - The cup product has no symmetry properties on the cochain level; nevertheless, it turns out for classes a 2 Hp \.X/ and b 2 Hq \.X/, we have a [ b D \.1/pq b [ a: That is to say, the induced product makes cohomology into an associative, gradedcommutative ring with unit\. (Philip Griffiths, John Morgan 27) - Homology has a co-associative, graded-commutative co-multiplication W Hk \.X/ ! ˚iCjDk Hi \.X/ ˝ Hj \.X /: \(2\) There is a cap product: H p \.X/ ˝ Hq \.X / ! Hqp \.X/: (Philip Griffiths, John Morgan 27) - Theorem 2\.8 \(Universal Coefficient Theorem\)\. 1\. There is a short exact sequence: 0 ! Ext\.Hn1 \.X/; Z/ ! Hn \.X/ ! Hom\.Hn \.X/; Z/ ! 0: 2\. For any abelian group G, there are short exact sequences: f0g ! Hn \.X/ ˝ G ! Hn \.XI G/ ! Tor\.Hn1 \.X/; G/ ! f0g; 0 ! Ext\.Hn1 \.X/; G/ ! Hn \.XI G/ ! Hom\.Hn \.X/; G/ ! 0: (Philip Griffiths, John Morgan 28) - Theorem 2\.9\. Let M be a closed, oriented manifold of dimension n\. Then there is a fundamental class ŒM 2 Hn \.M/ and cap product with this class induces an isomorphism \ŒMW Hq \.M/ ! Hnq \.M/: (Philip Griffiths, John Morgan 28) - 1\. Hn \.X/=Torsion and H n \.X/=Torsion are dual free abelian groups\. 2\. Torsion\.Hn1 \.X// and Torsion\.H n \.X// are Pontrjagen dual finite abelian groups\. (Philip Griffiths, John Morgan 28) - While (Philip Griffiths, John Morgan 29) - is perfect on the quotients of these homology groups by their torsion subgroups\. This is called the intersection pairing (Philip Griffiths, John Morgan 29) - \. For a smooth manifold, the fundamental class ŒM 2 Hn \.M/ can be described as follows (Philip Griffiths, John Morgan 29) - meet transversely\. This means that the intersection is a finite set of points and at each intersection point the tangent spaces of the two simplices are complementary subspaces in the tangent space of M (Philip Griffiths, John Morgan 29) - \. This pairing is called the linking pairing\. (Philip Griffiths, John Morgan 30) - We show that k \.S n / D 0 if k < n and 3 \.S2 / ¤ 0\. (Philip Griffiths, John Morgan 38) - Thus, we have f \.1/ 2 Hn \.X/\. This determines a natural transformation: HW n \.X/ ! Hn \.X/ Œf ! f \.1/ called the Hurewicz homomorphism\. (Philip Griffiths, John Morgan 38) - H\.f/ is also called the degree of f and sometimes denoted by deg\.f/\. (Philip Griffiths, John Morgan 38) - As one consequence of this and the calculations above, we see that excision is false for the homotopy groups\. (Philip Griffiths, John Morgan 39) - 4\.2 The Whitehead Theorem Theorem 4\.2\. Let X and Y be CW complexes with base points, x0 and y0 , being 0-cells\. Let fW \.X; x0 / ! \.Y; y0 / be a map inducing isomorphisms f W n \.X; x0 / Š ! n \.Y; y0 / for all n 0 Suppose that Y is connected\. Then fW X ! Y is a homotopy equivalence\. (Philip Griffiths, John Morgan 39) - Suppose dim X < 1 and n \.X/ D 0 for all n 0\. We shall show that X is contractible\. (Philip Griffiths, John Morgan 39) - There is a relative version of the theorem which states that if \.X; A/ is a CW pair and if n \.X; A/ D 0 for all n, then there is a deformation retraction f1 W X ! A; (Philip Griffiths, John Morgan 40) - In dealing with spaces that are not CW complexes, a map fW X ! Y is not necessarily a homotopy equivalence if it induces an isomorphism in all homotopy groups\. When it does, we say that f is a weak homotopy equivalence\. This is not an equivalence relation, but it generates an equivalence relation weak homotopy equivalence which when restricted to CW complexes is homotopy equivalence by Whitehead’s theorem\. (Philip Griffiths, John Morgan 41) - Theorem 4\.3\. The Hurewicz homomorphism n \.S n / ! Hn \.Sn / is an isomorphism\. (Philip Griffiths, John Morgan 41) - n Theorem 4\.4\. For n > 1, the map HW n \.\_i S / ! Hn \.\_i Sn / Š ˚i Z is an isomorphism\. (Philip Griffiths, John Morgan 43) - Theorem 4\.5\. Let X be a CW complex\. If k \.X/ D 0 for k < n, then \(i\) Q H Q k \.X/ D 0 for k < n \(recall that H H Q H Q is the reduced homology\), and \(ii\) HW n \.X/ ! Hn \.X/ is an isomorphism provided n > 1\. (Philip Griffiths, John Morgan 43) - Corollary 4\.6\. If X and Y are simply connected CW complexes and X f f ! Y induces an isomorphism on homology, then f is a homotopy equivalence\. (Philip Griffiths, John Morgan 45) - Corollary 4\.7\. If X has the homotopy type of an n-dimensional CW complex and if i \.X/ D 0 for i n, then X is contractible (Philip Griffiths, John Morgan 45) - S n \. \(Thus, a simply connected homology sphere \(a homology sphere is a space with the same homology as a sphere\) is homotopy equivalent to a sphere\.\) (Philip Griffiths, John Morgan 46) - If X is a topological manifold, then there is exactly one obstruction to it being triangulated in a homogeneous fashion, an obstruction in H4 \.XI Z=2Z/ (Philip Griffiths, John Morgan 47) - \. Thus, we see that if B is path connected, then all fibers are homotopy equivalent (Philip Griffiths, John Morgan 48) - Thus, we have a representation 1 1 \.B; b0 / ! Auto\. \.b0 // where Auto \. 1 \.b0 // is the group of homotopy classes of homotopy equivalences of 1 \.b0 /\. This representation is the action of 1 \.B; b0 / as homotopy classes of homotopy equivalences on the fiber\. There are induced actions on the homology and cohomology of the fiber\. (Philip Griffiths, John Morgan 48) - 1 Theorem 4\.11\. Let W E ! B be a fibration, and let F D \.b0 /\. There is an exact sequence: ! nC1 \.B; b0 / @ @ ! n \.F; e0 / i i ! n \.E; e0 / ! n \.B; b0 / ! where iW F ,! E is the inclusion\. (Philip Griffiths, John Morgan 48) - [To calculate the higher homotopy groups of S 1 , recall that R1 exp ! S1 is the universal cover\. In general, the unique path lifting property of Q X Q ! X implies that i \. X X Q X/ Q Š i \.X/ if i > 1 if X Q X Q is a covering 1 1 space of X\. Since R is contractible, i \.S (Philip Griffiths, John Morgan 49) - \. n 2 \.CP / Š Z and i \.CPn / D i \.S2nC1 / for i ¤ 2\. This follows immediately from the homotopy long exact sequence of the Hopf fibration (Philip Griffiths, John Morgan 49) - 2 3 \.S / Š Z\. (Philip Griffiths, John Morgan 50) - Let B be the loop space on B\. It is the fiber of PB ! B where PB is the path space of B\. Since PB is contractible, this gives i1 \.B/ Š i \.B/\. (Philip Griffiths, John Morgan 50) - Spectral Sequence of a Fibration (Philip Griffiths, John Morgan 51) - 5\.4 The Leray–Serre Spectral Sequence of a Fibration (Philip Griffiths, John Morgan 55) - That is to say the bundle of deRham cohomology along the fibers has a natural flat connection \(called the Gauss–Manin connection\) (Philip Griffiths, John Morgan 57) - 1\. Complex Projective Space\. We shall compute the cohomology ring of CP n D P n using the Hopf fibration S1 ! S2nC1 ! Pn \. (Philip Griffiths, John Morgan 58) - Such a space is denoted K\.Z; n/\. We shall prove the fundamental result here that H \.K\.Z; 2k/I Q/ Š QŒ’; and H \.K\.Z; 2k C 1/I Q/ Š Q\.“/: \(The first is a polynomial algebra and the second is an exterior algebra\.\) (Philip Griffiths, John Morgan 59) - We define the obstruction cochain Q O Q n / 2 CnC1 \.X; AI n \.Y// as follows O\.f (Philip Griffiths, John Morgan 64) - \. If we reverse the orientation (Philip Griffiths, John Morgan 64) - \(2\) It is 0 if, and only if, fn extends to a map fnC1 W X\.nC1/ [ A ! Y\. (Philip Griffiths, John Morgan 64) - Lemma 6\.2\. The obstruction cocycle Q O Q n / satisfies the following properties: O\.f (Philip Griffiths, John Morgan 64) - Theorem 6\.3\. Given fn W X\.n/ [ A ! Y with Y simply connected, there is a Q n /\. cohomology class O\.fn / 2 HnC1 \.X; AI n \.Y// constructed from the cocycle O\.f This class vanishes if and only if fn jX\.n1/ [ A can be extended to a map fW X\.nC1/ [ A ! Y\. (Philip Griffiths, John Morgan 66) - Thus, the obstructions to constructing a homotopy between two maps fW X ! Y and gW X ! Y, given a fixed homotopy on A, lie in Hn \.X; AI n \.Y//\. (Philip Griffiths, John Morgan 67) - The obstructions to constructing a section lie in H i \.BI i1 \.F//\. (Philip Griffiths, John Morgan 68) - Example A\. The Euler class: If E n ! B is an n-dimensional vector bundle, then a nowhere zero section of E n is the same as a section of the associated sphere bundle S n1 \(E\)\. The first obstruction to finding a section is in H n \.B; n1 \.S n1 // Š H n \.B; Z/\. It is called the Euler class of E n and is an unstable characteristic class of the vector bundle, unstable in the sense that taking the connected sum of the bundle with a trivial bundle kills the class\. (Philip Griffiths, John Morgan 68) - Notice that this shows that if dim B < n, then E n ! B always has a nonzero section, and thus, as a vector bundle E n Š En1 ˚ 1 , where 1 is the trivial line bundle\. (Philip Griffiths, John Morgan 68) - Now let us show that if Y is a CW complex with the same homotopy groups as X, then Y is homotopy equivalent to X (Philip Griffiths, John Morgan 69) - ith preassigned homotopy groups //\. Later we will show that any simply connected CW complex is homotopy equivalent to an iterated fibration of the K\. ; n/\. (Philip Griffiths, John Morgan 70) - Using the K\. ; n/, we can construct f i \.X/ D i g\. Simply take X D ct a i space with preassigned homotopy groups K\. i ; ni //\. Later we will show that any simply connected CW complex is homotopy equivalent to an iterated fibration of the K\. ; n/\. (Philip Griffiths, John Morgan 70) - record one important fact about obstruction classes: the first possible nonzero class is well defined and natural (Philip Griffiths, John Morgan 70) - Theorem 6\.4\. Let \.X; A/ be a CW-pair, and let fW A ! Y be given\. Suppose H i \.X; AI i1 \.Y// D 0 and Hi1 \.X; AI i1 \.Y// D 0 for i n\. Also suppose nC1 1 \.Y/ D 0\. The first obstruction O 2 H \.X; AI n \.Y// to extending f over X is well defined\. It is natural with respect to maps ®W \.X0 ; A0 / ! X; A/\. (Philip Griffiths, John Morgan 70) - The obstruction, denoted Œf; g, is the Whitehead product of f and g\. (Philip Griffiths, John Morgan 71) - In the last section, we saw that there was a natural transformation Œ\.X; A/; \.K\. ; n/; / ! Hn \.X; AI / which assigns to any map fW \.X; A/ ! \.K\. ; n/; /, the primary obstruction to deforming f to a constant map relative to A\. (Philip Griffiths, John Morgan 72) - Let 2 Hn \.K\. ; n/I / be the class corresponding to the identity homomorphism ! (Philip Griffiths, John Morgan 72) - \. It acts as a commutative, associative group multiplication up to homotopy\. Thus, K\. ; n/ is a homotopy commutative, associative H-space\. This map induces an abelian group structure on Œ\.X; A/I \.K\. ; n/; /\. (Philip Griffiths, John Morgan 73) - with this group structure becomes a group isomorphism (Philip Griffiths, John Morgan 73) - A K\. ; n/ fibration is said to be principal if the action of the fundamental group of the base on the fiber is trivial up to homotopy (Philip Griffiths, John Morgan 74) - Lemma 7\.2\. Let \.X; A/ be a CW pair, let pW E ! X be a principal K\. ; n/fibration, and let W A ! E be a section of E over A\. There is a unique obstruction O\.p; / 2 HnC1 \.AI / to extending over all of X\. Given any class O 2 H nC1 \.X; AI / is realized as the obstruction O\.p; / for some principal fibration pW E ! X and for some section W A ! E\. (Philip Griffiths, John Morgan 74) - Over K\. ; n C 1/ we have the principal fibration (Philip Griffiths, John Morgan 74) - Since K\. ; n C 1/ is simply connected, this fibration is principal (Philip Griffiths, John Morgan 74) - This proves all classes arise as obstructions (Philip Griffiths, John Morgan 74) - Suppose that E f f ! B and E0 f 0 ! B are principal K\. ; n/-fibrations (Philip Griffiths, John Morgan 74) - They are said to be equivalent if there is a map ˚W E ! E0 with f0 ı˚ D f and with ˚ inducing a map on the fibers which is compatible up to homotopy with the identification of the fibers with K\. ; n/, meaning that after these identifications, the map given by the restriction of ˚ induces the identity on the nth homotopy group of the fibers (Philip Griffiths, John Morgan 74) - Proposition 7\.3\. Let B and E be simply connected CW complexes\. A map fW E ! B is homotopy equivalent to a principal K\. ; n/-fibration if and only if the following two conditions hold: (Philip Griffiths, John Morgan 75) - Proposition 7\.4\. Let pW E Q EQ ! B be a principal K\. ; n/-fibration over a simplicial complex\. Then there is a simplicial complex E and a homotopy equivalence ®W E ! Q EQ so that the composition p ı ®W E ! B is a simplicial map\. (Philip Griffiths, John Morgan 75) - Definition 7\.5\. We say that §W E ! B is a simplicial model for the principal K\. ; n/-fibration over B (Philip Griffiths, John Morgan 76) - Fibrations and CW structures should be viewed as dual, and a Postnikov tower for a space is a decomposition dual to a cell decomposition (Philip Griffiths, John Morgan 77) - Remarks\. \(1\) For each n let X 0n 0n be the CW complex obtained from X by inductively attaching cells of dimension n C 2 so as to kill all homotopy groups in dimensions n C 1, so that i \.X0n / D 0 for i > n\. The inclusion X Xn induces an isomorphism on i for i n (Philip Griffiths, John Morgan 78) - If fn W X ! Xn is the nth stage of a Postnikov tower for X, then a simple application of obstruction theory shows that fn extends to a map f 0n 0n W X0n 0n ! Xn \. (Philip Griffiths, John Morgan 78) - j\. It is not true in general for inverse systems that taking homotopy groups commutes with taking inverse limits\. It is, however, true for this inverse system since jC1 \.XN / ! jC1 \.XN1 / is onto for all N\. (Philip Griffiths, John Morgan 79) - \) Let us consider the case X D S2 \. Then X2 D K\.Z; 2/ D CP1 ; H3 \.CP1 / D 0 and H4 \.CP1 / D Z\. Hence, 3 \.S2 / D Z\. If we form K\.Z; 3/ ! X3 ? y CP1 with k-invariant the identity H4 \.CP1 / ! Z, then H4 \.X3 / D 0 and H5 \.X3 / D Z=2\. (Philip Griffiths, John Morgan 80) - By the Serre spectral sequence for (Philip Griffiths, John Morgan 80) - spectral sequence for (Philip Griffiths, John Morgan 80) - Hence, 4 \.S2 / D Z=2\. (Philip Griffiths, John Morgan 80) - Continuing in this way gives a \(theoretical\) algorithm for calculating all the higher homotopy groups of S2 \. This calculation has never been done and seemingly is impossibly complicated to do\. It is an amazing theorem of E\. Curtis that i \.S2 / ¤ 0 for all i 2\. (Philip Griffiths, John Morgan 80) - Let A be an abelian group \(usually infinitely generated\)\. Then A may be given the structure of a Q vector space if and only if A Š A ˝Z Z ! A ˝Z Q\. \(This is equivalent to the equation ’x D “ having a unique solution x for all ’ 2 Z f0g and “ 2 A:/ (Philip Griffiths, John Morgan 81) - If 0 ! A1 ! A2 ! A3 ! 0 is a short exact sequence, then if two of the three terms are Q-vector spaces, so is the third one\. (Philip Griffiths, John Morgan 81) - If A is an abelian group and has a composition series A D A0 A1 : : : An 0 with successive quotients Q-vector spaces, then A is a Q-vector space (Philip Griffiths, John Morgan 81) - \(b\) H \.K\.Q; 2n/I Q/ is a Q-polynomial algebra on one generator, that generator being of degree 2n\. \(c\) H \.K\.Q; 2n C 1/I Q/ is a Q-exterior algebra on one generator, that generator being of degree 2n C 1\. (Philip Griffiths, John Morgan 82) - Since homotopy commutes with direct limits (Philip Griffiths, John Morgan 83) - Since homology also commutes with direct limits (Philip Griffiths, John Morgan 83) - There is a comparison theorem for spectral sequences which says that, given two spectral sequences and a map between them inducing an isomorphism on E ; 1 for \.; / ¤ \.0; 0/ and on E 0;q 2 for all q > 0, then it is an isomorphism on E p;q 2 for all \.p; q/ ¤ \.0; 0/\. (Philip Griffiths, John Morgan 83) - Corollary 8\.6\. K\.Z; n/ ! K\.Q; n/ induces an isomorphism on rational cohomology and thus on rational homology (Philip Griffiths, John Morgan 83) - Lemma 8\.8\. If X is a simply connected CW complex, then \.X/ ˝ Q D 0 if and only if Q H Q \.XI Q/ D 0\. H (Philip Griffiths, John Morgan 85) - Definition\. Given X and fW X ! X\.0/ with X\.0/ a Q-space and f satisfying \(1\), \(2\), or \(3\) above \(and hence all of them\), we call fW X ! X\.0/ the localization at 0 of X\. (Philip Griffiths, John Morgan 86) - The terminology comes from the fact that if we localize Z at 0 we get Q\. (Philip Griffiths, John Morgan 86) - We will not, in this course, consider any other localization, although it is possible to localize at any prime ideal\. In fact, there is a Hasse–Minkowski principle which allows one to recover the whole space from its various localizations\. (Philip Griffiths, John Morgan 86) - Lemma 8\.10\. Let Y be a topological space\. There is a CW complex X and a map W X ! Y which induces an isomorphism on all homotopy groups (Philip Griffiths, John Morgan 87) - proves that (Philip Griffiths, John Morgan 88) - This also shows the rational Bott periodicity theorem for the direct limit BU of the BU\.n/, namely, 2 BU\.0/ Š BU\.0/ \. (Philip Griffiths, John Morgan 89) - These examples illustrate a recurring theme—homotopy theory over Q is much simpler than homotopy theory over Z\. We have a chance of getting complete answers over Q, whereas over Z, this is seldom possible; e\.g\., \.S n /\. (Philip Griffiths, John Morgan 89) - Let C \.KI Q/ denote the simplicial cochain complex of K\. There is a map ¡W A \.K/ ! C \.KI Q/ defined by < ¡\.¨/; >D W n ¨\. This map is a map of cochain complexes by Stokes’ theorem \( (Philip Griffiths, John Morgan 91) - If M is a C 1 manifold, then associated to it is the differential algebra of C 1 forms\. It is an algebra over R\. The original theorem of deRham says that the cohomology of this differential algebra is naturally isomorphic \(as a ring\) to the singular cohomology with real coefficients\. The connection between forms on singular cochains is once again achieved by integration (Philip Griffiths, John Morgan 97) - Theorem 9\.7\. Let A C C1 \.M/ denote the DGA of C 1 forms on M\. Then, a map of cochain complexes induced by integration ¡W AC C1 \.M/ ! C \.M; R/ induces an isomorphism of cohomology rings\. (Philip Griffiths, John Morgan 97) - In this chapter, we shall study differential algebras in their own right\. What we are doing, actually, is studying the homotopy theory of differential algebras\. In fact, we shall construct an object \(the minimal model\) which should be considered the Postnikov tower of a differential algebra\. (Philip Griffiths, John Morgan 101) - Definition\. A differential graded algebra \(or DGA for short\), A , is a graded vector space over Q; R, or C, A D ˚p0 Ap ; having \(i\) A differentiation d W A ! AC1 with d2 D 0\. \(ii\) A multiplication A p ˝ Aq ! ApCq satisfying ’“ D \.1/pq “’: \(iii\) d\.’“/ D d’“ C \.1/p ’d“\. (Philip Griffiths, John Morgan 101) - The cohomology H \.X; Q/ of a space is a DGA \(with d D 0\), but the singular cochain complex C \.X; Q/ is not \(the signed commutativity fails\)\. (Philip Griffiths, John Morgan 101) - are considering is the problem of commutative cochains\. This was solved in an abstract manner by Quillen, and in an attempt to better understand this, Sullivan was led to the p\.l\. forms and the connection between differential forms and homotopy type\. (Philip Griffiths, John Morgan 101) - The commutative cochain problem\. Perhaps the main genesis of the theory we are considering is the problem of commutative cochains\. This was solved (Philip Griffiths, John Morgan 101) - one fundamental point was missing in that Whitney only constructs commutative cochains over R, and as already mentioned there is no way to build Postnikov towers over R and thus tie in the commutative cochains with homotopy type\. (Philip Griffiths, John Morgan 102) - Moreover, a somewhat grizzly computation shows that on the cohomology level, we have graded commutativity \(iii\) Œ’p [ Œ“q D \.1/pq Œ“q [ Œ’p \. However, it is obviously false that ’p [ “q D \.1/pq “q [ ’p on the cochain level\. Now one may attempt to modify the formula so as to have all three properties, but all such attempts are doomed to failure since, as realized by Steenrod 35 years ago, the failure to find commutative cochains over Z is reflected in the existence of cohomology operations, such as the Steenrod squares (Philip Griffiths, John Morgan 102) - Definition\. Commutative cochains assign functorially to each simplicial complex X a DGA defined over Q, C \.X/, satisfying \(i\)–\(iii\) above and such that: \(iv\) The cohomology of C \.X/ is H \.XI Q/\. \(v\) Given a subcomplex Y X, we have C \.X/ ! C \.Y/ ! 0: (Philip Griffiths, John Morgan 102) - Theorem\. Let C \.X/ be any solution to the commutative cochain problem over Q\. Then the minimal model M for C \.X/ gives the Q-homotopy type of X\. (Philip Griffiths, John Morgan 102) - This formula leads to the following properties: \(i\) •\.’p [ “q / D •’p [ “q C \.1/p ’p [ •“q \. \(ii\) ’p [ \.“q [ ”r / D \.’p [ “q / [ ”r (Philip Griffiths, John Morgan 102) - Theorem (Philip Griffiths, John Morgan 102) - The usual definition of the cup product ’’ [ “q between a p-cochain ’q and a q-cochain “q is < ’p [ “q ; pCq > D< ’p ; front p-face of pCq > < “q ; back q face of pCq > : (Philip Griffiths, John Morgan 102) - The problem of commutative cochains is equivalent to finding not only the cohomology but also the Q-homotopy type of a space from a cochain complex\. The p\.l\. forms explicitly solve this problem, and moreover, a simple comparison theorem shows that the C 1 forms give the R-homotopy type of a smooth manifold\. (Philip Griffiths, John Morgan 103) - , Whitney essentially showed that any solution to the commutative cochain problem over R satisfying a mild continuity condition is given by integration of suitable differential forms \(the flat forms\) over chains\. Now, almost 25 years later, we have finally understood what he was driving at\. (Philip Griffiths, John Morgan 103) - Definition\. A DGA A is said to be minimal if: \(i\) A is free as a graded-commutative algebra on generators of degrees 2\. \(ii\) d\.A / AC ^ AC where A D ˚k>0 Ak \. (Philip Griffiths, John Morgan 103) - Condition \(i\) means that A is a tensor product of polynomial algebras on generators of even degrees and exterior algebras on generators of odd degrees\. Condition \(ii\) says that d is decomposable\. There is a notion of minimal DGAs which have generators in degree 1 \( (Philip Griffiths, John Morgan 103) - One of the main results of this chapter is that every simply connected DGA has a minimal model\. (Philip Griffiths, John Morgan 103) - Remark\. The construction of M\.A / is motivated by the construction of the Postnikov tower of a space\. In fact, the parallel is quite precise, as we shall see in Chap\. 9\. (Philip Griffiths, John Morgan 103) - Actually the fundamental property of a minimal algebra is that it is an increasing sequence of subalgebras which are nicely related (Philip Griffiths, John Morgan 103) - Definition\. Let A be a DGA\. A Hirsch extension of A is an inclusion A ! A ˝d ƒ\.Vk /: (Philip Griffiths, John Morgan 103) - Given a DGA, A , we wish to construct a minimal model, M\.A /, for A \. By definition this means that M\.A / is a minimal DGA and there is a map ¡W M\.A / ! A of DGAs inducing an isomorphism on cohomology\. (Philip Griffiths, John Morgan 103) - Theorem 10\.3\. Every simply connected DGA has a minimal model (Philip Griffiths, John Morgan 106) - the k-invariant (Philip Griffiths, John Morgan 118) - the transgression in the Serre spectral sequence (Philip Griffiths, John Morgan 118) - principal fibration (Philip Griffiths, John Morgan 118) - Thus, a principal fibration where the fiber is a local Eilenberg–MacLane space \(and the group is of finite dimension over Q\) is completely determined by the homotopy group and ŒdnC1 2 H nC1 \.BI /\. (Philip Griffiths, John Morgan 118) - It is a simple argument to show that the elements of finite order in a nilpotent group form a subgroup Tor N (Philip Griffiths, John Morgan 127) - Mal’cev [14] proved that if N is a nilpotent group, then N=Tor N can be embedded in a uniquely divisible nilpotent group N\.0/ \. \(Uniquely divisible means x n D a has exactly one solution x 2 N for all a 2 N and n 2 ZC \.\) If we take N\.0/ to be minimal among all uniquely divisible nilpotent groups containing N , then N\.0/ is determined up to isomorphism by N\. It is called the Mal’cev completion of N\. (Philip Griffiths, John Morgan 127) - Thus, we have shown that assigning to based simplicial complexes their 1minimal models is a functor from based homotopy category of simplicial complexes to 1-minimal models and base point preserving maps between them (Philip Griffiths, John Morgan 130) - are both closed and oriented\. By the tubular neighborhood theorem there is a neighborhood \.M N/ which is diffeomorphic to a disk bundle over M: (Philip Griffiths, John Morgan 134) - receives an orientation\. By the Thom isomorphism theorem \(see the is a unique class UM 2 Hnk \.\.M N/; @\.M N/I Z/ so that r D0 xercises UM D 1 (Philip Griffiths, John Morgan 134) - There is a form of this duality for manifolds with boundary\. (Philip Griffiths, John Morgan 135) - which is the Lefschetz dual of the class (Philip Griffiths, John Morgan 135) - then the same construction yields a class Q U Q M 2 Hnk \.N/ which is the Lefschetz dual of the class ŒM; @M 2 Hk \. U (Philip Griffiths, John Morgan 135) - A space is said to be formal if the homotopy type of the DGA of forms on the space is the same as the homotopy type of the cohomology ring of the space\. (Philip Griffiths, John Morgan 136) - If X is a closed Riemannian manifold, then there is a canonical map \.H \.X/; d D 0/ ! A \.X/ which assigns to each cohomology class its unique harmonic representative (Philip Griffiths, John Morgan 137) - for X to admit a Riemannian metric in which the wedge product of harmonic forms is harmonic, it must be the case that X is formal \(over R\)\. (Philip Griffiths, John Morgan 137) - There is one class of Riemannian manifolds in which wedge product of harmonic forms is harmonic\. These are the Riemannian locally symmetric spaces\. (Philip Griffiths, John Morgan 137) - This shows that such homotopy types are classified by equivalence classes of symmetric bilinear pairings (Philip Griffiths, John Morgan 140) - It is unknown which pairings arise from closed simply connected smooth 4-manifolds but for closed, simply connected topological 4-manifolds all pairings satisfying Poincaré duality over the integers occur; see [5]\. (Philip Griffiths, John Morgan 140) - The diffeomorphism classification of simply connected 4-manifolds is much more complicated than the classification up to homotopy equivalence, see [4] (Philip Griffiths, John Morgan 140) - 14\.7 Q-Homotopy Type of BUn and Un (Philip Griffiths, John Morgan 140) - The Grassmannian of n-dimensional complex linear subspaces in C 1 is the classifying space for complex n-plane bundles and is denoted BUn \. \(Thus, BUn D limN!1 G\.n; N/\)\. Recall that H \.BUn I Z/ Š ZŒc1 ; c2 ; : : : ; cn (Philip Griffiths, John Morgan 140) - where c` 2 H2` \.BU˜ I Z/ is the `th Chern class of the universal vector bundle (Philip Griffiths, John Morgan 140) - Bott periodicity is the result 2i \.BU/ Z 2iC1 \.BU/ D0 together with the fact that cn \.n 1/Š W 2n \.BU/ !Z is an isomorphism (Philip Griffiths, John Morgan 141) - Let Un be the unitary group\. Recall that H \.Un ; Z/ Š Zfx1 ; x3 ; : : : ; x2n1 g is an exterior algebra with odd-dimensional generators (Philip Griffiths, John Morgan 141) - Here, S\.n; 2n/ is the Stiefel manifold of n-frames in 2n-space and Gr\.n; 2n/ is the Grassmannian of n-planes in 2n-space (Philip Griffiths, John Morgan 142) - It follows that MXY Š MX ˝ MY \. This should be viewed as a generalization of the Künneth theorem\. It includes the Künneth theorem \(by taking cohomology\)\. It also includes the rational or real form of the result that i \.XY/ Š i \.X/˚ i \.Y/, since I\.MX ˝ MY / Š I\.MX / ˚ I\.MY /\. (Philip Griffiths, John Morgan 142) - there is defined the Massey triple product (Philip Griffiths, John Morgan 143) - Definition\. A Hodge structure of weight m is given by Q-vector space HQ together with a Hodge decomposition HC D ˚pCqDm Hp; q H p;q D H q; p of its complexification HC WD HQ ˝ C D HQ ˝Q C\. (Philip Griffiths, John Morgan 143) - We shall show that the lemma implies that all Massey triple products are zero on a compact Kähler manifold (Philip Griffiths, John Morgan 144) - Remark\. The above is a special case of the theorem proved in [3] that the rational homotopy type of a compact Kähler manifold is a formal consequence of the cohomology ring \(cf\. Sect\. 14\.4 above for the definition of formal\)\. This result, in turn, is generalized to noncompact algebraic varieties in (Philip Griffiths, John Morgan 145) - In Chap\. 12, we made explicit the duality between minimal DGAs over the rationals and rational Postnikov towers\. We showed that the minimal model of the p\.l\. forms of a simplicial complex, A \.X/, is dual to the rational Postnikov tower of X\. (Philip Griffiths, John Morgan 146) - The obstruction to lifting fW X ! C0 is f \.k/ where k 2 HnC1 \.C0 I / is the k-invariant of the fibration (Philip Griffiths, John Morgan 149) - Definition\. Let C1 and C2 be categories and let FW C1 ! C2 be a functor\. F is said to be an equivalence of categories if the following hold: 1\. Every object B of C2 is isomorphic to an object F\.A/ for some object A of C1 \. 2\. For all objects A1 and A2 of C1 the map induced by F HomC1 \.A1 ; A2 / ! HomC2 \.F \.A1 /; F\.A2 // is a bijection (Philip Griffiths, John Morgan 154) - Given a category C and a set of morphisms S of C that include all isomorphisms and are closed under compositions, we define the localization of C at S by formally inverting all morphisms in S\. Thus, the objects of the new category are the same as the objects of C, but the morphisms from A to B in the new category consist of a string of morphisms: A D A0 ! A1 A2 ! Ak D B; where all the arrows to the ‘left’ are elements of S (Philip Griffiths, John Morgan 154) - That is to say, the rational homotopy category of simply connected spaces of finite type is isomorphic to the homotopy category of simply connected DGAs over Q with finite-dimensional cohomology in each degree, and the isomorphism between these rational homotopy categories is given by associating to a simplicial complex is p\.l\. forms (Philip Griffiths, John Morgan 154) - X\. The fact that the chain complex Q\.X/ computes the homology of X implies that this map is a weak homotopy equivalence; i\.e\., this map induces an isomorphism on the homology and homotopy groups\. If X is the homotopy type of a CW complex, then it is a homotopy equivalence\. (Philip Griffiths, John Morgan 157) - Let A be a connected DGA over Q with differential d, let n be a natural number, and let V be a finite-dimensional rational vector space\. Given a map “W V ! AnC1 whose image lies in the kernel of d, we have a Hirsch extension in degree n of DGA’s by forming A O A D d; and d\.v/ ! A ˝“ ƒ \.V /; with dj O D “\.v/: Here ƒ \.V / is the free graded-commutative algebra generated by V in degree n\. We denote the Hirsch extension by O A O A (Philip Griffiths, John Morgan 158) - Sullivan showed that if fW E ! B is a Serre fibration with fiber K\. ; n/ with 1 \.B/ acting trivially on the homology of the fiber, with V D ˝Q being finite dimensional, and if W A ! \.B/ is a map of DGA’s inducing an isomorphism on cohomology, then there is a Hirsch extension O A O A D \.A ˝ ƒ \.V /; d/ O and a O A map W O O A O A ! \.E/ with j O A D f ı inducing an isomorphism on cohomology\. The degree of the extension is n\. (Philip Griffiths, John Morgan 159) - Suppose that fW M ! B is a proper smooth submersion of smooth manifolds\. Then fW M ! B is a smoothly locally trivial fibration\. In this case, there is a filtration on the \(smooth\) differential forms \.M/ defined by letting F k \. n \.M// be the subspace of forms ¨ such that for any point p in M and any collection of tangent vectors £1 ; : : : ; £n with at least n k C 1 of the £i being vertical \(i\.e\., in the kernel of df\), we have ¨p \.£1 ; : : : ; £n / D 0\. This is a multiplicative filtration preserved by exterior differentiation, and hence, there is an induced spectral sequence whose E p;q 1 -term is identified with p \.BI Hq \.FI R// where the coefficients are the local system given by the cohomology of the fibers of the map\. (Philip Griffiths, John Morgan 160) - Before Sullivan’s work on differential forms and rational homotopy theory, in [19] Quillen had established algebraic models for rational homotopy theory\. Quillen worked dually from the way Sullivan does: Instead of using differential forms as the basic model, Quillen uses differential graded Lie algebras\. One can, as Quillen does, associate a differential graded, co-commutative co-algebra to a differential graded Lie algebra\. Dualizing this produces a differential graded algebra computing the rational homotopy type and thus in some sense solves the rational commutative cochain problem\. Sullivan’s construction was much more directly tied to differential forms on the space, whereas Quillen’s went through more homotopy theoretic constructions, i\.e\., the simplicial set \(of singular simplices\) associated with the loop space of the given space (Philip Griffiths, John Morgan 168) - Definition\. A graded Lie algebra \(GLA\) is a graded vector space L D ˚k Lk together with a homomorphism (Philip Griffiths, John Morgan 168) - 17\.3 The Bar Construction (Philip Griffiths, John Morgan 170) - There are many variants of the bar construction\. The one that is relevant for us here goes from a differential graded Lie algebra to a differential graded co-commutative, co-associative co-algebra\. (Philip Griffiths, John Morgan 170) - Theorem 17\.1\. Quillen assigns a reduced DGLA, \.L\.X/; @/, to a simply connected topological space X in a functorial way\. Applying the bar construction to \.L\.X/; @/ produces a co-commutative differential graded co-algebra Q \.X/\. Dualizing this produces a DGA Q \.X/ functorially associated to X\. Suppose that the rational homotopy groups of X in each degree are finite dimensional\. If K is a simplicial complex and fW K ! X is a weak homotopy equivalence, then the minimal model MK for the p\.l\. forms A \.K/ and the minimal model MX for the DGA Q \.X/ are identified by a map determined up to homotopy by the homotopy class of f\. (Philip Griffiths, John Morgan 172) - Some examples are in order: 1\. A simplicial set (Philip Griffiths, John Morgan 174) - The first and motivating example is that of the simplicial set of a topological space X\. The simplicial set associated to X has Sn \.X/ equal to the set of singular n-simplices of X, i\.e\., the set of continuous maps n ! X\. (Philip Griffiths, John Morgan 174) - As we remarked above, the homotopy groups of a simplicial set K are easiest to define when the simplicial set satisfies is a Kan complex (Philip Griffiths, John Morgan 174) - In general, to define the homotopy groups, one has to replace a simplicial set by an equivalent Kan complex, which can always be done\. (Philip Griffiths, John Morgan 174) - Any simplicial group is a Kan complex, and therefore, we can directly define the homotopy groups (Philip Griffiths, John Morgan 175) - Associated to a Lie algebra we have its Universal Enveloping Algebra which is a Hopf algebra, though the algebra structure is not commutative (Philip Griffiths, John Morgan 176) - The Poincaré–Birkhoff–Witt theorem says that L embeds in its Universal Enveloping Algebra as the space of primitive elements \(those satisfying \.a/ D a˝1C1˝a\)\. In fact, there is an equivalence of categories between Lie algebras and Hopf algebras (Philip Griffiths, John Morgan 176) - The primary example for us comes from the rational group ring QG of a group G\. (Philip Griffiths, John Morgan 177) - An element x is group-like if \.x/ D x ˝ x\. Group-like elements in a complete Hopf algebra are congruent to 1 modulo in the augmentation ideal (Philip Griffiths, John Morgan 178) - \. To a topological space X, we associated the simplicial group G\.X/ which is the analogue of the loop space applied to the simplicial set of \(1-trivial\) singular simplices in X\. (Philip Griffiths, John Morgan 178) - Quillen passes from the category of reduced simplicial groups to the category of reduced simplicial Hopf algebras over Q that are complete with respect to the augmentation ideal\. (Philip Griffiths, John Morgan 178) - \. This passage is the application of the simplicial functor to the b map that associates to a group the completion QG of the rational group ring QG with respect to the augmentation ideal\. Since G0 D feg, the simplicial Hopf algebra with respect to the augmentation ideal (Philip Griffiths, John Morgan 178) - There is an adjoint to this functor that associates to a reduced, complete Hopf algebra its group of group-like elements (Philip Griffiths, John Morgan 178) - Lastly, we pass from a simplicial Lie algebra to a differential graded Lie algebra as follows (Philip Griffiths, John Morgan 178) - This then is Quillen’s functorial assignment of a DGLA to a simply connected topological space\. As we have already remarked, we can apply the bar construction to pass from a DGLA to a co-commutative, co-associative co-algebra and then by duality to a DGA\. The resulting DGA has the same minimal model as the p\.l\. forms on a simplicial complex homotopy equivalent to X (Philip Griffiths, John Morgan 179) - These developments began with A1 -spaces and A1 -structures in the 1960s and have been developed in the intervening years to what now goes under the name of the theory of operads (Philip Griffiths, John Morgan 180) - intervening years to what now goes under the name of the theory of operads\. One of the first results of this theory was Stasheff’s characterization [24] of those topological spaces that are homotopy equivalent to loop spaces as being A1 -spaces\. \(Indeed, an A1 -structure on a space determines the homotopy type of its classifying space, that is to say, its de-looping\.\) (Philip Griffiths, John Morgan 180) - These ideas, originating in homotopy theory, have also been crucial recently for developments in symplectic geometry, in particular in formulating the symplectic side of the mirror symmetry conjecture which originated in physics\. One now formulates the appropriate object associated to a symplectic manifold as an A1 -category of Lagrangian submanifolds of the symplectic manifold, the Fukaya category (Philip Griffiths, John Morgan 180) - An \(nonsymmetric\) operad is a collection of sets fPn gn1 together with functions, defined for each \.n; j/ 1 and i with 1 i n, OŒ\.n; j/I iW Pj Pn ! PnCj1 and also a distinguished element 1 2 P1 that acts as a unit for the operations OŒ\.n; 1/I i for every 1 i n\. These operations are required to satisfy the following natural composition axioms (Philip Griffiths, John Morgan 181) - An (Philip Griffiths, John Morgan 181) - Definition\. Given an operad P D fPn g, the set X is an algebra over the operad if there is a morphism of operads from P ! End\.X/\. Said another way, for each pn 2 Pn determines a function f\.pn /W Xn ! X in such a way that under this correspondence the structure maps of the operad P are mapped to the corresponding compositions of functions (Philip Griffiths, John Morgan 182) - A space X is an algebra over the associahedron operad if there are continuous maps Kn Xn ! X compatible with the structure maps of the operad\. Said another way, End\.X/ is naturally an operad of topological spaces and X is an algebra over the associahedron operad if there is a map of topological operads K ! End\.X/\. These objects are also called A1 -spaces\. (Philip Griffiths, John Morgan 184) - Stasheff not only introduced the associahedra, he proved that an A1 -space X has a classifying space BX, so that an A1 -space is a loop space \(up to homotopy equivalence\)\. (Philip Griffiths, John Morgan 184) - There is an algebraic version of A1 -spaces called A1 -algebras\. These algebras can be defined as algebras \(in the category of graded vector spaces\) over the operad in the induced category of differential graded vector spaces consisting of cellular chain complexes of the associahedra with the induced structure maps from the associahedra operad \(thought of as an operad in the category of cell complexes and cellular maps\)\. (Philip Griffiths, John Morgan 184) - Definition\. An A1 -algebra is a graded vector space V, and for every n 1, a map (Philip Griffiths, John Morgan 184) - The motivating example occurs in [6]\. The objects formalized from what they did are called Fukaya categories (Philip Griffiths, John Morgan 185) - The motivating example was to formalize the algebraic structure associated with the Lagrangian Floer homology of a symplectic manifold\. (Philip Griffiths, John Morgan 185) - Given a symplectic manifold M, we form an A1 -category whose objects are Lagrangian submanifolds of M\. For Lagrangian submanifolds L and L 0 by definition, Hom\.L; L 0 / is the Floer cochain complex CF\.L; L 0 /\. (Philip Griffiths, John Morgan 186) - Under suitable hypotheses \(see [20] for one such context\), these structure maps produce an A1 -category structure, which is the Fukaya category of this symplectic manifold (Philip Griffiths, John Morgan 186) - To do this we introduce the graded-commutative analogue of A1 -algebras; see [10]\. These are the so-called C1 -algebras\. They are the quotient of A1 -algebras when one forces graded commutativity (Philip Griffiths, John Morgan 186) - \(The Kozul sign for a switch of two neighboring elements a˝b 7! b ˝a is \.1/jajjbj ; this generates the Kozul sign for arbitrary permutations\.\) (Philip Griffiths, John Morgan 187) - Let A be a DGA\. We can view this as a C1 -algebra with m1 D d, with m2 being the multiplication in the algebra, and with mi D 0 for all i > 2\. We have the following theorem of Kadeishvili [10]\. It says that up to homotopy equivalence, we can collapse A onto its cohomology at the expense of endowing the cohomology with a possibly nontrivial C1 -structure\. (Philip Griffiths, John Morgan 187) - There is an analogous theorem in the noncommutative case that says a \(not necessarily commutative\) DGA is equivalent in an appropriate homotopy category to an A1 -algebra structure on its cohomology (Philip Griffiths, John Morgan 188) - \. Using Bott periodicity and denoting by S n X the n-fold suspension of X, one obtains K-theory with K n \.X/ D defn: K \.S n X/ (Philip Griffiths, John Morgan 219) - The comparison theorem for spectral sequences (Philip Griffiths, John Morgan 219) - \(Hint: This problem is a good exercise in Postnikov towers\. (Philip Griffiths, John Morgan 221) - Homotopy of wedges of 2-spheres (Philip Griffiths, John Morgan 221) - Homotopy of wedges of 2-spheres (Philip Griffiths, John Morgan 221)