## Highlights - A treatment more closely attuned to the needs of algebraic geometers and analysts would include Č Čech cohomology on the one hand and de Rham cohomology and perhaps Morse homology on the other\. A treatment more closely attuned to the needs of algebraic topologists would include spectral sequences and an array of calculations with them (J. Peter May 9) - These changes reflect in part an enormous internal development of algebraic topology over this period, one which is largely unknown to most other mathematicians, even those working in such closely related fields as geometric topology and algebraic geometry (J. Peter May 9) - The study of generalized homology and cohomology theories pervades modern algebraic topology\. These theories satisfy the excision axiom\. One constructs most such theories homotopically, by constructing representing objects called spectra, and one must then prove that excision holds\. There is a way to do this in general that is no more difficult than the standard verification for singular homology and cohomology\. (J. Peter May 9) - A defect of nearly all existing texts is that they do not go far enough into the subject to give a feel for really substantial applications: the reader sees spheres and projective spaces, maybe lens spaces (J. Peter May 10) - A function p : X −→ Y is continuous if it takes nearby points to nearby points\. Precisely, p−1 \(U \) is open if U is open\. If X and Y are metric spaces, this means that, for any x ∈ X and ε > 0, there exists δ > 0 such that p\(Uδ \(x\)\) ⊂ Uε \(p\(x\)\)\. (J. Peter May 13) - The further one goes in the subject, the more elaborate become the constructions A and the more horrendous become the relevant calculational techniques (J. Peter May 13) - Let X be a space\. Two paths f, g : I −→ X from x to y are equivalent if they are homotopic through paths from x to y (J. Peter May 14) - Now suppose given two maps p, q : X −→ Y and a homotopy h : p ≃ q\. We would like to conclude that p∗ = q∗ , but this doesn’t quite make sense because homotopies needn’t respect basepoints (J. Peter May 15) - Theorem\. π1 \(S 1 , 1\) ∼ = Z\. (J. Peter May 16) - the first map induces the evident identification of I/∂I with S 1 (J. Peter May 17) - This lifting of paths works generally\. For any path f : I −→ S 1 with f \(0\) = 1, there is a unique path ˜ f˜ : I −→ R such that ˜ f˜\(0\) = 0 and p ◦ f ˜ f˜ = f \. (J. Peter May 17) - The Brouwer fixed point theorem (J. Peter May 18) - Proposition\. There is no continuous map r : D 2 −→ S 1 such that r ◦ i = id\. (J. Peter May 18) - Theorem \(Brouwer fixed point theorem\)\. Any continuous map f : D 2 −→ D 2 has a fixed point\. Proof\. Suppose that f \(x\) 6= x for all x\. Define r\(x\) ∈ S 1 to be the intersection with S 1 of the ray that starts at f \(x\) and passes through x\. Certainly r\(x\) = x if x ∈ S 1 \. By writing an equation for r in terms of f , we see that r is continuous\. This contradicts the proposition\. (J. Peter May 18) - Theorem \(Fundamental theorem of algebra\)\. Let f \(x\) = x n + c1 x n−1 + · · · + cn−1 x + cn be a polynomial with complex coefficients ci , where n > 0\. Then there is a complex number x such that f \(x\) = 0\. Therefore there are n such complex numbers \(counted with multiplicities\)\. Proof\. Using f \(x\)/\(x−c\) for a root c, we see that the last statement will follow by induction from the first (J. Peter May 18) - Then h is a homotopy from the constant map at f \(0\)/|f \(0\)| to f ˆ fˆ, and we conclude that deg\( ˆ fˆ\) = 0\. (J. Peter May 18) - 3\. Natural transformations A natural transformation α : F −→ G between functors C −→ D is a map of functors\. It consists of a morphism αA : F \(A\) −→ G\(A\) for each object A of C such that the following diagram commutes for each morphism f : A −→ B of C : (J. Peter May 22) - For example, if F : S −→ A b is the functor that sends a set to the free Abelian group that it generates and U : A b −→ S is the forgetful functor that sends an Abelian group to its underlying set, then we have a natural inclusion of sets S −→ U F \(S\)\. (J. Peter May 22) - The functors F and U are left adjoint and right adjoint to each other, in the sense that we have a natural isomorphism A b\(F \(S\), A\) ∼ = ∼ = S \(S, U \(A\)\) (J. Peter May 22) - for a set S and an Abelian group A\. This just expresses the “universal property” of free objects: a map of sets S −→ U \(A\) extends uniquely to a homomorphism of groups F \(S\) −→ A\. (J. Peter May 22) - The fundamental group is a homotopy invariant functor on T , in the sense that it factors through a functor hT −→ G \. (J. Peter May 22) - When we have a \(suitable\) relation of homotopy between maps in a category C , we define the homotopy category hC to be the category with the same objects as C but with morphisms the homotopy classes of maps (J. Peter May 22) - A homotopy equivalence in U is an isomorphism in hU \. (J. Peter May 22) - Less mysteriously, a map f : X −→ Y is a homotopy equivalence if there is a map g : Y −→ X such that both g ◦ f ≃ id and f ◦ g ≃ id\. (J. Peter May 22) - Functors carry isomorphisms to isomorphisms, so we see that a based homotopy equivalence induces an isomorphism of fundamental groups\. (J. Peter May 22) - Proposition\. If f : X −→ Y is a homotopy equivalence, then f∗ : π1 \(X, x\) −→ π1 \(Y, f \(x\)\) is an isomorphism for all x ∈ X\. (J. Peter May 22) - A space X is said to be contractible if it is homotopy equivalent to a point\. (J. Peter May 23) - For this purpose, we define the “fundamental groupoid” Π\(X\) of a space X to be the category whose objects are the points of X and whose morphisms x −→ y are the equivalence classes of paths from x to y\. Thus the set of endomorphisms of the object x is exactly the fundamental group π1 \(X, x\)\. (J. Peter May 23) - The term “groupoid” is used for a category all morphisms of which are isomorphisms\. (J. Peter May 23) - There is a useful notion of a skeleton skC of a category C \. This is a “full” subcategory with one object from each isomorphism class of objects of C , “full” meaning that the morphisms between two objects of skC are all of the morphisms between these objects in C \. The inclusion functor J : skC −→ C is an equivalence of categories (J. Peter May 23) - Let D be a small category and let C be any category\. A D-shaped diagram in C is a functor F : D −→ C \. A morphism F −→ F ′ of D-shaped diagrams is a natural transformation, and we have the category D[C ] of D-shaped diagrams in C (J. Peter May 24) - The colimit, colim F , of a D-shaped diagram F is an object of C together with a morphism of diagrams ι : F −→ colim F that is initial among all such morphisms\. (J. Peter May 24) - This means that if η : F −→ A is a morphism of diagrams, then there is a unique map η̃ : colim F −→ A in C such that η̃ ◦ ι = η\. Diagrammatically, this property is expressed by the assertion that, for each map d : D −→ D′ in D, we have a commutative diagram F (J. Peter May 24) - The limit of F is defined by reversing arrows: it is an object lim F of C together with a morphism of diagrams π : lim F −→ F that is terminal among all such morphisms (J. Peter May 24) - If D is a set regarded as a discrete category \(only identity morphisms\), then colimits and limits indexed on D are coproducts and products indexed on the set D\. Coproducts are disjoint unions in S or U , wedges \(or one-point unions\) in T , free products in G , and direct sums in A b\. Products are Cartesian products in all of these categories; more precisely, they are Cartesian products of underlying sets, with additional structure\. (J. Peter May 24) - If D is the category displayed schematically as eo /f or // d′ , d d where we have displayed all objects and all non-identity morphisms, then the colimits indexed on D are called pushouts or coequalizers, respectively\. Similarly, if D is displayed schematically as e /do f or // d′ , d (J. Peter May 24) - then the limits indexed on D are called pullbacks or equalizers, respectively (J. Peter May 25) - A category is said to be cocomplete if it has all colimits, complete if it has all limits\. The categories S , U , T , G , and A b are complete and cocomplete\. If a category has coproducts and coequalizers, then it is cocomplete, and similarly for completeness\. The proof is a worthwhile exercise\. (J. Peter May 25) - The proof well illustrates how to manipulate colimits formally\. (J. Peter May 25) - Any compact surface is homeomorphic to a sphere, or to a connected sum of tori T 2 = S 1 × S 1 , or to a connected sum of projective planes RP 2 = S 2 /Z2 \(where we write Z2 = Z/2Z\) (J. Peter May 28) - Lemma\. For based spaces X and Y , π1 \(X × Y \) ∼ = ∼ = π1 \(X\) × π1 \(Y \)\. (J. Peter May 28) - Proposition\. Let X = U ∪V , where U , V , and U ∩V are path connected open neighborhoods of the basepoint of X and V is simply connected\. Then π1 \(U \) −→ π1 \(X\) is an epimorphism whose kernel is the smallest normal subgroup of π1 \(U \) that contains the image of π1 \(U ∩ V \)\. (J. Peter May 28) - We shall later use the following application of the van Kampen theorem to prove that any group is the fundamental group of some space\. We need a definition\. Definition\. A space X is said to be simply connected if it is path connected and satisfies π1 \(X\) = 0\. (J. Peter May 28) - However, I know of no published source for the use that we shall make of the orbit category O\(π1 \(B, b\)\) in the classification of coverings of a space B (J. Peter May 29) - This point of view gives us the opportunity to introduce some ideas that are central to equivariant algebraic topology, the study of spaces with group actions (J. Peter May 29) - While the reader is free to think about locally contractible spaces, weaker conditions are appropriate for the full generality of the theory of covering spaces\. A space X is said to be locally path connected if for any x ∈ X and any neighborhood U of x, there is a smaller neighborhood V of x each of whose points can be connected to x by a path in U \. This is equivalent to the seemingly more stringent requirement that the topology of X have a basis consisting of path connected open sets\. In fact, if X is locally path connected and U is an open neighborhood of a point x, then the set V = {y | y can be connected to x by a path in U } is a path connected open neighborhood of x that is contained in U (J. Peter May 29) - Observe that if X is connected and locally path connected, then it is path connected (J. Peter May 29) - Definition\. A map p : E −→ B is a covering \(or cover, or covering space\) if it is surjective and if each point b ∈ B has an open neighborhood V such that each component of p−1 \(V \) is open in E and is mapped homeomorphically onto V by p (J. Peter May 29) - Any homeomorphism is a cover\. A product of covers is a cover\. The projection R −→ S 1 is a cover (J. Peter May 29) - Theorem \(Unique path lifting\) (J. Peter May 30) - Definition\. A covering p : E −→ B is regular if p∗ \(π1 \(E, e\)\) is a normal subgroup of π1 \(B, b\)\. It is universal if E is simply connected (J. Peter May 30) - As we shall explain in §4, for a universal cover p : E −→ B, the elements of Fb are in bijective correspondence with the elements of π1 \(B, b\)\. (J. Peter May 30) - Example\. For n ≥ 2, S n is a universal cover of RP n \. Therefore π1 \(RP n \) has only two elements\. There is a unique group with two elements, and this proves our earlier claim that π1 \(RP n \) = Z2 \. (J. Peter May 30) - As e′ runs through Fb , the groups p∗ \(π1 \(E, e′ \)\) run through all conjugates of p∗ \(π1 \(E, e\)\) in π1 \(B, b\)\. (J. Peter May 30) - A path f : I −→ B with f \(0\) = b lifts uniquely to a path g : I −→ E (J. Peter May 30) - Let p : E −→ B be a covering (J. Peter May 30) - The category x\C of objects under x has objects the maps f : x −→ y in C ; for objects f : x −→ y and g : x −→ z, the morphisms γ : f −→ g in x\C are the morphisms γ : y −→ z in C such that γ ◦ f = g : x −→ z\. (J. Peter May 30) - \(ii\) Let C be a small groupoid\. Define the star of x, denoted St\(x\) or StC \(x\), to be the set of objects of x\C , that is, the set of morphisms of C with source x\. (J. Peter May 31) - \(iii\) Let E and B be small connected groupoids\. A covering p : E −→ B is a functor that is surjective on objects and restricts to a bijection (J. Peter May 31) - For an object b of B, let Fb denote the set of objects of E such that p\(e\) = b\. Then p−1 \(St\(b\)\) is the disjoint union over e ∈ Fb of St\(e\)\. (J. Peter May 31) - Proposition\. If p : E −→ B is a covering of spaces, then the induced functor Π\(p\) : Π\(E\) −→ Π\(B\) is a covering of groupoids (J. Peter May 31) - The fibers Fb of a covering of groupoids are related by translation functions (J. Peter May 31) - Proposition\. Any two fibers Fb and Fb′ of a covering of groupoids have the same cardinality\. Therefore any two fibers of a covering of spaces have the same cardinality\. (J. Peter May 31) - The isotropy group Gs of a point s is the subgroup {g|gs = s} of G\. An action is free if gs = s implies g = e, that is, if Gs = e for every s ∈ S\. (J. Peter May 32) - An action is transitive if for every pair s, s′ of elements of S, there is an element g of G such that gs = s′ \. (J. Peter May 32) - When G acts transitively on a set S, we obtain an isomorphism of G-sets between S and the G-set G/Gs for any fixed s ∈ S by sending gs to the coset gGs (J. Peter May 32) - For a subgroup H of G, let N H denote the normalizer of H in G and define W H = N H/H\. Such quotient groups W H are sometimes called Weyl groups (J. Peter May 32) - Lemma\. Let G act transitively on a set S, choose s ∈ S, and let H = Gs \. Then W H is isomorphic to the group AutG \(S\) of automorphisms of the G-set S\. (J. Peter May 32) - Lemma\. A G-map α : G/H −→ G/K has the form α\(gH\) = gγK, where the element γ ∈ G satisfies γ −1 hγ ∈ K for all h ∈ H\. (J. Peter May 32) - Definition\. The category O\(G\) of canonical orbits has objects the G-sets G/H and morphisms the G-maps of G-sets\. (J. Peter May 32) - Lemma\. The category O\(G\) is isomorphic to the category G whose objects are the subgroups of G and whose morphisms are the distinct subconjugacy relations γ −1 Hγ ⊂ K for γ ∈ G\. (J. Peter May 32) - A covering of groupoids is universal if and only if π\(B, b\) acts freely on Fb , and then Fb is isomorphic to π\(B, b\) as a π\(B, b\)-set\. Specializing to covering spaces, this sharpens our earlier claim that the elements of Fb and π1 \(B, b\) are in bijective correspondence\. (J. Peter May 33) - Definition\. A map g : E −→ E ′ of coverings of B is a functor g such that the following diagram of functors is commutative (J. Peter May 34) - Lemma\. A map g : E −→ E ′ of coverings is itself a covering (J. Peter May 34) - Corollary\. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover\. (J. Peter May 34) - We have given an algebraic classification of all possible covers of B: there is at most one isomorphism class of covers corresponding to each conjugacy class of subgroups of π\(B, b\) (J. Peter May 35) - We begin with the following result, which deserves to be called the fundamental theorem of covering space theory and has many other applications (J. Peter May 36) - Definition\. A map g : E −→ E ′ of coverings over B is a map g such that the following diagram is commutative: (J. Peter May 36) - In particular, two maps of covers g, g ′ : E −→ E ′ coincide if g\(e\) = g ′ \(e\) for any one e ∈ E\. (J. Peter May 37) - Corollary\. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover\. (J. Peter May 37) - We hasten to add that the theorem above is atypical of algebraic topology\. It is not usually the case that algebraic invariants like the fundamental group totally determine the existence and uniqueness of maps of topological spaces with prescribed properties (J. Peter May 37) - Corollary\. Let p : E −→ B be a covering and choose b ∈ B and e ∈ Fb \. Write G = π1 \(B, b\) and H = p∗ \(π1 \(E, e\)\)\. Then Aut\(E\) is isomorphic to the group of automorphisms of the G-set Fb and therefore to the group W H\. If p is regular, then Aut\(E\) ∼ = ∼ = G/H\. If p is universal, then Aut\(E\) ∼ = ∼ = G (J. Peter May 37) - We have now given an algebraic classification of all possible covers of B: there is at most one isomorphism class of covers corresponding to each conjugacy class of subgroups of π1 \(B, b\)\. (J. Peter May 38) - We show here that all of these possibilities are actually realized\. (J. Peter May 38) - Again, while it suffices to think in terms of locally contractible spaces, appropriate generality demands a weaker hypothesis\. We say that a space B is semi-locally simply connected if every point b ∈ B has a neighborhood U such that π1 \(U, b\) −→ π1 \(B, b\) is the trivial homomorphism (J. Peter May 38) - Theorem\. If B is connected, locally path connected, and semi-locally simply connected, then B has a universal cover\. (J. Peter May 38) - Definition\. A G-space X is a space X that is a G-set with continuous action map G × X −→ X\. Define the orbit space X/G to be the set of orbits {Gx|x ∈ X} with its topology as a quotient space of X\. (J. Peter May 39) - The definition makes sense for general topological groups G\. However, our interest here is in discrete groups G, for which the continuity condition just means that action by each element of G is a homeomorphism (J. Peter May 39) - Proposition\. Let p : E −→ B be a cover such that Aut\(E\) acts transitively on Fb \. Then the cover p is regular and E/ Aut\(E\) is homeomorphic to B\. (J. Peter May 39) - A map f : X −→ Y is said to be a local homeomorphism if every point of X has an open neighborhood that maps homeomorphically onto an open set in Y \. (J. Peter May 40) - Let X be a G-space, where G is a \(discrete\) group\. For a subgroup H of G, define X H = {x|hx = x for all h ∈ H} ⊂ X; X H is the H-fixed point subspace of X (J. Peter May 40) - We define graphs, describe their homotopy types, and use them to show that a subgroup of a free group is free and that any group is the fundamental group of some space (J. Peter May 43) - Observe that a graph is a locally contractible space: any neighborhood of any point contains a contractible neighborhood of that point\. Therefore a connected graph has all possible covers\. (J. Peter May 43) - A graph is finite if it has only finitely many vertices and edges or, equivalently, if it is a compact space (J. Peter May 43) - A graph is locally finite if each vertex is a boundary point of only finitely many edges or, equivalently, if it is a locally compact space\. (J. Peter May 43) - A subspace A of a space X is a deformation retract if there is a homotopy h : X × I −→ X such that h\(x, 0\) = x, h\(a, t\) = a, and h\(x, 1\) ∈ A for all x ∈ X, a ∈ A, and t ∈ I\. Such a homotopy is called a deformation of X onto A\. (J. Peter May 43) - for a suitably nice inclusion, called a “cofibration,” of a contractible space T in a space X, the quotient map X −→ X/T is a homotopy equivalence (J. Peter May 44) - Theorem\. Let X be a connected graph with maximal tree T \. Then the quotient space X/T is the wedge of one circle for each edge of X not in T , and the quotient map q : X −→ X/T is a homotopy equivalence (J. Peter May 44) - types of graphs has the following immediate implication\. Corollary\. If X is a connected graph, then π1 \(X\) is a free group with one generator for each edge not in a given maximal tree\. If X is finite, then π1 \(X\) is free on 1 − χ\(X\) generators; in particular, χ\(X\) ≤ 1, with equality if and only if X is a tree\. (J. Peter May 45) - Theorem\. If B is a connected graph with vertex set B 0 and p : E −→ B is a covering, then E is a connected graph with vertex set E 0 = p−1 \(B 0 \) and with one edge for each edge j of B and point e ∈ Fj\(0\) \. Therefore, if B is finite and p is a finite cover whose fibers have cardinality n, then E is finite and χ\(E\) = nχ\(B\)\. (J. Peter May 45) - Theorem\. A subgroup H of a free group G is free\. If G is free on k generators and H has finite index n in G, then H is free on 1 − n + nk generators\. (J. Peter May 45) - Theorem\. For any group G, there is a connected space X such that π1 \(X\) is isomorphic to G\. (J. Peter May 45) - Define the \(unreduced\) cone on E to be CE = \(E × I\)/\(E × {1}\) and define X = B ∪p CE/\(∼\), where \(e, 0\) ∼ p\(e\)\. (J. Peter May 46) - The space X constructed in the proof is called the “homotopy cofiber” of the map p\. (J. Peter May 46) - The homotopy and homology groups of spaces are supported on compact subspaces, and it turns out that if one assumes a separation property that is a little weaker than the Hausdorff property, then one can refine the point-set topology of spaces to eliminate such pathology without changing these invariants\. (J. Peter May 47) - We shall understand compact spaces to be both compact and Hausdorff (J. Peter May 47) - A space X is said to be “weak Hausdorff” if g\(K\) is closed in X for every map g : K −→ X from a compact space K into X (J. Peter May 47) - A space X is “compactly generated” if it is a weak Hausdorff k-space (J. Peter May 47) - Lemma\. If X is a compactly generated space and Y is any space, then a function f : X −→ Y is continuous if and only if its restriction to each compact subspace K of X is continuous (J. Peter May 47) - A subspace A of X is said to be “compactly closed” if g −1 \(A\) is closed in K for any map g : K −→ X from a compact space K into X\. When X is weak Hausdorff, this holds if and only if the intersection of A with each compact subset of X is closed\. A space X is a “k-space” if every compactly closed subspace is closed (J. Peter May 47) - By definition, a space X is Hausdorff if the diagonal subspace ∆X = {\(x, x\)} is closed in X ×c X\. The weak Hausdorff property admits a similar characterization\. Lemma\. If X is a k-space, then X is weak Hausdorff if and only if ∆X is closed in X × X\. (J. Peter May 48) - The interpretation is that a quotient space of a compactly generated space by a “closed equivalence relation” is compactly generated (J. Peter May 48) - Proposition\. If X and Y are compactly generated spaces, A is a closed subspace of X, and f : A −→ Y is any continuous map, then the pushout Y ∪f X is compactly generated\. (J. Peter May 48) - Another source of pathology is passage to colimits over sequences of maps Xi −→ Xi+1 \. (J. Peter May 48) - Proposition\. If {Xi } is a sequence of compactly generated spaces and inclusions Xi −→ Xi+1 with closed images, then colim Xi is compactly generated\. (J. Peter May 48) - Limits of weak Hausdorff spaces are weak Hausdorff, but limits of k-spaces need not be k-spaces (J. Peter May 48) - We construct limits of compactly generated spaces by applying the functor k to their limits as spaces\. (J. Peter May 48) - Point-set level colimits of weak Hausdorff spaces need not be weak Hausdorff (J. Peter May 48) - However, if a point-set level colimit of compactly generated spaces is weak Hausdorff, then it is a k-space and therefore compactly generated\. We shall only be interested in colimits in those cases where this holds\. (J. Peter May 48) - Proposition\. For spaces X, Y , and Z in U , the canonical bijection Z \(X×Y \) ∼ = ∼ = \(Z Y \) X is a homeomorphism\. (J. Peter May 49) - Observe in particular that a homotopy X × I −→ Y can equally well be viewed as a map X −→ Y I (J. Peter May 49) - Exact sequences that feature in the study of homotopy, homology, and cohomology groups all can be derived homotopically from the theory of cofiber and fiber sequences that we present in this and the following two chapters (J. Peter May 51) - The theories of cofiber and fiber sequences illustrate an important, but informal, duality theory, known as Eckmann-Hilton duality (J. Peter May 51) - It is based on the adjunction between Cartesian products and function spaces (J. Peter May 51) - Definition\. A map i : A −→ X is a cofibration if it satisfies the homotopy extension property \(HEP\)\. This means that if h ◦ i0 = f ◦ i in the diagram (J. Peter May 51) - Pushouts of cofibrations are cofibrations (J. Peter May 51) - Namely, we can let Y in our original test diagram be the “mapping cylinder (J. Peter May 52) - As a matter of point-set topology, left as an exercise, it follows that a cofibration is an inclusion with closed image\. (J. Peter May 52) - M i ≡ X ∪i \(A × I\), which is the pushout of i and (J. Peter May 52) - We can use the mapping cylinder construction to decompose an arbitrary map f : X −→ Y as the composite of a cofibration and a homotopy equivalence (J. Peter May 53) - To see this, recall that M f = Y ∪f \(X × I\) and observe that f coincides with the composite X j − → Mf r r − → Y, (J. Peter May 53) - where j\(x\) = \(x, 1\) and where r\(y\) = y on Y and r\(x, s\) = f \(x\) on X × I\. (J. Peter May 53) - We shall often consider pairs \(X, A\) consisting of a space X and a subspace A\. Cofibration pairs will be those pairs that “behave homologically” just like the associated quotient spaces X/A\. (J. Peter May 53) - Definition\. A pair \(X, A\) is an NDR-pair \(= neighborhood deformation retract pair\) if there is a map u : X −→ I such that u−1 \(0\) = A and a homotopy h : X × I −→ X such that h0 = id, h\(a, t\) = a for a ∈ A and t ∈ I, and h\(x, 1\) ∈ A if u\(x\) < 1; \(X, A\) is a DR-pair if u\(x\) < 1 for all x ∈ X, in which case A is a deformation retract of X\. (J. Peter May 53) - Theorem\. Let A be a closed subspace of X\. Then the following are equivalent: \(i\) \(X, A\) is an NDR-pair\. \(ii\) \(X × I, X × {0} ∪ A × I\) is a DR-pair\. \(iii\) X × {0} ∪ A × I is a retract of X × I\. \(iv\) The inclusion i : A −→ X is a cofibration (J. Peter May 53) - It is often important to work in the category of spaces under a given space A, and we shall later need a basic result about homotopy equivalences in this category\. We shall also need a generalization concerning homotopy equivalences of pairs\. (J. Peter May 54) - A homotopy between maps under A is a homotopy that at each time t is a map under A\. We then write h : f ≃ f ′ rel A (J. Peter May 54) - A\. Such an equivalence is called a “cofiber homotopy equivalence\.” The name is suggested by the following result, whose proof illustrates a more substantial use of the HEP than we have seen before (J. Peter May 54) - Proposition\. Let i : A −→ X and j : A −→ Y be cofibrations and let f : X −→ Y be a map such that f ◦i = j\. Suppose that f is a homotopy equivalence\. Then f is a cofiber homotopy equivalence\. (J. Peter May 54) - Definition\. A surjective map p : E −→ B is a fibration if it satisfies the covering homotopy property \(CHP\) (J. Peter May 57) - This notion of a fibration is due to Hurewicz\. There is a more general notion of a Serre fibration, in which the test spaces Y are restricted to be cubes I n \. Serre fibrations are more appropriate for many purposes (J. Peter May 57) - With this formulation, we can “dualize” the proof that pushouts of cofibrations are cofibrations to show that pullbacks of fibrations are fibrations\. We often write A ×g E for the pullback of a given fibration p : E −→ B and a map g : A −→ B\. (J. Peter May 57) - Lemma\. If p : E −→ B is a fibration and g : A −→ B is any map, then the induced map A ×g E −→ A is a fibration (J. Peter May 57) - Although the CHP is expressed in terms of general test diagrams, there is a certain universal test diagram\. Namely, we can let Y in our original test diagram be the “mapping path space” N p ≡ E ×p B I = {\(e, β\)|β\(0\) = p\(e\)} ⊂ E × B I \. (J. Peter May 57) - In general, path lifting functions are not unique\. In fact, we have already studied the special kinds of fibrations for which they are unique\. Lemma\. If p : E −→ B is a covering, then p is a fibration with a unique path lifting function s\. (J. Peter May 58) - A map s : N p −→ E I such that k ◦ s = id, where k : E I −→ N p has coordinates p0 and pI , is called a path lifting function (J. Peter May 58) - Lemma\. If i : A −→ X is a cofibration and B is a space, then the induced map p = B i : B X −→ B A is a fibration\. (J. Peter May 58) - We can use the mapping path space construction to decompose an arbitrary map f : X −→ Y as the composite of a homotopy equivalence and a fibration\. (J. Peter May 58) - When restricted to the spaces U in a well chosen open cover O of the base space B, a covering is homeomorphic to the projection U × F −→ U , where F is a fixed discrete set (J. Peter May 59) - The obvious generalization of this is the notion of a bundle\. A map p : E −→ B is a bundle if, when restricted to the spaces U in a well chosen open cover O of B, there are homeomorphisms φ : U × F −→ p−1 \(U \) such that p ◦ φ = π1 , where F is a fixed topological space\. (J. Peter May 59) - that every bundle is a fibration (J. Peter May 59) - Theorem\. Let p : E −→ B be a map and let O be a numerable open cover of B\. Then p is a fibration if and only if p : p−1 \(U \) −→ U is a fibration for every U ∈ O\. (J. Peter May 59) - Since pullbacks of fibrations are fibrations (J. Peter May 59) - B\. A space over B is a map p : E −→ B\. A map of spaces over B is a commutative diagram (J. Peter May 60) - A homotopy between maps over B is a homotopy that at each time t is a map over B\. There results a notion of a homotopy equivalence over B\. Such an equivalence is called a “fiber homotopy equivalence\.” (J. Peter May 60) - Proposition\. Let p : D −→ B and q : E −→ B be fibrations and let f : D −→ E be a map such that q ◦ f = p\. Suppose that f is a homotopy equivalence\. Then f is a fiber homotopy equivalence (J. Peter May 60) - Translation of fibers along paths in the base space played a fundamental role in our study of covering spaces (J. Peter May 61) - Theorem\. Lifting of equivalence classes of paths in B to homotopy classes of maps of fibers specifies a functor λ : Π\(B\) −→ hU \. Therefore, if B is path connected, then any two fibers of B are homotopy equivalent\. (J. Peter May 62) - Just as the fundamental group π1 \(B, b\) of the base space of a covering acts on the fiber Fb , so the fundamental group π1 \(B, b\) of the base space of a fibration acts “up to homotopy” on the fiber (J. Peter May 62) - The composite of homotopy equivalences is a homotopy equivalence, and composition defines a continuous product on Aut\(X\)\. With this product, Aut\(X\) is a “topological monoid,” namely a space with a continuous and associative multiplication with a two-sided identity element, but it is not a group\. However, the path components of Aut\(X\) are the homotopy classes of homotopy equivalences of X, and these do form a group under composition (J. Peter May 62) - Corollary\. Lifting of equivalence classes of loops specifies a homomorphism π1 \(B, b\) −→ π0 \(Aut\(Fb \)\)\. (J. Peter May 62) - We use cofibrations and fibrations in the category T of based spaces to generate two “exact sequences of spaces” from a given map of based spaces (J. Peter May 65) - For based spaces X and Y , we let [X, Y ] denote the set of based homotopy classes of based maps X −→ Y \. This set has a natural basepoint, namely the homotopy class of the constant map from X to the basepoint of Y (J. Peter May 65) - The appropriate analogue of the Cartesian product in the category of based spaces is the “smash product” X ∧ Y defined by X ∧ Y = X × Y /X ∨ Y (J. Peter May 65) - With these definitions, we have a natural homeomorphism of based spaces F \(X ∧ Y, Z\) ∼ = ∼ = F \(X, F \(Y, Z\)\) for based spaces X and Y \. (J. Peter May 65) - We define the cone on X to be CX = X ∧ I, where I is given the basepoint 1\. That is, CX = X × I/\({∗} × I ∪ X × {1}\)\. (J. Peter May 65) - We view S 1 as I/∂I, denote its basepoint by 1, and define the suspension of X to be ΣX = X ∧ S 1 \. That is, ΣX = X × S 1 /\({∗} × S 1 ∪ X × {1}\)\. (J. Peter May 65) - These are sometimes called the reduced cone and suspension, to distinguish them from the unreduced constructions, in which the line {∗} × I through the basepoint of X is not identified to a point (J. Peter May 65) - Dually, we define the path space of X to be P X = F \(I, X\), where I is given the basepoint 0\. Thus the points of P X are the paths in X that start at the basepoint\. We define the loop space of X to be ΩX = F \(S 1 , X\)\. Its points are the loops at the basepoint\. (J. Peter May 66) - We have the adjunction F \(ΣX, Y \) ∼ = ∼ = F \(X, ΩY \)\. Passing to π0 , this gives that [ΣX, Y ] ∼ = ∼ = [X, ΩY ]\. (J. Peter May 66) - Lemma\. [ΣX, Y ] is a group and [Σ2 X, Y ] is an Abelian group (J. Peter May 66) - We say that X is “nondegenerately based,” or “well pointed,” if the inclusion of its basepoint is a cofibration in the unbased sense\. (J. Peter May 66) - Write Y+ for the union of a space Y and a disjoint basepoint and observe that we can identify X ∧ Y+ with X × Y /{∗} × Y \. (J. Peter May 66) - The space X ∧ I+ is called the reduced cylinder on X, and a based homotopy X × I −→ Y is the same thing as a based map X ∧ I+ −→ Y \. We change notations and write M f for the based mapping cylinder Y ∪f \(X ∧ I+ \) of a based map (J. Peter May 66) - As in the unbased case, we conclude that a based map i : A −→ X is a cofibration if and only if M i is a retract of X ∧ I+ \. (J. Peter May 66) - For a based map f : X −→ Y , define the “homotopy cofiber” Cf to be Cf = Y ∪f CX = M f /j\(X\), (J. Peter May 67) - where j : X −→ M f sends x to \(x, 1\) (J. Peter May 67) - Let i : Y −→ Cf be the inclusion\. It is a cofibration since it is the pushout of f and the cofibration X −→ CX that sends x to \(x, 0\) (J. Peter May 67) - is called the cofiber sequence generated by the map f ; here (J. Peter May 67) - These “long exact sequences of based spaces” give rise to long exact sequences of pointed sets, where a sequence S ′ f − →S g → S − ′′ of pointed sets is said to be exact if g\(s\) = ∗ if and only if s = f \(s′ \) for some s (J. Peter May 67) - Theorem\. For any based space Z, the induced sequence · · · −→ [ΣCf, Z] −→ [ΣY, Z] −→ [ΣX, Z] −→ [Cf, Z] −→ [Y, Z] −→ [X, Z] is an exact sequence of pointed sets, or of groups to the left of [ΣX, Z], or of Abelian groups to the left of [Σ2 X, Z]\. (J. Peter May 67) - Lemma\. If i : A −→ X is a cofibration, then the quotient map ψ : Ci −→ Ci/CA ∼ = ∼ = X/A is a based homotopy equivalence (J. Peter May 68) - For a based map f : X −→ Y , define the “homotopy fiber” F f to be F f = X ×f P Y = {\(x, χ\)|f \(x\) = χ\(1\)} ⊂ X × P Y\. Equivalently, F f is the pullback displayed in the diagram (J. Peter May 69) - Theorem\. For any based space Z, the induced sequence · · · −→ [Z, ΩF f ] −→ [Z, ΩX] −→ [Z, ΩY ] −→ [Z, F f ] −→ [Z, X] −→ [Z, Y ] is an exact sequence of pointed sets, or of groups to the left of [Z, ΩY ], or of Abelian groups to the left of [Z, Ω2 Y ]\. (J. Peter May 70) - It is often useful to know that cofiber sequences and fiber sequences can be connected to one another\. The adjunction between loops and suspension has “unit” and “counit” maps η : X −→ ΩΣX and ε : ΣΩX −→ X (J. Peter May 71) - Observe that πn \(X\) = πn−1 \(ΩX\) = · · · = π0 \(Ω n X\)\. (J. Peter May 73) - For ∗ ∈ A ⊂ X, the \(homotopy\) fiber of the inclusion A −→ X may be identified with the space P \(X; ∗, A\) of paths in X that begin at the basepoint and end in A (J. Peter May 73) - This is a group if n ≥ 2 and an Abelian group if n ≥ 3\. Again, πn \(X, A\) = π0 \(Ω n−1 P \(X; ∗, A\)\)\. These are called relative homotopy groups\. (J. Peter May 73) - Using φ∗ to identify π∗ F with π∗ \(F p\), we may rewrite the long exact sequence of the bottom row of the diagram as · · · −→ πn \(F \) −→ πn \(E\) −→ πn \(B\) ∂ ∂ − → πn−1 \(F \) −→ · · · −→ π0 \(E\) −→ {∗}\. (J. Peter May 74) - This is one of the main tools for the computation of homotopy groups (J. Peter May 74) - Lemma\. If p : E −→ B is a covering, then p∗ : πn \(E\) −→ πn \(B\) is an isomorphism for all n ≥ 2\. (J. Peter May 74) - Lemma\. For all spaces X and Y and all n, πn \(X × Y \) ∼ = ∼ = πn \(X\) × πn \(Y \)\. (J. Peter May 75) - There are three standard bundles, called the Hopf bundles, that can be used to obtain a bit more information about the homotopy groups of spheres\. (J. Peter May 75) - That is, CP 1 = \(C × C − {0}\)/\(∼\), where \(z1 , z2 \) ∼ \(λz1 , λz2 \) for complex numbers λ, z1 , and z2 \. Write [z1 , z2 ] for the equivalence class of \(z1 , z2 \)\. We obtain a homeomorphism CP 1 −→ S 2 by identifying S 2 with the one-point compactification of C and mapping [z1 , z2 ] to z2 /z1 if z1 6= 0 and to the point at ∞ if z1 = 0\. The Hopf map η : S 3 −→ S 2 is specified by η\(z1 , z2 \) = [z1 , z2 ], where S 3 is identified with the unit sphere in the complex plane C2 \. (J. Peter May 75) - Since we have complete information on the homotopy groups of S 1 , the long exact sequence of homotopy groups associated to η has the following direct consequence\. Lemma\. π2 \(S 2 \) = ∼ = Z and πn \(S ∼ 3 \) ∼ = ∼ = πn \(S 2 \) for n ≥ 3\. (J. Peter May 75) - It is left as an exercise to show that the long exact sequence associated to ν implies that π7 \(S 4 \) contains an element of infinite order, and σ can be used similarly to show the same for π15 \(S 8 \)\. (J. Peter May 75) - In fact, the homotopy groups πq \(S n \) for q > n > 1 are all finite except for π4n−1 \(S 2n \), which is the direct sum of Z and a finite group\. (J. Peter May 75) - The difficulty of computing homotopy groups is well illustrated by the fact that there is no non-contractible simply connected compact manifold \(or finite CW complex\) all of whose homotopy groups are known\. We shall find many non-compact spaces whose homotopy groups we can determine completely (J. Peter May 75) - Lemma\. If X is the colimit of a sequence of inclusions Xi −→ Xi+1 of based spaces, then the natural map colimi πn \(Xi \) −→ πn \(X\) is an isomorphism for each n\. (J. Peter May 75) - Corollary\. A homotopy equivalence of spaces or of pairs of spaces induces an isomorphism on all homotopy groups\. (J. Peter May 77) - Definition\. A map e : Y −→ Z is an n-equivalence if, for all y ∈ Y , the map e∗ : πq \(Y, y\) −→ πq \(Z, e\(y\)\) is an injection for q < n and a surjection for q ≤ n; e is said to be a weak equivalence if it is an n-equivalence for all n\. (J. Peter May 77) - Observe that πn+1 \(X, x\) can be viewed as the set of relative homotopy classes of maps \(CS n , S n \) −→ \(X, x\)\. (J. Peter May 77) - It gives a useful criterion for determining when a given map is an n-equivalence (J. Peter May 77) - We introduce a large class of spaces, called CW complexes, between which a weak equivalence is necessarily a homotopy equivalence\. Thus, for such spaces, the homotopy groups are, in a sense, a complete set of invariants\. Moreover, we shall see that every space is weakly equivalent to a CW complex (J. Peter May 81) - iv\) A map of pairs f : \(X, A\) −→ \(Y, B\) between relative CW complexes is said to be “cellular” if f \(X n \) ⊂ Y n for all n\. (J. Peter May 81) - Of course, pushouts and unions are understood in the topological sense, with the compactly generated topologies\. A subspace of X is closed if and only if its intersection with each X n is closed\. (J. Peter May 81) - \(ii\) Via a homeomorphism I × I ∼ = ∼ = D2 , the standard presentations of the torus T = S × S , the projective plane RP 2 , and the Klein bottle K as quotients of a 1 1 square display these spaces as CW complexes with one or two vertices, two edges, and one 2-cell: (J. Peter May 81) - Lemma\. If \(X, A\) is a relative CW complex, then the quotient space X/A is a CW complex with a vertex corresponding to A and one n-cell for each relative n-cell of \(X, A\)\. (J. Peter May 82) - Lemma\. If A is a subcomplex of a CW complex X, Y is a CW complex, and f : A −→ Y is a cellular map, then the pushout Y ∪f X is a CW complex that contains Y as a subcomplex and has one cell for each cell of X that is not in A\. The quotient complex \(Y ∪f X\)/Y is isomorphic to X/A\. (J. Peter May 82) - Lemma\. The colimit of a sequence of inclusions of subcomplexes Xn −→ Xn+1 in CW complexes is a CW complex that contains each of the Xi as a subcomplex\. (J. Peter May 82) - In particular, if we take e to be the identity map of Y , we see that the inclusion A −→ X is a cofibration\. Observe that, by passage to colimits, we are free to take n = ∞ in the theorem\. (J. Peter May 83) - Theorem \(Whitehead\)\. If X is a CW complex and e : Y −→ Z is an nequivalence, then e∗ : [X, Y ] −→ [X, Z] is a bijection if dim X < n and a surjection if dim X = n\. (J. Peter May 83) - Theorem \(Whitehead\)\. An n-equivalence between CW complexes of dimension less than n is a homotopy equivalence\. A weak equivalence between CW complexes is a homotopy equivalence (J. Peter May 84) - If X is a finite CW complex, in the sense that it has finitely many cells, and if dim X > 1 and X is not contractible, then it is known that X has infinitely many non-zero homotopy groups\. The Whitehead theorem is thus surprisingly strong: in its first statement, if low dimensional homotopy groups are mapped isomorphically, then so are all higher homotopy groups (J. Peter May 84) - Fortunately, any map between CW complexes is homotopic to a cellular map (J. Peter May 84) - Definition\. A space X is said to be n-connected if πq \(X, x\) = 0 for 0 ≤ q ≤ n and all x\. A pair \(X, A\) is said to be n-connected if π0 \(A\) −→ π0 \(X\) is surjective and πq \(X, A, a\) = 0 for 1 ≤ q ≤ n and all a\. It is equivalent that the inclusion A −→ X be an n-equivalence (J. Peter May 84) - Lemma\. A relative CW complex \(X, A\) with no m-cells for m ≤ n is nconnected\. In particular, \(X, X n \) is n-connected for any CW complex X\. (J. Peter May 84) - Theorem \(Cellular approximation\)\. Any map f : \(X, A\) −→ \(Y, B\) between relative CW complexes is homotopic relative to A to a cellular map\. (J. Peter May 84) - Corollary\. For CW complexes X and Y , any map X −→ Y is homotopic to a cellular map, and any two homotopic cellular maps are cellularly homotopic\. (J. Peter May 85) - Theorem \(Approximation by CW complexes\)\. For any space X, there is a CW complex ΓX and a weak equivalence γ : ΓX −→ X\. For a map f : X −→ Y and another such CW approximation γ : ΓY −→ Y , there is a map Γf : ΓX −→ ΓY , unique up to homotopy, such that the following diagram is homotopy commutative (J. Peter May 85) - If X is n-connected, n ≥ 1, then ΓX can be chosen to have a unique vertex and no q-cells for 1 ≤ q ≤ n\. (J. Peter May 85) - A triad \(X; A, B\) is said to be excisive if X is the union of the interiors of A and B\. Such triads play a fundamental role in homology and cohomology theory (J. Peter May 87) - A CW triad \(X; A, B\) is a CW complex X with subcomplexes A and B such that X = A ∪ B\. (J. Peter May 87) - Define the double mapping cylinder M \(i, j\) = A ∪ \(C × I\) ∪ B (J. Peter May 88) - to be the space obtained from C × I by gluing A to C × {0} along i and gluing B to C × {1} along j\. (J. Peter May 88) - Lemma\. For a cofibration i : C −→ A and any map j : C −→ B, the quotient map q : M \(i, j\) −→ A ∪C B is a homotopy equivalence (J. Peter May 88) - Because i is a cofibration, the retraction r : M i −→ A is a cofiber homotopy equivalence (J. Peter May 88) - That is, there is a homotopy inverse map and a pair of homotopies under C\. (J. Peter May 88) - The fundamental obstruction to the calculation of homotopy groups is the failure of excision: for an excisive triad \(X; A, B\), the inclusion \(A, A ∩ B\) −→ \(X, B\) fails to induce an isomorphism of homotopy groups in general (J. Peter May 91) - However, we do have such an isomorphism in a range of dimensions\. This implies the Freudenthal suspension theorem, which gives that πn+q \(Σn X\) is independent of n if q is small relative to n (J. Peter May 91) - Theorem \(Homotopy excision\)\. Let \(X; A, B\) be an excisive triad such that C = A ∩ B is non-empty\. Assume that \(A, C\) is \(m − 1\)-connected and \(B, C\) is \(n − 1\)-connected, where m ≥ 2 and n ≥ 1\. Then the inclusion \(A, C\) −→ \(X, B\) is an \(m + n − 2\)-equivalence (J. Peter May 91) - This specializes to give a relationship between the homotopy groups of pairs \(X, A\) and of quotients X/A and to prove the Freudenthal suspension theorem (J. Peter May 91) - Theorem\. Let f : X −→ Y be an \(n−1\)-equivalence between \(n−2\)-connected spaces, where n ≥ 2; thus πn−1 \(f \) is an epimorphism\. Then the quotient map π : \(M f, X\) −→ \(Cf, ∗\) is a \(2n − 2\)-equivalence\. In particular, Cf is \(n − 1\)connected\. If X and Y are \(n − 1\)-connected, then π : \(M f, X\) −→ \(Cf, ∗\) is a \(2n − 1\)-equivalence (J. Peter May 91) - Corollary\. Let f : X −→ Y be a based map between \(n − 1\)-connected nondegenerately based spaces, where n ≥ 2\. Then Cf is \(n − 1\)-connected and πn \(M f, X\) −→ πn \(Cf, ∗\) is an isomorphism\. Moreover, the canonical map η : F f −→ ΩCf induces an isomorphism πn−1 \(F f \) −→ πn \(Cf \)\. (J. Peter May 92) - Theorem\. Let i : A −→ X be a cofibration and an \(n − 1\)-equivalence between \(n − 2\)-connected spaces, where n ≥ 2\. Then the quotient map \(X, A\) −→ \(X/A, ∗\) is a \(2n − 2\)-equivalence, and it is a \(2n − 1\)-equivalence if A and X are \(n − 1\)connected\. (J. Peter May 92) - A specialization of the last result gives the Freudenthal suspension theorem\. For a based space X, define the suspension homomorphism Σ : πq \(X\) −→ πq+1 \(ΣX\) by letting Σf = f ∧ id : S q+1 ∼ = ∼ = S q ∧ S 1 −→ X ∧ S 1 = ΣX\. (J. Peter May 93) - Theorem \(Freudenthal suspension\)\. Assume that X is nondegenerately based and \(n − 1\)-connected, where n ≥ 1\. Then Σ is a bijection if q < 2n − 1 and a surjection if q = 2n − 1\. (J. Peter May 93) - Theorem\. For all n ≥ 1, πn \(S n \) = Z and Σ : πn \(S n \) −→ πn+1 \(S n+1 \) is an isomorphism\. (J. Peter May 93) - We saw by use of the Hopf bundle S 3 −→ S 2 that π2 \(S 2 \) = Z, and the suspension theorem applies to give the conclusion for n ≥ 2\. A little extra argument is needed to check that Σ is an isomorphism for n = 1; one can inspect the connecting homomorphism of the Hopf bundle or refer ahead to the observation that the Hurewicz homomorphism commutes with the corresponding suspension isomorphism in homology (J. Peter May 93) - The dimensional range of the suspension theorem is sharp\. We saw before that π3 \(S 2 \) = π3 \(S 3 \), which is Z\. The suspension theorem applies to show that Σ : π3 \(S 2 \) −→ π4 \(S 3 \) is an epimorphism, and it is known that π4 \(S 3 \) = Z2 \. (J. Peter May 93) - Applying suspension repeatedly, we can form a colimit π s qs \(X\) = colim πq+n \(Σ n X\)\. (J. Peter May 93) - This group is called the qth stable homotopy group of X\. For q < n − 1, the maps of the colimit system are isomorphisms and therefore π s qs \(X\) = πq+n \(Σ n X\) if q < n − 1\. (J. Peter May 94) - The calculation of the stable homotopy groups of spheres, πqs qs \(S 0 \), is one of the deepest and most studied problems in algebraic topology\. Important problems of geometric topology, such as the enumeration of the distinct differential structures on S q for q ≥ 5, have been reduced to the determination of these groups (J. Peter May 94) - 3\. Proof of the homotopy excision theorem This is a deep result, and it is remarkable that a direct homotopical proof, in principle an elementary one, is possible\. Most standard texts, if they treat this topic at all, give a far more sophisticated proof of a significantly weaker result\. (J. Peter May 94) - We are trying to show that a certain map of pairs induces an isomorphism in a range of dimensions\. We capture the relevant map as part of a long exact sequence, and we prove that the third term in the long exact sequence vanishes in the required range\. (J. Peter May 94) - The long exact sequence of the pair in the first form of the definition is · · · −→ πq+1 \(X; A, B\) −→ πq \(A, C\) −→ πq \(X, B\) −→ πq \(X; A, B\) −→ · · · \. (J. Peter May 94) - Definition\. For a triad \(X; A, B\) with basepoint ∗ ∈ C = A ∩ B, define πq \(X; A, B\) = πq−1 \(P \(X; ∗, B\), P \(A; ∗, C\)\), (J. Peter May 94) - A chain complex over R is a sequence of maps of R-modules · · · −→ Xi+1 di+1 −−−→ Xi d i d i −→ Xi−1 −→ · · · such that di ◦ di+1 = 0 for all i (J. Peter May 99) - An element of the kernel of di is called a cycle and an element of the image of di+1 is called a boundary\. We say that two cycles are “homologous” if their difference is a boundary\. We write Bi \(X\) ⊂ Zi \(X\) ⊂ Xi (J. Peter May 99) - for the submodules of boundaries and cycles, respectively, and we define the ith homology group Hi \(X\) to be the quotient module Zi \(X\)/Bi \(X\)\. We write H∗ \(X\) for the sequence of Rmodules Hi \(X\)\. We understand “graded R-modules” to be sequences of R-modules such as this \(and we never take the sum of elements in different gradings\)\. (J. Peter May 99) - A chain homotopy s : f ≃ g between chain maps f, g : X −→ X ′ is a sequence ′ of homomorphisms si : Xi −→ Xi+1 such that d′ ′i ′i+1 ◦ si + si−1 ◦ di = fi − gi (J. Peter May 100) - Lemma\. Chain homotopic maps induce the same homomorphism of homology groups\. (J. Peter May 100) - d = d ⊗ id + id ⊗ d\. (J. Peter May 100) - Lemma\. A chain homotopy s : f ≃ g between chain maps f, g : X −→ X ′ determines and is determined by a chain map h : X ⊗I −→ X ′ such that h\(x, [0]\) = f \(x\) and h\(x, [1]\) = g\(x\)\. (J. Peter May 100) - We regard R-modules M as chain complexes concentrated in degree zero, and thus with zero differential\. For a chain complex X, there results a chain complex X ⊗ M ; H∗ \(X ⊗ M \) is called the homology of X with coefficients in M \. (J. Peter May 100) - Define a chain complex I by letting I0 be the free Abelian group with two generators [0] and [1], letting I1 be the free Abelian group with one generator [I] such that d\([I]\) = [0] − [1], and letting Ii = 0 for all other i\. (J. Peter May 100) - A sequence M ′ f − →M g g → M ′′ of modules is exact if im f = ker g\. If M ′ = 0, this − means that g is a monomorphism; if M ′′ = 0, it means that f is an epimorphism (J. Peter May 101) - Proposition\. A short exact sequence of chain complexes naturally gives rise to a long exact sequence of R-modules · · · −→ Hq \(X ′ \) f f∗ −→ Hq \(X\) g g∗ −→ Hq \(X ′′ \) ∂ ∂ − → Hq−1 \(X ′ \) −→ · · · \. (J. Peter May 101) - We define the “connecting homomorphism” ∂ : Hq \(X ′′ \) −→ Hq−1 \(X ′ \) by ∂[x′′ ] = [x′ ], where f \(x′ \) = d\(x\) for some x such that g\(x\) = x′′ \. There is such an x since g is an epimorphism, and there is such an x′ since gd\(x\) = dg\(x\) = 0\. It is a standard exercise in “diagram chasing” to verify that ∂ is well defined and the sequence is exact (J. Peter May 101) - Let 0 −→ π f − →ρ g − → σ −→ 0 be an exact sequence of Abelian groups and let C be a chain complex of flat \(= torsion free\) Abelian groups\. Write H∗ \(C; π\) = H∗ \(C ⊗ π\)\. Construct a natural long exact sequence · · · −→ Hq \(C; π\) f f∗ −→ Hq \(C; ρ\) g g∗ −→ Hq \(C; σ\) β − → Hq−1 \(C; π\) −→ · · · \. The connecting homomorphism β is called a Bockstein operation\. (J. Peter May 101) - Unlike homotopy groups, these are stable invariants, the same for a space and its suspension, and it is this that makes them computable\. (J. Peter May 103) - Fix an Abelian group π and consider pairs of spaces \(X, A\)\. We shall see that π determines a “homology theory on pairs \(X, A\)\.” (J. Peter May 103) - 1\. Axioms for homology (J. Peter May 103) - Clearly, up to canonical isomorphism, this construction of a homology theory on pairs of spaces is independent of the choice of our CW approximation functor Γ\. The reader may have seen singular homology before\. As we shall explain later, the classical construction of singular homology amounts to a choice of a particularly nice CW approximation functor, one that is actually functorial on the point-set level, before passage to homotopy categories (J. Peter May 104) - Let X be a CW complex\. We shall define the cellular chain complex C∗ \(X\)\. We let Cn \(X\) be the free Abelian group with one generator [j] for each n-cell j\. We must define a differential dn : Cn \(X\) −→ Cn−1 \(X\) (J. Peter May 105) - It will be convenient to work with unreduced cones, cofibers, and suspensions in this section; that is, we do not choose basepoints and so we do not collapse out lines through basepoints (J. Peter May 105) - We still have the basic result that if i : A −→ X is a cofibration, then collapsing the cone on A to a point gives a homotopy equivalence ψ : Ci −→ X/A\. (J. Peter May 105) - Our first definition of dn involves the calculation of the degrees of maps between spheres (J. Peter May 105) - However, there are three models of S n that are needed in our discussion: the standard sphere S n ⊂ Dn+1 , the quotient Dn /S n−1 , and the \(unreduced\) suspension ΣS n−1 \. We must fix suitably compatible homeomorphisms relating these “n-spheres\.” We define a homeomorphism (J. Peter May 105) - For a subcomplex A of X, define C∗ \(X, A\) = C∗ \(X\)/C∗ \(A\) ∼ = ∼ = C ̃ C̃∗ \(X/A\) (J. Peter May 108) - and define H∗ \(X, A\) = H∗ \(C∗ \(X, A\)\) ∼ = ∼ = H ̃ H̃∗ \(X/A\)\. (J. Peter May 109) - We have dealt so far with the case of integral homology\. For more general coefficient groups π, we define C∗ \(X, A; π\) = C∗ \(X, A\) ⊗ π (J. Peter May 109) - A nice fact about cellular homology is that the definition leads directly to an algebraic procedure for the calculation of the homology of Cartesian products (J. Peter May 109) - Theorem\. If X and Y are CW complexes, then X × Y is a CW complex such that C∗ \(X × Y \) ∼ = ∼ = C∗ \(X\) ⊗ C∗ \(Y \)\. (J. Peter May 109) - We define an isomorphism of graded Abelian groups κ : C∗ \(X\) ⊗ C∗ \(Y \) −→ C∗ \(X × Y \) by setting κ\([i] ⊗ [j]\) = \(−1\) pq [i × j]\. (J. Peter May 110) - Cellular chains make some computations quite trivial\. For example, since S n is a CW complex with one vertex and one n-cell, we see immediately that ̃ H̃n \(S n ; π\) ∼ = ∼ =π and ̃ H̃q \(S n ; π\) = 0 for q 6= n\. (J. Peter May 111) - if we look back at the CW decompositions of the torus T , the projective plane RP 2 , and the Klein bottle K and if we let j denote the unique 2-cell in each case, then we find the following descriptions of the cellular chains and integral homologies by quick direct inspections (J. Peter May 111) - H∗ \(T ; Z\) = C∗ \(T \)\. (J. Peter May 111) - H0 \(RP 2 ; Z\) = Z (J. Peter May 111) - H1 \(RP 2 ; Z\) = Z2 (J. Peter May 111) - Therefore H0 \(K; Z\) = Z (J. Peter May 111) - H1 \(K; Z\) = Z ⊕ Z2 with Z generated by the class (J. Peter May 111) - While homology groups are far easier to compute than homotopy groups, direct chain level calculation is seldom the method of choice\. Rather, one uses chains as a tool for developing more sophisticated algebraic techniques, notably spectral sequences (J. Peter May 112) - We give an illustration that both shows that chain level calculations are sometimes practicable even when there are many non-zero differentials to determine and indicates why one might not wish to attempt such calculations for really complicated spaces\. (J. Peter May 112) - Lemma\. The degree of the antipodal map an : S n −→ S n is \(−1\)n+1 \. (J. Peter May 112) - We shall use cellular chains to compute the homology of RP n (J. Peter May 112) - The nth integral homology group of such a manifold M is Z if M is orientable and zero if M is not orientable\. The nth mod 2 homology group of M is Z2 whether or not M is orientable\. (J. Peter May 113) - This calculation well illustrates general facts about the homology of compact connected closed n-manifolds M (J. Peter May 113) - A “generalized homology theory” E∗ is defined to be a system of functors Eq \(X, A\) and natural transformations ∂ : Eq \(X, A\) −→ Eq−1 \(A\) that satisfy all of our axioms except for the dimension axiom\. (J. Peter May 115) - following our proposal that E∗ \(X, A\) be taken as an alternative notation for H∗ \(X, A; π\)\. (J. Peter May 115) - One of the themes of this chapter is the relationship between homology theories on pairs of spaces and reduced homology theories on based spaces\. The latter are more convenient in most advanced work in algebraic topology (J. Peter May 115) - For a based space X, we define the reduced homology of X to be ̃ Ẽq \(X\) = Eq \(X, ∗\)\. (J. Peter May 115) - Since the basepoint is a retract of X, there results a direct sum decomposition E∗ \(X\) ∼ = ∼ = E ̃ Ẽ∗ \(X\) ⊕ E∗ \(∗\) (J. Peter May 115) - the exactness axiom implies that there is a reduced long exact sequence · · · −→ E ̃ Ẽq \(A\) −→ E ̃ Ẽq \(X\) −→ Eq \(X, A\) ∂ ∂ − → E ̃ Ẽq−1 \(A\) −→ · · · \. (J. Peter May 115) - We can obtain the unreduced homology groups as special cases of the reduced ones\. For an unbased space X, we define a based space X+ by adjoining a disjoint basepoint to X\. By the additivity axiom, we see immediately that E∗ \(X\) = E ̃ Ẽ∗ \(X+ \)\. (J. Peter May 115) - In fact, the unreduced cone on a space Y coincides with the reduced cone on Y+ : the line through the disjoint basepoint is identified to the cone point when constructing the reduced cone on Y+ \. Therefore the unreduced cofiber of an unbased map f coincides with the reduced cofiber of the based map f+ \. (J. Peter May 116) - The observant reader will have noticed that the unreduced suspension of X is not the reduced suspension on X+ \. Rather, under either interpretation of suspension, Σ\(X+ \) is homotopy equivalent to the wedge of Σ\(X\) and a circle\. (J. Peter May 116) - a special case of the reduced homology of spaces\. Theorem\. For any cofibration i : A −→ X, the quotient map q : \(X, A\) −→ \(X/A, ∗\) induces an isomorphism E∗ \(X, A\) −→ E∗ \(X/A, ∗\) = E ̃ Ẽ∗ \(X/A\)\. (J. Peter May 116) - Recall that a basepoint ∗ ∈ X is nondegenerate if the inclusion {∗} −→ X is a cofibration (J. Peter May 117) - This ensures that the inclusion of the line through the basepoint in the unreduced suspension of X is a cofibration, so that the map from the unreduced suspension to the suspension that collapses out the line through the basepoint is a homotopy equivalence\. (J. Peter May 117) - Theorem\. For a nondegenerately based space X, there is a natural isomorphism Σ : E ̃ Ẽq \(X\) ∼ = ∼ = E ̃ Ẽq+1 \(ΣX\)\. (J. Peter May 117) - Corollary\. For any n and q, ̃ Ẽq \(S n \) ∼ = ∼ = Eq−n \(∗\)\. (J. Peter May 117) - In the study of generalized homology theories, it is most convenient to restrict attention to reduced homology theories defined on nondegenerately based spaces\. (J. Peter May 118) - Definition\. A reduced homology theory ̃ Ẽ∗ consists of functors ̃ Ẽq from the homotopy category of nondegenerately based spaces to the category of Abelian groups that satisfy the following axioms (J. Peter May 118) - For the additivity axiom, we note that the cofiber of a disjoint union of maps is the wedge of the cofibers of the given maps (J. Peter May 119) - The first is the long exact sequence of a triple \(X, A, B\) of spaces B ⊂ A ⊂ X, which is just like its analogue for homotopy groups\. Proposition\. For a triple \(X, A, B\), the following sequence is exact: · · · −→ Eq \(A, B\) i i −→∗ Eq \(X, B\) j j∗ −→ Eq \(X, A\) ∂ ∂ − → Eq−1 \(A, B\) −→ · · · \. Here i : \(A, B\) −→ \(X, B\) and j : \(X, B\) −→ \(X, A\) are inclusions and ∂ is the composite Eq \(X, A\) ∂ ∂ − → Eq−1 \(A\) −→ Eq−1 \(A, B\)\. (J. Peter May 120) - Theorem \(Mayer-Vietoris sequence\)\. Let \(X; A, B\) be an excisive triad and set C = A ∩ B\. The following sequence is exact: · · · −→ Eq \(C\) ψ − → Eq \(A\) ⊕ Eq \(B\) φ − → Eq \(X\) ∆ ∆ −→ Eq−1 \(C\) −→ · · · \. (J. Peter May 120) - Alternatively, one can use CW approximation\. For a CW triad, there is a short exact sequence 0 −→ C∗ \(C\) −→ C∗ \(A\) ⊕ C∗ \(B\) −→ C∗ \(X\) −→ 0 whose associated long exact sequence is the Mayer-Vietoris sequence\. (J. Peter May 121) - We have seen that the compactness of spheres S n and cylinders S n × I implies that, for any choice of basepoint in X0 , the natural map colim π∗ \(Xi \) −→ π∗ \(X\) is an isomorphism\. We shall use the additivity and weak equivalence axioms and the Mayer-Vietoris sequence to prove the analogue for homology (J. Peter May 122) - We record an algebraic description of the colimit of a sequence for use in the proof\. Lemma\. Let fi : Ai −→ Ai+1 be a sequence of homomorphisms of Abelian groups\. Then there is a short exact sequence s a short 0 −→ i Ai ct seq − → i Ai β − → colim Ai −→ 0, where α\(ai \) = ai − fi \(ai \) for ai ∈ Ai and the restriction of β to Ai is the canonical map given by the definition of a colimit (J. Peter May 123) - The proof makes use of a useful general construction called the “telescope” of the Xi , denoted tel Xi (J. Peter May 123) - Inductively, let Y0 = X0 × {0} and suppose that we have constructed Yi ⊃ Xi × {i}\. Define Yi+1 to be the double mapping cylinder Yi ∪ Mi+1 obtained by identifying \(x, i\) ∈ Yi with \(x, i\) ∈ Mi+1 for x ∈ Xi \. Define tel Xi to be the union of the Yi , with the colimit topology (J. Peter May 123) - Remark\. There is a general theory of “homotopy colimits,” which are up to homotopy versions of colimits\. The telescope is the homotopy colimit of a sequence\. The double mapping cylinder that we used in approximating excisive triads by CW triads is the homotopy pushout of a diagram of the shape • ←− • −→ •\. We implicitly used homotopy coequalizers in constructing CW approximations of spaces (J. Peter May 124) - Definition\. For based spaces X, define the Hurewicz homomorphism h : πn \(X\) −→ H ̃ H̃n \(X\) by h\([f ]\) = f∗ \(in \)\. (J. Peter May 125) - The dimension axiom implicitly fixes a generator i0 of ̃ H̃0 \(S 0 \), and we choose generators in of ̃ H̃n \(S n \) inductively by setting Σin = in+1 \. (J. Peter May 125) - Lemma\. Let X be a wedge of n-spheres\. Then h : πn \(X\) −→ H ̃ H̃n \(X\) is the Abelianization homomorphism if n = 1 and is an isomorphism if n > 1\. (J. Peter May 126) - Theorem \(Hurewicz\)\. Let X be any \(n − 1\)-connected based space\. Then h : πn \(X\) −→ H ̃ H̃n \(X\) is the Abelianization homomorphism if n = 1 and is an isomorphism if n > 1\. (J. Peter May 126) - Here the groups on the left are defined in terms of homotopy groups and were used in our construction of cellular chains, while the groups on the right are those of our given homology theory\. We use the groups on the right to construct cellular chains in our given theory, and we find that the isomorphism is compatible with differentials (J. Peter May 127) - Thus let X be a CW complex\. For each integer n, define Cn \(X\) = Hn \(X n , X n−1 \) ∼ = ∼ = H ̃ H̃n \(X n /X n−1 \)\. Define d : Cn \(X\) −→ Cn−1 \(X\) to be the composite Hn \(X n , X n−1 \) ∂ ∂ − → Hn−1 \(X n−1 \) −→ Hn−1 \(X n−1 , X n−2 \)\. It is not hard to check that d ◦ d = 0\. (J. Peter May 127) - Theorem\. There is a natural isomorphism H∗ \(X, A\) ∼ = ∼ = H∗ \(C∗ \(X, A\)\) under which the natural transformation ∂ agrees with the natural transformation induced by the connecting homomorphisms associated to the short exact sequences 0 −→ C∗ \(A\) −→ C∗ \(X\) −→ C∗ \(X, A\) −→ 0\. (J. Peter May 127) - By the dimension and additivity axioms, we know the homology of wedges of spheres (J. Peter May 128) - \(Hint: construct M \(π, n\) as the cofiber of a map between wedges of spheres\.\) The spaces M \(π, n\) are called Moore spaces\. (J. Peter May 129) - \(Hint: start with M \(π, n\), using the Hurewicz theorem, and kill its higher homotopy groups\.\) The spaces K\(π, n\) are called Eilenberg-Mac Lane spaces (J. Peter May 129) - The standard topological n-simplex is the subspace (J. Peter May 131) - There are “face maps” (J. Peter May 131) - degeneracy maps (J. Peter May 131) - A map f : ∆n −→ X is called a singular n-simplex (J. Peter May 132) - The singular homology of X is usually defined in terms of this chain complex: H∗ \(X; π\) = H∗ \(C∗ \(X\) ⊗ π\)\. (J. Peter May 132) - We define a space ΓX, called the “geometric realization of the total singular complex of X,” (J. Peter May 132) - We may define the geometric realization |K∗ | of general simplicial sets exactly as we defined the geometric realization ΓX = |S∗ X| of the total singular complex of a topological space\. In fact, the total singular complex and geometric realization functors are adjoint (J. Peter May 134) - One can define a simplicial object in any category C as a sequence of objects Kn of C connected by face and degeneracy maps in C that satisfy the commutation relations that we have displayed\. Thus we have simplicial groups, simplicial Abelian groups, simplicial spaces, and so forth\. We can think of simplicial sets as discrete simplicial spaces, and we then see that geometric realization generalizes directly to a functor | − | from the category S U of simplicial spaces to the category U of spaces (J. Peter May 135) - We note one of the principal features of geometric realization (J. Peter May 135) - The projections induce maps of simplicial spaces from X∗ × Y∗ to X∗ and Y∗ \. On passage to geometric realization, these give the coordinates of a map |X∗ × Y∗ | −→ |X∗ | × |Y∗ |\. It turns out that this map is always a homeomorphism (J. Peter May 135) - Now restrict attention to simplicial sets K∗ and L∗ \. Then the homeomorphism just specified is a map between CW complexes\. However, it is not a cellular map; rather, it takes the n-skeleton of |K∗ × L∗ | to the 2n-skeleton of |K∗ | × |L∗ |\. It is homotopic to a cellular map, no longer a homeomorphism, and there results a chain homotopy equivalence C∗ \(|K∗ × L∗ |\) −→ C∗ \(|K∗ |\) ⊗ C∗ \(|L∗ |\) (J. Peter May 135) - In particular, for spaces X and Y , there is a natural chain homotopy equivalence from the singular chain complex C∗ \(X × Y \) to the tensor product C∗ \(X\) ⊗ C∗ \(Y \)\. (J. Peter May 135) - The space BG is called the classifying space of G (J. Peter May 136) - We may view Bn \(G\) as the orbit space En \(G\)/G\. (J. Peter May 136) - It is less intuitive, but true, that the space E\(G\) is contractible\. By the long exact homotopy sequence, these facts imply that πq+1 \(BG\) ∼ = ∼ = πq \(G\) (J. Peter May 136) - We illustrate these ideas by defining the “classifying spaces” and “universal bundles” associated to topological groups G (J. Peter May 136) - Since geometric realization commutes with products (J. Peter May 137) - This allows us to iterate the construction, setting B 0 \(G\) = G and B n \(G\) = B\(B n−1 \(G\)\) for n ≥ 1\. Specializing to a discrete Abelian group π, we define K\(π, n\) = B n \(π\)\. (J. Peter May 137) - t is called a “transfer homomorphism\.” (J. Peter May 137) - Indeed, if one focuses on singular chains, then one eschews chain level computations in principle as well as in practice\. (J. Peter May 139) - We here recall some classical results in homological algebra that explain how to calculate H∗ \(X; π\) from H∗ \(X\) ≡ H∗ \(X; Z\) and how to calculate H∗ \(X × Y \) from H∗ \(X\) ⊗ H∗ \(Y \)\. (J. Peter May 139) - Recall that an R-module M is said to be flat if the functor M ⊗ N is exact \(that is, preserves exact sequences in the variable N \)\. (J. Peter May 139) - For a principal ideal domain \(PID\) R, the only torsion product is the first one, denoted Tor R 1 \(M, N \) (J. Peter May 139) - It can be computed by constructing a short exact sequence 0 −→ F1 −→ F0 −→ M −→ 0 and tensoring with N to obtain an exact seqence 0 −→ Tor R 1 \(M, N \) −→ F1 ⊗ N −→ F0 ⊗ N −→ M ⊗ N −→ 0, where F1 and F0 are free R-modules\. That is, we choose an epimorphism F0 −→ M and note that, since R is a PID, its kernel F1 is also free\. (J. Peter May 139) - Theorem \(Universal coefficient\)\. Let R be a P ID and let X be a flat chain complex over R\. Then, for each n, there is a natural short exact sequence 0 −→ Hn \(X\) ⊗ M α α − → Hn \(X ⊗ M \) β → Tor − R 1 \(Hn−1 \(X\), M \) −→ 0\. The sequence splits, so that Hn \(X ⊗ M \) ∼ = ∼ = \(Hn \(X\) ⊗ M \) ⊕ Tor R 1 \(Hn−1 \(X\), M \), but the splitting is not natural\. (J. Peter May 140) - In Chapter 20 §3, we shall see an important class of examples in which the splitting is very far from being natural (J. Peter May 140) - Corollary\. If R is a field, then α : H∗ \(X\) ⊗ M −→ H∗ \(X; M \) is a natural isomorphism\. (J. Peter May 140) - 2\. The Künneth theorem The universal coefficient theorem in homology is a special case of the Künneth theorem\. (J. Peter May 140) - Theorem \(Künneth\)\. Let R be a P ID and let X be a flat chain complex and Y be any chain complex\. Then, for each n, there is a natural short exact sequence Y be any c 0 −→ p+q=n Hp \(X\)⊗Hq \(Y \) α α − → Hn \(X⊗Y \) β , there is β −→ p+q=n−1 Tor R 1 \(Hp \(X\), Hq \(Y \)\) −→ 0\. The sequence splits, so that Hn \(X ⊗ Y \) ∼ = ∼ =\( o t p+q=n Hp \(X\) ⊗ Hq \(Y \)\) ⊕ \( p+q=n−1 Tor R 1 \(Hp \(X\), Hq \(Y \)\)\), but the splitting is not natural\. (J. Peter May 140) - Corollary\. If R is a field, then α : H∗ \(X\) ⊗ H∗ \(Y \) −→ H∗ \(X ⊗ Y \) is a natural isomorphism\. (J. Peter May 140) - Since all modules over a field are free and thus flat (J. Peter May 141) - \. We assume that the reader has seen Ext modules, which measure the failure of Hom to be an exact functor\. For a PID R, the only Ext module is the first one, denoted Ext1 R R \(M, N \)\. It can be computed by constructing a short exact sequence 0 −→ F1 −→ F0 −→ M −→ 0 and applying Hom to obtain an exact seqence 0 −→ Hom\(M, N \) −→ Hom\(F0 , N \) −→ Hom\(F1 , N \) −→ Ext1 R R \(M, N \) −→ 0, where F1 and F0 are free R-modules (J. Peter May 142) - Theorem \(Universal coefficient\)\. Let R be a P ID and let X be a free chain complex over R\. Then, for each n, there is a natural short exact sequence 0 −→ Ext1 R R \(Hn−1 \(X\), M \) β − → H n \(X; M \) α α − → Hom\(Hn \(X\), M \) −→ 0\. The sequence splits, so that H n \(X; M \) ∼ = ∼ = Hom\(Hn \(X\), M \) ⊕ Ext1 R R \(Hn−1 \(X\), M \), but the splitting is not natural\. (J. Peter May 142) - Corollary\. If R is a field, then α : H ∗ \(X; M \) −→ Hom\(H∗ \(X\), M \) is a natural isomorphism\. (J. Peter May 142) - Second, when R is a PID, a short exact sequence 0 −→ L ′ −→ L −→ L ′′ −→ 0 of R-modules gives rise to a six-term exact sequence (J. Peter May 143) - For Rmodules L, M , and N , we have an adjunction Hom\(L ⊗ M, N \) ∼ = ∼ = Hom\(L, Hom\(M, N \)\)\. (J. Peter May 144) - We also have a natural homomorphism Hom\(L, M \) ⊗ N −→ Hom\(L, M ⊗ N \), and this is an isomorphism if either L or N is a finitely generated projective Rmodule (J. Peter May 144) - Again, we have a natural map Hom\(L, M \) ⊗ Hom\(L ′ , M ′ \) −→ Hom\(L ⊗ L ′ , M ⊗ M ′ \), which is an isomorphism if L and L′ are finitely generated and projective or if L is finitely generated and projective and M = R\. (J. Peter May 144) - For its flatness hypothesis, it is useful to remember that, for any Noetherian ring R, the dual Hom\(F, R\) of a free R-module is a flat R-module (J. Peter May 144) - The essential new feature is the cup product structure that makes the cohomology of X with coefficients in a commutative ring R a commutative graded R-algebra\. (J. Peter May 145) - For general spaces X, we can use ΓX = |S∗ X| as a canonical CW approximation functor (J. Peter May 146) - By our observations about cochain complexes (J. Peter May 147) - This product makes H ∗ \(X; R\) into a graded unital, associative, and “commutative” R-algebra\. Here commutativity is understood in the appropriate graded sense (J. Peter May 147) - In both diagrams, τ \(x ⊗ y\) = \(−1\) pq y ⊗ x if deg x = p and deg y = q\. The reason is that, on the topological level, t permutes p-cells past q-cells and, on the level of cellular chains, this involves the transposition S p+q = S p ∧ S q −→ S q ∧ S p = S p+q \. We leave it as an exercise that this map has degree \(−1\)pq \. It is this fact that forces the cup product to be commutative in the graded sense (J. Peter May 148) - In principle, the way to compute cup products is to pass to cellular chains from a cellular approximation to the diagonal map ∆\. The point is that ∆ fails to be cellular since it carries the n-skeleton of X to the 2n-skeleton of X × X\. In practice, this does not work very well and more indirect means of computation must be used\. (J. Peter May 148) - 4\. An example: RP n and the Borsuk-Ulam theorem (J. Peter May 148) - We shall later use Poincaré duality to give a quick proof that the cohomology algebra H ∗ \(RP n ; Z2 \) is a truncated polynomial algebra Z2 [x]/\(xn+1 \), where deg x = 1\. (J. Peter May 148) - We use this fact together with covering space theory to prove a celebrated result known as the Borsuk-Ulam theorem\. A map g : S m −→ S n is said to be antipodal if it takes pairs of antipodal points to pairs of antipodal points\. It then induces a map f : RP m −→ RP n such that the following diagram commutes: (J. Peter May 149) - Theorem\. If m > n ≥ 1, then there exist no antipodal maps S m −→ S n \. (J. Peter May 149) - Theorem \(Borsuk-Ulam\)\. For any continuous map f : S n −→ Rn , there exists x ∈ S n such that f \(x\) = f \(−x\)\. (J. Peter May 149) - Proof\. Suppose for a contradiction that f \(x\) 6= f \(−x\) for all x\. We could then define a continuous antipodal map g : S n −→ S n−1 by letting g\(x\) be the point at which the vector from 0 through f \(x\) − f \(−x\) intersects S n−1 \. (J. Peter May 149) - We give an outline of one of the most striking features of cohomology: the cohomology groups of a space X with coefficients in the homotopy groups of a space Y control the construction of homotopy classes of maps X −→ Y \. As a matter of motivation, this helps explain why one is interested in general coefficient groups\. (J. Peter May 150) - Definition\. Fix n ≥ 1\. A connected space X is said to be n-simple if π1 \(X\) is Abelian and acts trivially on the homotopy groups πq \(X\) for q ≤ n; X is said to be simple if it is n-simple for all n\. (J. Peter May 150) - We ask when f can be extended to a map X n+1 −→ Y that restricts to the given map on A\. (J. Peter May 150) - Let f : X n −→ Y be a map (J. Peter May 150) - If we compose the attaching maps S n → X of cells of X \ A with f , we obtain elements of πn \(Y \)\. These elements specify a well defined “obstruction cocycle” cf ∈ C n+1 \(X, A; πn \(Y \)\)\. (J. Peter May 150) - Clearly, by considering extensions cell by cell, f extends to X n+1 if and only if cf = 0 (J. Peter May 150) - considering extensions cell by cell, f extends to X n+1 if and only if cf = 0\. This is not a computable criterion\. However, if we allow ourselves to modify f a little, then we can refine the criterion to a cohomological one that often is computable\. (J. Peter May 150) - Theorem\. For f : X n −→ Y , the restriction of f to X n−1 extends to a map n+1 X → Y if and only if [cf ] = 0 in H n+1 \(X, A; πn \(Y \)\)\. (J. Peter May 150) - Theorem\. Given maps f, f ′ : X n → Y and a homotopy rel A of their restrictions to X n−1 , there is an obstruction class in H n \(X, A; πn \(Y \)\) that vanishes if and only if the restriction of the given homotopy to X n−2 extends to a homotopy f ≃ f ′ rel A\. (J. Peter May 150) - Theorem \(Mayer-Vietoris sequence\)\. Let \(X; A, B\) be an excisive triad and set C = A ∩ B\. The following sequence is exact: · · · −→ E q−1 \(C\) ∆∗ ∆∗ −−→ E q \(X\) φ∗ −→ E q \(A\) ⊕ E q \(B\) ψ∗ −−→ E q \(C\) −→ · · · \. Here, if i : C −→ A, j : C −→ B, k : A −→ X, and ℓ : B −→ X are the inclusions, then φ ∗ \(χ\) = \(k ∗ \(χ\), ℓ ∗ \(χ\)\) and ψ ∗ \(α, β\) = i ∗ \(α\) − j ∗ \(β\) and ∆∗ is the composite E q−1 \(C\) δ → δ → E − q \(A, C\) ∼ = ∼ = E q \(X, B\) −→ E q \(X\)\. (J. Peter May 155) - Theorem \(Relative Mayer-Vietoris sequence\)\. The following sequence is exact: · · · −→ E q−1 \(Y, C\) ∆∗ ∆∗ −−→ E q \(Y, X\) φ∗ −→ E q \(Y, A\) ⊕ E q \(Y, B\) ψ∗ −−→ E q \(Y, C\) −→ · · · \. Here, if i : \(Y, C\) −→ \(Y, A\), j : \(Y, C\) −→ \(Y, B\), k : \(Y, A\) −→ \(Y, X\), and ℓ : \(Y, B\) −→ \(Y, X\) are the inclusions, then φ ∗ \(χ\) = \(k ∗ \(χ\), ℓ ∗ \(χ\)\) and ψ ∗ \(α, β\) = i ∗ \(α\) − j ∗ \(β\) and ∆∗ is the composite E q−1 \(Y, C\) −→ E q−1 \(A, C\) ∼ = ∼ = E q−1 \(X, B\) δ δ − → E q \(Y, X\)\. (J. Peter May 155) - We shall use the additivity and weak equivalence axioms and the MayerVietoris sequence to explain how to compute E ∗ \(X\)\. The answer is more subtle than in homology because, algebraically, limits are less well behaved than colimits: they are not exact functors from diagrams of Abelian groups to Abelian groups\. (J. Peter May 156) - That is, we may as well define lim 1 Ai to be the displayed cokernel\. We then have the following result\. Theorem\. For each q, there is a natural short exact sequence 0 −→ lim 1 E q−1 \(Xi \) −→ E q \(X\) π π − → lim E q \(Xi \) −→ 0, where π is induced by the inclusions Xi −→ X\. (J. Peter May 156) - Rather than go into the general theory, we simply display how the “first right derived functor” lim 1 of an inverse sequence of Abelian groups can be computed (J. Peter May 156) - We say that an inverse sequence fi : Ai+1 −→ Ai satisfies the Mittag-Leffler condition if, for each fixed i, there exists j ≥ i such that, for every k > j, the image of the composite Ak −→ Ai is equal to the image of the composite Aj −→ Ai \. For example, this holds if all but finitely many of the fi are epimorphisms or if the Ai are all finite\. As a matter of algebra, we have the following vanishing result\. Lemma\. If the inverse sequence fi : Ai+1 −→ Ai satisfies the Mittag-Leffler condition, then lim 1 Ai = 0\. (J. Peter May 157) - If we assume given a theory that satisfies the axioms, we see that the cochains with coefficients in π of a CW complex X can be redefined by C n \(X; π\) = H n \(X n , X n−1 ; π\), with differential d : C n \(X; π\) −→ C n+1 \(X; π\) the composite H n \(X n , X n−1 ; π\) −→ H n \(X n \) δ δ − → H n+1 \(X n+1 , X n \)\. That is, the following result holds\. Theorem\. C ∗ \(X; π\) as just defined is isomorphic to Hom\(C∗ \(X\), π\)\. (J. Peter May 157) - Cup products are “unstable,” in the sense that they vanish on suspensions\. This is an indication of how much more information they carry than the mere additive groups\. The proof given by this sequence of exercises actually applies to any “multiplicative” cohomology theory, that is, any theory that has suitable cup products (J. Peter May 158) - Commentary: Additively, cohomology groups are “stable,” in the sense that ̃ H̃ p \(Y \) ∼ = ∼ = H ̃ H̃ p+1 \(ΣY \)\. (J. Peter May 158) - It is apparent that there is a kind of duality relating the construction of homology and cohomology\. In its simplest form, this is reflected by the fact that evaluation of cochains on chains gives a natural homomorphism C p \(X; π\) ⊗ Cp \(X; ρ\) −→ π ⊗ ρ\. This passes to homology and cohomology to give an evaluation pairing H p \(X; π\) ⊗ Hp \(X; ρ\) −→ π ⊗ ρ\. (J. Peter May 159) - When R is a field and the Hp \(X; R\) are finite dimensional vector spaces, the adjoint of this pairing is an isomorphism H p \(X; R\) ∼ = ∼ = HomR \(Hp \(X; R\), R\)\. (J. Peter May 159) - That is, the cohomology groups of X are the vector space duals of the homology groups of X\. (J. Peter May 159) - We shall study manifolds without boundary in this chapter, turning to manifolds with boundary in the next\. We do not assume that M is differentiable\. It is known that M can be given the structure of a finite CW complex, and its homology and cohomology groups are therefore finitely generated\. When M is differentiable, it is not hard to prove this using Morse theory, but it is a deep theorem in the general topological case (J. Peter May 159) - If R is a field and M is “R-orientable,” then there is an “R-fundamental class” z ∈ Hn \(M ; R\)\. The composite of the cup product and evaluation on z gives a cup product pairing H p \(M ; R\) ⊗ H n−p \(M ; R\) −→ R\. (J. Peter May 159) - One version of the Poincaré duality theorem asserts that this pairing is nonsingular, so that its adjoint is an isomorphism H p \(M ; R\) = ∼ = HomR \(H ∼ n−p \(M ; R\), R\) ∼ = ∼ = Hn−p \(M ; R\)\. (J. Peter May 159) - Theorem \(Poincaré duality\)\. Let M be a compact R-oriented n-manifold\. Then, for an R-module π, there is an isomorphism D : H p \(M ; π\) −→ Hn−p \(M ; π\)\. (J. Peter May 160) - Proposition\. If M is a compact n-manifold, then an R-orientation of M determines and is determined by an R-fundamental class z ∈ Hn \(M ; R\)\. (J. Peter May 160) - Corollary\. Let Tp ⊂ H p \(M \) be the torsion subgroup\. The cup product pairing α ⊗ β −→ hαβ, zi induces a nonsingular pairing H p \(M \)/Tp ⊗ H n−p \(M \)/Tn−p −→ Z\. (J. Peter May 160) - Corollary\. As a graded ring, H ∗ \(CP n \) is the truncated polynomial algebra Z[α]/\(αn+1 \), where deg α = 2\. That is, H 2q \(CP n \) is the free Abelian group with generator αq for 1 ≤ q ≤ n\. Proof (J. Peter May 161) - We shall see that an oriented manifold is R-oriented for any commutative ring R (J. Peter May 161) - a field\. Then α ⊗ β −→ hα ∪ β, zi defines a nonsingular pairing H p \(M ; R\) ⊗R H n−p \(M ; R\) −→ R\. We shall see that every manifold is Z2 -oriented, and an argument exactly like that for CP n allows us to compute the cup products in H ∗ \(RP n ; Z2 \)\. We used this information in our proof of the Borsuk-Ulam theorem (J. Peter May 161) - Corollary\. Let M be a connected compact R-oriented n-manifold, where R is (J. Peter May 161) - Corollary\. As a graded ring, H ∗ \(RP n ; Z2 \) is the truncated polynomial algebra Z2 [α]/\(α n+1 \), where deg α = 1\. That is, α q is the non-zero element of H q \(RP n ; Z2 \) for 1 ≤ q ≤ n\. (J. Peter May 161) - Using the evident natural map from the tensor product of homologies to the homology of a tensor product, we see that ∩ passes to homology to induce a pairing ∩ : H ∗ \(X; π\) ⊗R H∗ \(X; R\) −→ H∗ \(X; π\)\. (J. Peter May 162) - Inspecting definitions, we see that, on elements, these observations prove the fundamental identity hα ∪ β, xi = hβ, α ∩ xi\. (J. Peter May 163) - For use in the proof of the Poincaré duality theorem, we observe that the cap product generalizes to relative cap products (J. Peter May 163) - By excision, exactness, and homotopy invariance, we have isomorphisms Hi \(M, M − x\) = ∼ = Hi \(U, U − x\) ∼ = ∼ = H ∼ ̃ H̃i−1 \(U − x\) ∼ = ∼ = H ̃ H̃i−1 \(S n−1 \)\. (J. Peter May 163) - We think of Hn \(M, M − x\) as a free R-module on one generator, but the generator \(which corresponds to a unit of the ring R\) is unspecified\. Intuitively, an R-orientation of M is a consistent choice of generators\. Definition\. An R-fundamental class of M at a subspace X is an element z ∈ Hn \(M, M − X\) such that, for each x ∈ X, the image of z under the map Hn \(M, M − X\) −→ Hn \(M, M − x\) induced by the inclusion \(M, M − X\) −→ \(M, M − x\) is a generator\. If X = M , we refer to z ∈ Hn \(M \) as a fundamental class of M \. An R-orientation of M is an open cover {Ui } and R-fundamental classes zi of M at Ui such that if Ui ∩ Uj is non-empty, then zi and zj map to the same element of Hn \(M, M − Ui ∩ Uj \)\. (J. Peter May 164) - We say that M is R-orientable if it admits an R-orientation (J. Peter May 164) - Theorem \(Vanishing\)\. Let M be an n-manifold\. For any coefficient group π, Hi \(M ; π\) = 0 if i > n, and ̃ H̃n \(M ; π\) = 0 if M is connected and is not compact\. (J. Peter May 164) - We can use this together with Mayer-Vietoris sequences to construct R-fundamental classes at compact subspaces from R-orientations (J. Peter May 164) - Theorem\. Let K be a compact subset of M \. Then, for any coefficient group π, Hi \(M, M − K; π\) = 0 if i > n, and an R-orientation of M determines an Rfundamental class of M at K\. In particular, if M is compact, then an R-orientation of M determines an R-fundamental class of M (J. Peter May 164) - Corollary\. Let M be a connected compact n-manifold, n > 0\. Then either M is not orientable and Hn \(M ; Z\) = 0 or M is orientable and the map Hn \(M ; Z\) −→ Hn \(M, M − x; Z\) ∼ = ∼ =Z is an isomorphism for every x ∈ M \. (J. Peter May 165) - As an aside, the corollary leads to a striking example of the failure of the naturality of the splitting in the universal coefficient theorem (J. Peter May 165) - Thus the left and right vertical arrows are zero\. If the splittings of the rows were natural, this would imply that the middle vertical arrow is also zero (J. Peter May 165) - We begin with the general observation that homology is “compactly supported” in the sense of the following result\. Lemma\. For any space X and element x ∈ Hq \(X\), there is a compact subspace K of X and an element k ∈ Hq \(K\) that maps to x\. (J. Peter May 166) - the much more subtle statement that Hn \(M \) = 0 if M is connected and is not compact\. (J. Peter May 166) - Remember that homology is a covariant functor with compact supports\. Cohomology is a contravariant functor, and it does not have compact supports (J. Peter May 168) - We would like to prove the Poincaré duality theorem by inductive comparisons of Mayer-Vietoris sequences, and the opposite variance of homology and cohomology makes it unclear how to proceed\. To get around this, we introduce a variant of cohomology that does have compact supports and has enough covariant functoriality to allow us to proceed by comparisons of Mayer-Vietoris sequences (J. Peter May 168) - Consider the set K of compact subspaces K of M \. This set is directed under inclusion; to conform with our earlier discussion of colimits, we may view K as a category whose objects are the compact subspaces K and whose maps are the inclusions between them\. We define H q cq \(M \) = colim H q \(M, M − K\), where the colimit is taken with respect to the homomorphisms H q \(M, M − K\) −→ H q \(M, M − L\) (J. Peter May 168) - Intuitively, thinking in terms of singular cohomology, its elements are represented by cocycles that vanish off some compact subspace (J. Peter May 168) - A map f : M −→ N is said to be proper if f −1 \(L\) is compact in M when L is compact in N \. This holds, for example, if f is the inclusion of a closed subspace (J. Peter May 169) - Theorem \(Poincaré duality\)\. Let M be an R-oriented n-manifold\. Then D : Hcp cp \(M \) −→ Hn−p \(M \) is an isomorphism\. (J. Peter May 169) - We shall prove a generalization to not necessarily compact manifolds\. (J. Peter May 169) - 6\. The orientation cover There is an orientation cover of a manifold that helps illuminate the notion of orientability (J. Peter May 171) - Proposition\. Let M be a connected n-manifold\. Then there is a 2-fold cover p : ̃ M̃ −→ M such that ̃ M̃ is connected if and only if M is not orientable (J. Peter May 171) - Corollary\. If M is simply connected, or if π1 \(M \) contains no subgroup of index 2, then M is orientable\. If M is orientable, then M admits exactly two orientations (J. Peter May 172) - We can use homology with coefficients in a commutative ring R to construct an analogous R-orientation cover\. It depends on the units of R\. For example, if R = Z2 , then the R-orientation cover is the identity map of M since there is a unique unit in R\. This reproves the obvious fact that any manifold is Z2 oriented\. The evident ring homomorphism Z −→ R induces a natural homomorphism H∗ \(X; Z\) −→ H∗ \(X; R\), and we see immediately that an orientation of M induces an R-orientation of M for any R\. (J. Peter May 172) - The Poincaré duality theorem imposes strong constraints on the Euler characteristic of a manifold\. It also leads to new invariants, most notably the index\. (J. Peter May 173) - Moreover, there is a relative version of Poincaré duality in the context of manifolds with boundary, and this leads to necessary algebraic conditions on the cohomology of a manifold that must be satisfied if it is to be a boundary\. (J. Peter May 173) - We shall later outline the theory of cobordism, which leads to necessary and sufficient algebraic conditions for a manifold to be a boundary\. (J. Peter May 173) - The Euler characteristic χ\(X\) of a space with finitely generated homology is defined by istic χ\(X\) χ\(X\) = i \(−1\)i rank Hi \(X; Z\)\. P The universal coefficient theorem implies that P heorem im χ\(X\) = i \(−1\)i dim Hi \(X; F \) for any field of coefficients F (J. Peter May 173) - We may take F = Z2 , and so dispense with the requirement that M be oriented\. If n is odd, the summands of χ\(M \) cancel in pairs, and we obtain the following conclusion\. Proposition\. If M is a compact manifold of odd dimension, then χ\(M \) = 0\. (J. Peter May 173) - This pairing is nonsingular\. Since α ∪ β = \(−1\)m β ∪ α, it is skew symmetric if m is odd and is symmetric if m is even\. When m is odd, we obtain the following conclusion\. Proposition\. If M is a compact oriented n-manifold, where n ≡ 2 mod 4, then χ\(M \) is even\. (J. Peter May 174) - Lemma\. Let F be a field of characteristic 6= 2, V be a finite dimensional vector space over F , and φ : V × V −→ F be a nonsingular skew symmetric bilinear form\. Then V has a basis {x1 , \. \. \., xr , y1 , \. \. \., yr } such that φ\(xi , yi \) = 1 for 1 ≤ i ≤ r and φ\(z, w\) = 0 for all other pairs of basis elements \(z, w\)\. Therefore the dimension of V is even (J. Peter May 174) - Lemma\. Let V be a finite dimensional real vector space and φ : V × V −→ R be a nonsingular symmetric bilinear form\. Define q\(x\) = φ\(x, x\)\. Then V has a basis {x1 , \. \. \., xr , y1 , \. \. \., ys } such that φ\(z, w\) = 0 for all pairs \(z, w\) of distinct basis elements, q\(xi \) = 1 for 1 ≤ i ≤ r and q\(yj \) = −1 for 1 ≤ j ≤ s\. The number r − s is an invariant of φ, called the signature of (J. Peter May 174) - Definition\. Let M be a compact oriented n-manifold\. If n = 4k, define the index of M , denoted I\(M \), to be the signature of the cup product form H 2k \(M ; R\)⊗ H 2k \(M ; R\) −→ R\. If n 6≡ 0 mod 4, define I\(M \) = 0\. (J. Peter May 174) - Proposition\. For any compact oriented n-manifold, χ\(M \) ≡ I\(M \) mod 2\. (J. Peter May 174) - The Euler characteristic and index are related by the following congruence\. (J. Peter May 174) - Lemma\. If M and M ′ are compact oriented n-manifolds, then I\(M ∐ M ′ \) = I\(M \) + I\(M ′ \), where M ∐ M ′ is given the evident orientation induced from those of M and M ′ \. (J. Peter May 175) - Lemma\. Let M be a compact oriented m-manifold and N be a compact oriented n-manifold\. Then I\(M × N \) = I\(M \) · I\(N \), where M × N is given the orientation induced from those of M and N \. (J. Peter May 175) - Let Hn = {\(x1 , \. \. \., xn \)|xn ≥ 0} be the upper half-plane in Rn \. Recall that an n-manifold with boundary is a Hausdorff space M having a countable basis of open sets such that every point of M has a neighborhood homeomorphic to an open subset of H n \. A point x is an interior point if it has a neighborhood homeomorphic to an open subset of Hn − ∂Hn ∼ = Rn ; otherwise it is a boundary point\. It is a fact called “invariance of domain” that if U and V are homeomorphic subspaces of Rn and U is open, then V is open\. Therefore, a homeomorphism of an open subspace of Hn onto an open subspace of Hn carries boundary points to boundary points\. (J. Peter May 176) - We denote the boundary of an n-manifold M by ∂M \. Thus M is a manifold without boundary if ∂M is empty; M is said to be closed if, in addition, it is compact\. The space ∂M is an \(n − 1\)-manifold without boundary\. (J. Peter May 176) - It is a fundamental question in topology to determine which closed manifolds are boundaries (J. Peter May 176) - we can ask whether or not a smooth \(= differentiable\) closed manifold is the boundary of a smooth manifold \(with the induced smooth structure\) (J. Peter May 176) - Remember that χ\(M \) = 0 if M is a closed manifold of odd dimension\. (J. Peter May 176) - Proposition\. If M = ∂W , where W is a compact \(2m + 1\)-manifold, then χ\(M \) = 2χ\(W \)\. (J. Peter May 176) - Corollary\. If M = ∂W for a compact manifold W , then χ\(M \) is even\. (J. Peter May 176) - For example, since χ\(RP 2m \) = 1 and χ\(CP n \) = n+ 1, this criterion shows that 2m RP and CP 2m cannot be boundaries\. Notice that we have proved that these are not boundaries of topological manifolds, let alone of smooth ones\. (J. Peter May 177) - The index gives a more striking criterion: if a closed oriented 4k-manifold M is the boundary of a \(topological\) manifold, then I\(M \) = 0\. (J. Peter May 177) - In the case of smooth manifolds, it can be seen in terms of inward-pointing unit vectors of the normal line bundle of the embedding ∂M −→ M \. (J. Peter May 177) - We let M be an n-manifold with boundary, n > 0, throughout this section, and we let R be a given commutative ring\. We say that M is R-orientable \(or orientable if R = Z\) if its interior ̊ M̊ = M − ∂M is R-orientable; similarly, an R-orientation of M is an R-orientation of its interior (J. Peter May 177) - Theorem \(Topological collaring\)\. There is an open neighborhood V of ∂M in M such that the identification ∂M = ∂M × {0} extends to a homeomorphism V ∼ = ∼ = ∂M × [0, 1\)\. (J. Peter May 177) - It follows that the inclusion M̊ M̊ −→ M is a homotopy equivalence and the inclusion ∂M −→ M is a cofibration (J. Peter May 177) - Proposition\. An R-orientation of M determines an R-orientation of ∂M \. (J. Peter May 177) - Proposition\. If M is compact and R-oriented and z∂M ∈ Hn−1 \(∂M \) is the fundamental class determined by the induced R-orientation on ∂M , then there is a unique element z ∈ Hn \(M, ∂M \) such that ∂z = z∂M ; z is called the R-fundamental class determined by the R-orientation of M \. (J. Peter May 178) - Theorem \(Relative Poincaré duality\)\. Let M be a compact R-oriented nmanifold with R-fundamental class z ∈ Hn \(M, ∂M ; R\)\. Then, with coefficients taken in any R-module π, capping with z specifies duality isomorphisms (J. Peter May 178) - Theorem\. If M is the boundary of a compact oriented \(4k + 1\)-manifold, then I\(M \) = 0\. (J. Peter May 179) - We first give an algebraic criterion for the vanishing of the signature of a form and then show that the cup product form on the middle dimensional cohomology of M satisfies the criterion\. Lemma\. Let W be a n-dimensional subspace of a 2n-dimensional real vector space V \. Let φ : V × V −→ R be a nonsingular symmetric bilinear form such that φ : W × W −→ R is identically zero\. Then the signature of φ is zero\. (J. Peter May 179) - We here give a homotopical way of constructing ordinary theories that makes no use of chains, whether cellular or singular\. We also show how to construct cup and cap products homotopically\. This representation of homology and cohomology in terms of Eilenberg-Mac Lane spaces is the starting point of the modern approach to homology and cohomology theory, and we shall indicate how theories that do not satisfy the dimension axiom can be represented (J. Peter May 183) - indicate how theories that do not satisfy the dimension axiom can be represented\. We shall also describe Postnikov systems, which give a way to approximate general \(simple\) spaces by weakly equivalent spaces built up out of Eilenberg-Mac Lane spaces\. This is conceptually dual to the way that CW complexes allow the approximation of spaces by weakly equivalent spaces built up out of spheres (J. Peter May 183) - Finally, we present the important notion of cohomology operations and relate them to the cohomology of Eilenberg-Mac Lane spaces\. (J. Peter May 183) - Recall that a reduced homology theory on based CW complexes is a sequence of functors E ̃ Ẽq from the homotopy category of based CW complexes to the category of Abelian groups\. (J. Peter May 183) - By a result of Milnor, if X has the homotopy type of a CW complex, then so does ΩX\. By the Whitehead theorem, we therefore have a homotopy equivalence σ̃ : K\(π, n\) −→ ΩK\(π, n + 1\)\. (J. Peter May 183) - This map is the adjoint of a map σ : ΣK\(π, n\) −→ K\(π, n + 1\)\. (J. Peter May 183) - Theorem\. For CW complexes X, Abelian groups π and integers n ≥ 0, there are natural isomorphisms ̃ H̃q \(X; π\) ∼ = ∼ colimn πq+n \(X ∧ K\(π, n\)\)\. = (J. Peter May 184) - Definition\. A prespectrum is a sequence of based spaces Tn , n ≥ 0, and based maps σ : ΣTn −→ Tn+1 \. (J. Peter May 184) - If X = S 0 , then X ∧ K\(π, n\) = K\(π, n\)\. (J. Peter May 184) - The example at hand is the Eilenberg-Mac Lane prespectrum {K\(π, n\)}\. Another example is the “suspension prespectrum” {Σn X} of a based space X; the required maps Σ\(Σn X\) −→ Σn+1 X are the evident identifications\. When X = S 0 , this is called the sphere prespectrum\. (J. Peter May 184) - Theorem\. Let {Tn } be a prespectrum such that Tn is \(n − 1\)-connected and of the homotopy type of a CW complex for each n\. Define ̃ Ẽq \(X\) = colimn πq+n \(X ∧ Tn \), where the colimit is taken over the maps πq+n \(X ∧ Tn \) Σ Σ → πq+n+1 \(Σ\(X ∧ Tn \)\) − ∼ = ∼ = πq+n+1 \(X ∧ ΣTn \) id ∧σ id ∧σ −−−→ πq+n+1 \(X ∧ Tn+1 \)\. Then the functors ̃ Ẽq define a reduced homology theory on based CW complexes\. (J. Peter May 184) - We need some preliminaries to prove the ad Definition\. Define the weak product i i Yi of a set of based spaces Yi to Definition\. Def be the subspace of i Qw Yi consisting of those points all but finitely many of whose Q coordinates are basepoints (J. Peter May 184) - Since passage to colimits preserves exact sequences (J. Peter May 184) - Lemma\. For a set of based spaces {Yi }, the canonical map i πq \(Yi \) −→ πq \( i Yi \) i spaces {Yi }, the c πq \(Yi \) −→ πq \( i i Yi \) is an isomorphism\. (J. Peter May 185) - Example\. Applying the theorem to the sphere prespectrum, we find that the stable homotopy groups πqs qs \(X\) give the values of a reduced homology theory; it is called “stable homotopy theory (J. Peter May 185) - Theorem\. For CW complexes X, Abelian groups π, and integers n ≥ 0, there are natural isomorphisms ̃ H̃ n \(X; π\) ∼ = ∼ = [X, K\(π, n\)]\. (J. Peter May 185) - If Z has a multiplication φ : Z × Z −→ Z such that the basepoint ∗ of Z is a two-sided unit up to homotopy, so that Z is an “H-space,” then φ induces an “addition” [X, Z] × [X, Z] −→ [X, Z]\. (J. Peter May 186) - \. We say that Z is “grouplike” if there is a map χ : Z −→ Z such that φ\(id × χ\)∆ : Z −→ Z is homotopic to the trivial map, and then χ∗ : [X, Z] −→ [X, Z] sends an element x ∈ [X, Z] to x−1 \. (J. Peter May 186) - \. If Z is a grouplike homotopy associative and commutative H-space, then the functor [X, Z] takes values in Abelian groups\. (J. Peter May 186) - Actually, the existence of inverses can be deduced if Z is only “grouplike” in the weaker sense that π0 \(X\) is a group, but we shall not need the extra generality\. Now consider the multiplication on a loop space ΩY given by composition of loops\. Our proof that π1 \(Y \) is a group and π2 \(Y \) is an Abelian group amounts to a proof of the following result\. Lemma\. For any based space Y , ΩY is a grouplike homotopy associative Hspace and Ω 2 Y is a grouplike homotopy associative and commutative H-space\. (J. Peter May 186) - Definition\. An Ω-prespectrum is a sequence of based spaces Tn and weak homotopy equivalences σ̃ : Tn −→ ΩTn+1 \. (J. Peter May 186) - It is a consequence of a general result called the Brown representability theorem that every reduced cohomology theory is represented in this fashion by an Ω-prespectrum\. (J. Peter May 187) - We can also construct cap products homotopically\. To do so, it is convenient to bring function spaces into play, using the obvious isomorphisms [X, Y ] ∼ = ∼ = π0 F \(X, Y \) and evaluation maps ε : F \(X, Y \) ∧ X −→ Y\. (J. Peter May 188) - We have implicitly studied the represented functors k\(X\) = [X, Y ] by decomposing X into cells\. This led in particular to the calculation of ordinary represented cohomology [X, K\(π, n\)] by means of cellular chains (J. Peter May 190) - There is an Eckmann-Hilton dual way of studying [X, Y ] by decomposing Y into “cocells\.” We briefly describe this decomposition of spaces into their “Postnikov systems” here (J. Peter May 190) - This decomposition answers a natural question: how close are the homotopy groups of a CW This deco groups of a C Q type? Since n complex X to being a complete set of invariants for its homotopy K\(πn \(X\), n\) has the same homotopy groups as X but is generally not weakly homotopy equivalent to it, some added information is needed\. If X is simple, it turns out that the homotopy groups together with an inductively defined sequence of cohomology classes give a complete set of invariants\. (J. Peter May 190) - Recall that a connected space X is said to be simple if π1 \(X\) is Abelian and acts trivially on πn \(X\) for n ≥ 2\. A Postnikov system for a simple based space X consists of based spaces Xn together with based maps αn : X −→ Xn and pn+1 : Xn+1 −→ Xn , (J. Peter May 190) - n ≥ 1, such that pn+1 ◦αn+1 = αn , X1 is an Eilenberg-Mac Lane space K\(π1 \(X\), 1\), pn+1 is the fibration induced from the path space fibration over an EilenbergMac Lane space K\(πn+1 \(X\), n + 2\) by a map k n+2 : Xn −→ K\(πn+1 \(X\), n + 2\), (J. Peter May 191) - and αn induces an isomorphism πq \(X\) → πq \(Xn \) for q ≤ n\. It follows that πq \(Xn \) = 0 for q > n\. (J. Peter May 191) - The system can be displayed diagrammatically as follows (J. Peter May 191) - The maps αn induce a weak equivalence X → lim Xn , but the inverse limit generally will not have the homotopy type of a CW complex\. The a CW complex\. The maps αn induce a weak equivalence X → lim Xn , but the inverse limit generally will not have the homotopy type of a CW complex\. The “k-invariants” k n+2 that specify the system are to be regarded as cohomology classes k n+2 ∈ H n+2 \(Xn ; πn+1 \(X\)\)\. (J. Peter May 191) - By our definition of a Postnikov system, we must define Xn+1 to be the homotopy fiber of k n+2 \. Thus its points are pairs \(ω, x\) consisting of a path ω : I → K\(πn+1 \(X\), n+2\) and a point x ∈ Xn such that ω\(0\) = ∗ and ω\(1\) = k n+2 \(x\)\. (J. Peter May 192) - Lemma \(Yoneda\)\. There is a canonical bijection between natural transformations Φ : k −→ k ′ and elements φ ∈ k ′ \(Z\)\. (J. Peter May 192) - Consider a “represented functor” k\(X\) = [X, Z] and another contravariant functor k ′ from the homotopy category of based CW complexes to the category of sets (J. Peter May 192) - Corollary\. There is a canonical bijection between natural transformations Φ : [−, Z] −→ [−, Z ′ ] and elements φ ∈ [Z, Z ′ ]\. (J. Peter May 192) - Theorem\. Cohomology operations ̃ H̃ q \(−; π\) −→ H ̃ H̃ q+n \(−; ρ\) are in canonical bijective correspondence with elements of ̃ H̃ q+n \(K\(π, q\); ρ\)\. (J. Peter May 192) - In general, cohomology operations are only natural transformations of setvalued functors\. However, stable operations are necessarily homomorphisms of cohomology groups (J. Peter May 192) - \. To determine all cohomology operations, we need only compute the cohomology of all EilenbergMac Lane spaces (J. Peter May 193) - We have described an explicit construction of these spaces as topological Abelian groups in Chapter 16 §5, and this construction leads to an inductive method of computation (J. Peter May 193) - Theorem\. For n ≥ 0, there are stable cohomology operations Sq n : H q \(X; Z2 \) −→ H q+n \(X; Z2 \), called the Steenrod operations\. They satisfy the following properties (J. Peter May 193) - There are also formulas, called the Adem relations, describing Sq i Sq j , as a linear combination of operations Sq i+j−k Sq k , 2k (J. Peter May 193) - It turns out that the Steenrod operations generate all mod 2 cohomology operations\. In fact, the identity map of K\(Z2 , q\) specifies a fundamental class ιq ∈ H q \(K\(Z2 , q\); Z2 \), and the following theorem holds\. Theorem\. H ∗ \(K\(Z2 , q\); Z2 \) is a polynomial algebra whose generators are certain iterates of Steenrod operations applied to the fundamental class ιq \. Explicitly, writing Sq I = Sq i1 · · · Sq ij for a sequence of positive integers I = {i1 , \. \. \., ij }, the generators are the Sq I ιq for those sequences I such that ir ≥ 2ir+1 for 1 ≤ r < j and i1 < i2 + · · · + ij + q\. (J. Peter May 193) - The β are called Bockstein operations\. (J. Peter May 193) - For Abelian groups π and ρ, show that [K\(π, n\), K\(ρ, n\)] ∼ = ∼ = Hom\(π, ρ\)\. (J. Peter May 193) - We shall require our open covers to be numerable, as can always be arranged when B is paracompact (J. Peter May 195) - In the case of non-connected base spaces, the fibers over points in different components may have different dimension (J. Peter May 195) - A map \(g, f \) of vector bundles is an isomorphism if and only if f is a homeomorphism and g restricts to an isomorphism on each fiber\. (J. Peter May 195) - We say that two vector bundles over B are equivalent if they are isomorphic over B, so that there is an isomorphism \(g, id\) between them\. We let En \(B\) denote the set of equivalence classes of n-plane bundles over B (J. Peter May 195) - Thus we have a contravariant set-valued functor En \(−\) on spaces (J. Peter May 195) - Vector bundles should be thought of as rather rigid geometric objects, and the equivalence relation between them preserves that rigidity\. Nevertheless, equivalence classes of n-plane bundles can be classified homotopically\. (J. Peter May 195) - In turn, the starting point of the classification theorem is the observation that the functor En \(−\), like homology and cohomology, is homotopy invariant in the sense that it factors (J. Peter May 195) - through the homotopy category hU \. In less fancy language, this amounts to the following result\. Proposition\. The pullbacks of an n-plane bundle p : E −→ B along homotopic maps f0 , f1 : A −→ B are equivalent\. (J. Peter May 196) - It can be verified on general abstract nonsense grounds, using Brown’s representability theorem, that the functor En \(−\) is representable in the form [−, BO\(n\)] (J. Peter May 196) - It is far more useful to have an explicit concrete construction of the relevant “classifying space” BO\(n\)\. (J. Peter May 196) - We construct a particular n-plane bundle γn : En −→ BO\(n\), called the “universal n-plane bundle\.” By pulling back γn along \(homotopy classes of\) maps f : B −→ BO\(n\), we obtain a natural transformation of functors [−, BO\(n\)] −→ En \(−\)\. We show that this natural transformation is a natural isomorphism of functors by showing how to construct a map \(g, f \), unique up to homotopy, from any given n-plane bundle E over any space B to the universal n-plane bundle En ; it is in this sense that En is “universal\.” (J. Peter May 196) - Let Vn \(R q \) be the Stiefel variety of orthonormal n-frames in R q \. Its points are n-tuples of orthonormal vectors in Rq , and it is topologized as a subspace of \(Rq \)n or, equivalently, as a subspace of \(S q−1 \)n \. It is a compact manifold\. Let Gn \(Rq \) be the Grassmann variety of n-planes in Rq \. Its points are the n-dimensional subspaces of Rq \. Sending an n-tuple of orthonormal vectors to the n-plane they span gives a surjective function Vn \(Rq \) −→ Gn \(Rq \), and we topologize Gn \(Rq \) as a quotient space of Vn \(Rq \)\. It too is a compact manifold (J. Peter May 196) - We define the classifying space BO\(n\) to be Gn \(R∞ \)\. (J. Peter May 196) - Let Enq nq be the subbundle of the trivial bundle Gn \(Rq \) × Rq whose points are the pairs \(x, v\) such that v is a vector in the plane x; denote the projection of Enq nq by γnq nq , so that γnq nq \(x, v\) = x\. When n = 1, γ is called the “canonical line bundle” over RP q−1 \. (J. Peter May 196) - We may let q go to infinity\. We let En = En∞ and let γn = γn∞ : En −→ BO\(n\) (J. Peter May 196) - Theorem\. The natural transformation Φ : [−, BO\(n\)] −→ En \(−\) obtained by sending the homotopy class of a map f : B −→ BO\(n\) to the equivalence class of the n-plane bundle f ∗ En is a natural isomorphism of functors\. (J. Peter May 197) - it is called the Gauss map of the tangent bundle of M (J. Peter May 197) - Similarly, using the orthogonal complements of tangent planes, we obtain the Gauss map E\(ν\) −→ E q q−n of the q normal bundle ν of the embedding of M in R \. (J. Peter May 197) - Definition\. Let k ∗ be a cohomology theory, such as H ∗ \(−; π\) for an Abelian group π\. A characteristic class c of degree q for n-plane bundles is a natural assignment of a cohomology class c\(ξ\) ∈ k q \(B\) to bundles ξ with base space B\. (J. Peter May 197) - Thus, if \(g, f \) is a map from a bundle ζ over A to a bundle ξ over B, so that ζ is equivalent to f ∗ ξ, then f ∗ c\(ξ\) = c\(ζ\)\. (J. Peter May 198) - Since the functor En is represented by BO\(n\), the Yoneda lemma specializes to give the following result\. Lemma\. Evaluation on γn specifies a canonical bijection between characteristic classes of n-plane bundles and elements of k ∗ \(BO\(n\)\)\. (J. Peter May 198) - classes of n-plane bundles and elements of k ∗ \(BO\(n\)\)\. The formal similarity to the definition of cohomology operations is obvious (J. Peter May 198) - Moreover, the behavior of characteristic classes with respect to operations on bundles can be determined by calculating the maps on cohomology induced by maps between classifying spaces (J. Peter May 198) - We are particularly interested in Whitney sums of bundles (J. Peter May 198) - The internal sum, or Whitney sum, of two bundles over the same base space B is obtained by pulling back their external sum along the diagonal map of B (J. Peter May 198) - For example, let ε denote the trivial line bundle over any space\. We have the operation that sends an n-plane bundle ξ over B to the \(n + 1\)-plane bundle ξ ⊕ ε over B\. There is a classifying map in : BO\(n\) −→ BO\(n + 1\) that is characterized up to homotopy by i∗n n \(γn+1 \) = γn ⊕ ε\. (J. Peter May 198) - Theorem\. For n-plane bundles ξ over base spaces B, n ≥ 0, there are characteristic classes wi \(ξ\) ∈ H i \(B; Z2 \), i ≥ 0, called the Stiefel-Whitney classes\. They satisfy and are uniquely characterized by the following axioms\. (J. Peter May 199) - Every mod 2 characteristic class for n-plane bundles can be written uniquely as a polynomial in the Stiefel-Whitney classes {w1 , \. \. \., wn }\. (J. Peter May 199) - The mod 2 cohomology H ∗ \(BO\(n\); Z2 \) is the polynomial algebra Z2 [w1 , \. \. \., wn ]\. (J. Peter May 199) - P to consider as formal sums xi , deg xi = i guarantee that the sum isPfinite\. a vector bundle ξ to be wi \(ξ\); (J. Peter May 199) - Suppose that M immerses in Rq with normal bundle ν\. Then τ \(M \) ⊕ ν ∼ = ∼ = εq and we have the “Whitney duality formula” w\(M \) ∪ w\(ν\) = 1, (J. Peter May 199) - which shows how to calculate tangential Stiefel-Whitney classes in terms of normal Stiefel-Whitney classes, and conversely\. This formula can be used to prove non-immersion results when we know w\(M \) (J. Peter May 199) - Since the normal bundle of the standard embedding S q −→ Rq+1 is trivial, w\(S q \) = 1\. A manifold is said to be parallelizable if its tangent bundle is trivial\. For some manifolds M , we can show that M is not parallelizable by showing that one of its Stiefel-Whitney classes is non-zero, but this strategy fails for M = S q \. (J. Peter May 200) - We describe some standard computations in the cohomology of projective spaces that give less trivial examples\. Write ζq for the canonical line bundle over RP q in this section (J. Peter May 200) - For example, w\(RP q \) = 1 if and only if q = 2k − 1 for some k \(as the reader should check\) and therefore RP q can be parallelizable only if q is of this form (J. Peter May 200) - If R q+1 admits a bilinear product without zero divisors, then it is not hard to prove that τ \(RP q \) ∼ = ∼ = Hom\(ζq , ζ ⊥ q⊥ \) admits q linearly independent cross-sections and is therefore trivial\. We conclude that Rq+1 can admit such a product only if q+1 = 2k for some k\. The real numbers, complex numbers, quaternions, and Cayley numbers show that there is such a product for q + 1 = 1, 2, 4, and 8\. As we shall explain in the next chapter, these are in fact the only q for which Rq+1 admits such a product (J. Peter May 200) - While the calculation of w\(RP q \) just given is quite special, there is a remarkable general recipe, called the “Wu formula,” for the computation of w\(M \) in terms of Poincaré duality and the Steenrod operations in H ∗ \(M ; Z2 \)\. In analogy with w\(M \), general recipe, called the “Wu formula,” for the computation of w\( Poincaré duality and the Steenrod operations in H ∗ \(M ; Z2 \)\. In anal we define the total Steenrod square of an element x by Sq\(x\) = i Sq i \(x\)\. (J. Peter May 200) - Theorem \(Wu formula\)\. Let M be a smooth closed n-manifold with funda\(M ; Z2 \)\. Then the total Stiefel-Whitney class w\(M \) is equal Theorem \(Wu formula\)\. Let M be a smooth closed n-manifold with funda mental class z ∈ HnP \(M ; Z2 \)\. Then the total Stiefel-Whitney class w\(M \) is equ to Sq\(v\), where v = vi ∈ H ∗∗ \(M ; Z2 \) is the unique cohomology class such that hv ∪ x, zi = hSq\(x\), zi for all x ∈ H \(M ; Z2 \)\. Thus, for k ≥ 0, vk ∪ x = Sq k \(x\) for all x ∈ H n−k \(M ; Z2 \), ∗ and wk \(M \) = vk i+j=k Sq i \(vj \)\. (J. Peter May 201) - The basic reason that such a formula holds is that the StiefelWhitney classes can be defined in terms of the Steenrod operations (J. Peter May 201) - Since the tangent bundle of M depends on its smooth structure, this is rather surprising (J. Peter May 201) - Characteristic classes determine important numerical invariants of manifolds, called their characteristic numbers\. (J. Peter May 201) - Definition\. Let M be a smooth closed R-oriented n-manifold with fundamental class z ∈ Hn \(M ; R\)\. For a characteristic class c of degree n, define the tangential characteristic number c[M ] ∈ R by c[M ] = hc\(τ \(M \)\), zi\. Similarly, define the normal characteristic number c[ν\(M \)] by c[ν\(M \)] = hc\(ν\(M \)\), zi, where ν\(M \) is the normal bundle associated to an embedding of M in Rq for q sufficiently large\. \(These numbers are well defined because any two embeddings of M in Rq for large q are isotopic and have equivalent normal bundles\.\) (J. Peter May 201) - arge q are isotopic and have equivalent normal bundles\.\) In particular, if ri are integers such that iri = n, then the monomial w r1 · · · wnrn nrn P is a characteristic class of degree n, and all mod 2 characteristic classes of degree n are linear combinations of these\. Different manifolds can have the same Stiefel-Whitney numbers (J. Peter May 201) - Lemma\. If M is the boundary of a smooth compact \(n + 1\)-manifold W , then all tangential Stiefel-Whitney numbers of M are zero\. (J. Peter May 201) - Lemma\. All tangential Stiefel-Whitney numbers of a smooth closed manifold M are zero if and only if all normal Stiefel-Whitney numbers of M are zero\. (J. Peter May 201) - Theorem \(Thom\)\. If M is a smooth closed n-manifold all of whose normal Stiefel-Whitney numbers are zero, then M is the boundary of a smooth \(n + 1\)manifold\. (J. Peter May 202) - Thus we need only compute the Stiefel-Whitney numbers of M to determine whether or not it is a boundary\. By Wu’s formula, the computation only requires knowledge of the mod 2 cohomology of M , with its Steenrod operations (J. Peter May 202) - There are several ways to construct the Stiefel-Whitney classes\. The most illuminating one depends on a simple, but fundamentally important, construction on vector bundles, namely their “Thom spaces\.” (J. Peter May 202) - Definition\. Let ξ : E −→ B be an n-plane bundle\. Apply one-point compactification to each fiber of ξ to obtain a new bundle Sph\(E\) over B whose fibers are spheres S n with given basepoints, namely the points at ∞\. These basepoints specify a cross-section B −→ Sph\(E\)\. Define the Thom space T ξ to be the quotient space T \(ξ\) = Sph\(E\)/B\. That is, T \(ξ\) is obtained from E by applying fiberwise one-point compactification and then identifying all of the points at ∞ to a single basepoint \(denoted ∞\)\. (J. Peter May 202) - Remark\. If we give the bundle ξ a Euclidean metric and let D\(E\) and S\(E\) denote its unit disk bundle and unit sphere bundle, then there is an evident homeomorphism between T ξ and the quotient space D\(E\)/S\(E\) (J. Peter May 202) - In turn, D\(E\)/S\(E\) is homotopy equivalent to the cofiber of the inclusion S\(E\) −→ D\(E\) and therefore to the cofiber of the projection S\(E\) −→ B\. (J. Peter May 202) - If the bundle ξ is trivial, so that E = B × Rn , then Sph\(E\) = B × S n \. Quotienting out B amounts to the same thing as giving B a disjoint basepoint and then forming the smash product B+ ∧ S n \. That is, in this case the Thom complex is Σn B+ \. Therefore, for any cohomology theory k ∗ , k q \(B\) = k̃ k̃ q \(B+ \) ∼ = ∼ = k̃ k̃ n+q \(T ξ\)\. (J. Peter May 202) - which is called the “Thom diagonal\. (J. Peter May 203) - This should look very similar to the problem of patching local fundamental classes to obtain a global one; that is, it looks like a question of orientation (J. Peter May 203) - Definition\. Let ξ : E −→ B be an n-plane bundle\. An R-orientation, or Thom class, of ξ is an element µ ∈ ̃ H̃ n \(T ξ; R\) such that, for every point b ∈ B, i∗b b \(µ\) is a generator of the free R-module ̃ H̃ n \(Sbn bn \)\. (J. Peter May 203) - We leave it as an instructive exercise to verify that an R-orientation of a closed n-manifold M determines and is determined by an R-orientation of its tangent bundle τ \(M \)\. (J. Peter May 203) - Theorem \(Thom isomorphism theorem\)\. Let µ ∈ H ̃ H̃ n \(T ξ; R\) be a Thom class for an n-plane bundle ξ : E −→ B\. Define Φ : H q \(B; R\) −→ H ̃ H̃ n+q \(T ξ; R\) by Φ\(x\) = x ∪ µ\. Then Φ is an isomorphism\. (J. Peter May 203) - However, much the best proof from the point of view of anyone seriously interested in algebraic topology is to apply the Serre spectral sequence of the bundle Sph\(E\)\. (J. Peter May 203) - \. Use of a field ensures that the cohomology of the relevant direct limits is the inverse limit of the cohomologies (J. Peter May 203) - The Serre spectral sequence is a device for computing the cohomology of the total space E of a fibration from the cohomologies of its base B and fiber F \. It measures the cohomological deviation of H ∗ \(E\) from H ∗ \(B\)⊗H ∗ \(F \) (J. Peter May 203) - Just as in orientation theory for manifolds, the question of orientability depends on the structure of the units of the ring R, and this leads to the following conclusion\. Proposition\. Every vector bundle admits a unique Z2 -orientation\. This can be proved along with the Thom isomorphism theorem by a MayerVietoris argument (J. Peter May 204) - First, taking the characteristic class point of view, we define the Stiefel-Whitney classes in terms of the Steenrod operations by setting wi \(ξ\) = Φ −1 Sq i Φ\(1\) = Φ −1 Sq i (J. Peter May 204) - Here S\(γ1 \) is the infinite sphere S ∞ , which is the universal cover of RP ∞ and is therefore contractible (J. Peter May 204) - We indicate two constructions of the Stiefel-Whitney classes (J. Peter May 204) - As we shall explain in §8, passage from topological groups to their classifying spaces is a product-preserving functor, at least up to homotopy\. (J. Peter May 204) - Indeed, up to homotopy, inner conjugation by an element of G induces the identity map on BG for any topological group G\. (J. Peter May 204) - \. The subring H ∗ \(\(RP ∞ \)n ; Z2 \)Σn of elements invariant under the action is the polynomial algebra on the elementary symmetric functions σi , 1 ≤ i ≤ n, in the variables αi \. (J. Peter May 205) - By the Künneth theorem, we see that H ∗ \(\(RP ∞ \) n ; Z2 \) = ⊗ i i=1 H ∗ \(RP ∞ ; Z2 \) = Z2 [α1 , \. \. \., αn ], (J. Peter May 205) - The resulting map ω ∗ : H ∗ \(BO\(n\); Z2 \) −→ H ∗ \(\(RP ∞ \) n ; Z2 \) Σn is a ring homomorphism between polynomial algebras on generators of the same degrees\. It turns out to be a monomorphism and therefore an isomorphism (J. Peter May 205) - One advantage of this approach is that, since we know the Steenrod operations on H ∗ \(RP ∞ ; Z2 \) and can read them off on H ∗ \(\(RP ∞ \) n ; Z2 \) by the Cartan formula, it leads to a purely algebraic calculation of the Steenrod operations in H ∗ \(BO\(n\); Z2 \)\. Explicitly, the following “Wu formula” holds: Sq i \(wj \) = fol i t=0 lowing “Wu for j+t−i−1 t u wi−t wj+t \. (J. Peter May 205) - The proof of the classification theorem for complex n-plane bundles works in exactly the same way as for real n-plane bundles, using complex Grassmann varieties\. (J. Peter May 205) - In fact, the fundamental groups of the real Grassmann varieties are Z2 , and their universal covers are their orientation covers\. (J. Peter May 205) - These covers are the oriented Grassmann varieties ̃ G̃n \(Rq \)\. (J. Peter May 205) - We write BU \(n\) = Gn \(C∞ \) and BSO\(n\) = G̃n \(R∞ \), (J. Peter May 205) - and we construct universal complex n-plane bundles γn : EUn −→ BU \(n\) and oriented n-plane bundles γ̃n : E ̃ Ẽn −→ BSO\(n\) as in the first section (J. Peter May 205) - Let E Un \(B\) denote the set of equivalence classes of complex nplane bundles over B and let E Ẽ Ẽn \(B\) denote the set of equivalence classes of oriented real n-plane bundles over B; (J. Peter May 205) - Theorem\. The natural transformation Φ : [−, BU \(n\)] −→ E Un \(−\) obtained by sending the homotopy class of a map f : B −→ BU \(n\) to the equivalence class of the n-plane bundle f ∗ EUn is a natural isomorphism of functors\. (J. Peter May 205) - Theorem\. The natural transformation Φ : [−, BSO\(n\)] −→ Ẽ Ẽn \(−\) obtained by sending the homotopy class of a map f : B −→ BSO\(n\) to the equivalence class of the oriented n-plane bundle f ∗ Ẽn is a natural isomorphism of functors\. (J. Peter May 206) - Lemma\. Evaluation on γn specifies a canonical bijection between characteristic classes of complex n-plane bundles and elements of k ∗ \(BU \(n\)\)\. Lemma\. Evaluation on γ̃n specifies a canonical bijection between characteristic classes of oriented n-plane bundles and elements of k ∗ \(BSO\(n\)\)\. (J. Peter May 206) - Theorem\. H ∗ \(BSO\(n\); Z2 \) ∼ = Z2 [w2 , \. \. \., wn ]\. (J. Peter May 206) - The Thom space of a complex or oriented real vector bundle is the Thom space of its underlying real vector bundle\. We obtain characteristic classes in cohomology with any coefficients by applying cohomology operations to Thom classes, but it is rarely the case that the resulting characteristic classes generate all characteristic classes: the cases H ∗ \(BO\(n\); Z2 \) and H ∗ \(BSO\(n\); Z2 \) are exceptional (J. Peter May 206) - Theorem\. For n ≥ 1, there are elements ci ∈ H 2i \(BU \(n\); Z\), i ≥ 0, called the Chern classes\. They satisfy and are uniquely characterized by the following axioms\. \(1\) c0 = 1 and ci = 0 if i > n\. \(2\) c1 is the canonical generator of H 2 \(BU \(1\); Z\) when n = 1\. \(3\) i∗n ∗n \(ci \) = ci \. \(4\) p∗m m,n \) = ci \. \(ci \) = j j=0 cj ⊗ ci−j \. The integral cohomology H ∗ \(BU \(n\); Z\) is the polynomial algebra Z[c1 , \. \. \., cn ]\. (J. Peter May 207) - The reader deserves to be warned about a basic inconsistency in the literature\. Remark\. With the discussion above, c1 \(γ n+1 1n \) is the canonical generator of H \(CP n ; Z\), where γ1n+1 2 1n is the canonical line bundle of lines in Cn+1 and points on the line\. This is the standard convention in algebraic topology\. In algebraic geometry, it is more usual to define Chern classes so that the first Chern class of the dual of γ n+1 1n is the canonical generator of H 2 \(CP n ; Z\)\. With this convention, the nth Chern class would be \(−1\)n cn \. It is often unclear in the literature which convention is being followed\. (J. Peter May 207) - Turning to oriented real vector bundles, we define the Pontryagin and Euler classes as follows, taking cohomology with coefficients in any commutative ring R\. Definition\. Define the Pontryagin classes pi ∈ H 4i \(BO\(n\); R\) by pi = \(−1\) i c ∗ \(c2i \), c ∗ : H 4i \(BU \(n\); R\) −→ H 4i \(BO\(n\); R\); also write pi for πn∗ \(pi \) ∈ H 4i \(BSO\(n\); R\)\. (J. Peter May 207) - Definition\. Define the Euler class e\(ξ\) ∈ H n \(B; R\) of an R-oriented n-plane bundle ξ over the base space B by e\(ξ\) = Φ−1 µ2 , where µ ∈ H n \(T ξ; R\) is the Thom class\. Giving the universal oriented n-plane bundle over BSO\(n\) the R-orientation induced by its integral orientation, this defines the Euler class e ∈ H n \(BSO\(n\); R\)\. (J. Peter May 207) - The name “Euler class” is justified by the following classical result, which well illustrates the kind of information that characteristic numbers can encode\. 1 Theorem\. If M is a smooth closed oriented manifold, then the characteristic number e[M ] = he\(τ \(M \)\), zi ∈ Z is the Euler characteristic of M \. (J. Peter May 208) - The presence of 2-torsion makes the description of the integral cohomology rings of BO\(n\) and BSO\(n\) quite complicated, and these rings are almost never used in applications\. Rather, one uses the mod 2 cohomology rings and the following description of the cohomology rings that result by elimination of 2-torsion\. (J. Peter May 208) - Theorem\. Take coefficients in a ring R in which 2 is a unit\. Then H ∗ \(BO\(2n\)\) ∼ = ∼ H = ∗ \(BO\(2n + 1\)\) ∼ = ∼ = H ∗ \(BSO\(2n + 1\)\) ∼ = ∼ = R[p1 , \. \. \., pn ] and H ∗ \(BSO\(2n\)\) ∼ = ∼ = R[p1 , \. \. \., pn−1 , e], with e 2 = pn \. (J. Peter May 208) - Consider bundles ξ : Y −→ B with fiber G\. For spaces U in a numerable open cover O of B, there are homeomorphisms φ : U ×G −→ p−1 \(U \) such that p◦φ = π1 \. We say that Y is a principal G-bundle if Y has a free right action by G, B is the orbit space Y /G, ξ is the quotient map, and the φ are maps of right G-spaces\. We say that ξ : Y −→ B is a universal principal G-bundle if Y is a contractible space\. In particular, for any topological group G whose identity element is a nondegenerate basepoint, such as any Lie group G, the map p : EG −→ BG constructed in Chapter 16 §5 is a universal principal G-bundle (J. Peter May 208) - Observe that the long exact sequence of homotopy groups of a universal principal G-bundle gives isomorphisms πq \(BG\) ∼ = ∼ = πq−1 \(G\) for q ≥ 1\. (J. Peter May 208) - There is a classification theorem for principal G-bundles\. Let PG\(B\) denote the set of equivalence classes of principal G-bundles over B, where two principal G-bundles over B are equivalent if there is a G-homeomorphism over B between them\. Via pullback of bundles, this is a contravariant set-valued functor on the homotopy category of spaces\. Theorem\. Let γ : Y −→ Y /G be any universal principal G-bundle\. The natural transformation Φ : [−, Y /G] −→ PG\(−\) obtained by sending the homotopy class of a map f : B −→ Y /G to the equivalence class of the principal G-bundle f ∗ Y is a natural isomorphism of functors\. (J. Peter May 209) - Here an action is effective if gf = f for every f ∈ F implies g = e\. (J. Peter May 209) - For a principal Gbundle Y , let G act on Y × F by g\(y, f \) = \(yg −1 , gf \) and let Y ×G F be the orbit space \(Y × F \)/G\. With the correct formal definition of a fiber bundle with group G and fiber F , every such fiber bundle p : E −→ B is equivalent to one of the form Y ×G F −→ Y /G ∼ = ∼ = B for some principal G-bundle Y over B (J. Peter May 209) - In fact, the “associated principal G-bundle” Y can be constructed as the function space of all maps ψ : F −→ E such that ψ is an admissible homeomorphism onto some fiber Fb = p −1 \(b\)\. (J. Peter May 209) - We conclude that, for any F , PG\(B\) is naturally isomorphic to the set of equivalence classes of bundles with group G and fiber F over B\. Fiber bundles with group O\(n\) and fiber Rn are real n-plane bundles, fiber bundles with group U \(n\) and fiber C n are complex n-plane bundles, and fiber bundles with group SO\(n\) and fiber Rn are oriented real n-plane bundles\. Thus the classification theorems of the previous sections could all be rederived as special cases of the general classification theorem for principal G-bundles stated in this section (J. Peter May 209) - The fact that it is a generalized cohomology theory is a consequence of the Bott periodicity theorem, which is one of the most important and influential theorems in all of topology (J. Peter May 211) - we explain how the Adams operations in K-theory allow a quick solution to the “Hopf invariant one problem\.” (J. Peter May 211) - One implication is the purely algebraic theorem that the only possible dimensions of a real \(not necessarily associative\) division algebra are 1, 2, 4, and 8 (J. Peter May 211) - We shall only discuss complex Ktheory, although there is a precisely analogous construction of real K-theory KO (J. Peter May 211) - From the point of view of algebraic topology, real K-theory is a substantially more powerful invariant, but complex K-theory is usually more relevant to applications in other fields\. (J. Peter May 211) - We consider the set V ect\(X\) of equivalence classes of vector bundles over a space X\. (J. Peter May 211) - \. The set V ect\(X\) forms an Abelian monoid \(= semi-group\) under Whitney sum, and it forms a semi-ring with multiplication given by the \(internal\) tensor product of vector bundles over X\. (J. Peter May 211) - There is a standard construction, called the Grothendieck construction, of an Abelian group G\(M \) associated to an Abelian monoid M : one takes the quotient of the free Abelian group generated by the elements of M by the subgroup generated by the set of elements of the form m + n − m ⊕ n, where ⊕ is the sum in M \. The evident morphism of Abelian monoids i : M −→ G\(M \) is universal: for any homomorphism of monoids f : M −→ G, where G is an Abelian group, there is a unique homomorphism of groups f ˜ f˜ : G\(M \) −→ G such that f ˜ f˜ ◦ i = f (J. Peter May 211) - If M is a semi-ring, then its multiplication induces a multiplication on G\(M \) such that G\(M \) is a ring, called the Grothendieck ring of M \. If the semi-ring M is commutative, then the ring G\(M \) is commutative (J. Peter May 211) - Definition\. The K-theory of X, denoted K\(X\), is the Grothendieck ring of the semi-ring V ect\(X\)\. An element of K\(X\) is called a virtual bundle over X\. We write [ξ] for the element of K\(X\) determined by a vector bundle ξ\. (J. Peter May 211) - we have the function d : V ect\(X\) −→ Z that sends a vector bundle to the dimension of its restriction to the component of the basepoint ∗\. Since d is a homomorphism of semi-rings, it induces a dimension function d : K\(X\) −→ Z, which is a homomorphism of rings\. Since d is an isomorphism when X is a point, d can be identified with the induced map K\(X\) −→ K\(∗\)\. Definition\. The reduced K-theory ̃ K̃\(X\) of a based space X is the kernel of d : K\(X\) −→ Z\. It is an ideal of K\(X\) and thus a ring without identity\. Clearly K\(X\) ∼ = ∼ = K ̃ K̃\(X\) × Z\. (J. Peter May 212) - We say that bundles ζ and ξ are stably equivalent if, for a sufficiently large q, the bundles ζ ⊕ εq−m and ξ ⊕ εq−n are equivalent, where m = d\(ζ\) and n = d\(ξ\)\. (J. Peter May 212) - E U \(X\) ∼ = ∼ = [X+ , BU ]\. (J. Peter May 212) - Let E U \(X\) be the set of stable equivalence classes of vector bundles over X (J. Peter May 212) - Proposition\. If ξ : E −→ X is a vector bundle over X, then there is a bundle η over X such that ξ ⊕ η is equivalent to εq for some q\. (J. Peter May 212) - The space ΓE of sections of E is a vector space under fiberwise addition and scalar multiplication (J. Peter May 212) - The resulting short exact sequence of vector bundles, like any other short exact sequence of vector bundles, splits as a direct sum (J. Peter May 212) - Corollary\. Every virtual bundle over X can be written in the form [ξ] − q for some bundle ξ and non-negative integer q\. (J. Peter May 212) - Corollary\. There is a natural isomorphism E U \(X\) −→ K ̃ K̃\(X\)\. (J. Peter May 212) - Corollary\. Give Z the discrete topology\. For compact spaces X, there is a natural isomorphism K\(X\) ∼ = ∼ = [X+ , BU × Z]\. (J. Peter May 212) - Proposition\. The space BU × Z is a ring space up to homotopy\. That is, there are additive and multiplicative H-space structures on BU × Z such that the associativity, commutativity, and distributivity diagrams required of a ring commute up to homotopy\. (J. Peter May 213) - The study of ring spaces such as this is a relatively new, and quite deep, part of algebraic topology\. However, the reader should feel reasonably comfortable with the additive H-space structure on BU \. (J. Peter May 213) - Theorem\. K\(S 2 \) is generated as a ring by [H] subject to the single relation \([H] − 1\)2 = 0\. Therefore, as Abelian groups, K\(S 2 \) is free on the basis {1, [H]} and ̃ K̃\(S 2 \) is free on the basis {1 − [H]}\. (J. Peter May 214) - Conversely, an isomorphism f from the trivial bundle over S 1 to itself gives a way to glue together the trivial bundles over D and D′ to reconstruct a bundle over S 2 \. Say that two such “clutching functions” f are equivalent if the bundles they give rise to are equivalent\. A careful analysis of the form of the possible clutching functions f leads to a canonical example in each equivalence class and (J. Peter May 214) - Theorem \(Bott periodicity\)\. For compact spaces X, α : K\(X\) ⊗ K\(S 2 \) −→ K\(X × S 2 \) is an isomorphism\. (J. Peter May 214) - The following useful observation applies to any representable functor, not just K-theory\. Lemma\. For nondegenerately based spaces X and Y , the projections of X × Y on X and on Y and the quotient map X×Y −→ X∧Y induce a natural isomorphism ̃ K̃\(X ∧ Y \) ⊕ K ̃ K̃\(X\) ⊕ K ̃ K̃\(Y \) ∼ = ∼ = K ̃ K̃\(X × Y \), and ̃ K̃\(X ∧ Y \) is the kernel of the map ̃ K̃\(X × Y \) −→ ̃ K̃\(X\) ⊕ K ̃ K̃\(Y \) induced by the inclusions of X and Y in X × Y \. (J. Peter May 214) - Theorem \(Bott periodicity\)\. For nondegenerately based compact spaces X, β : ̃ K̃\(X\) ⊗ K ̃ K̃\(S 2 \) −→ K ̃ K̃\(X ∧ S 2 \) = K ̃ K̃\(Σ 2 X\) is an isomorphism\. (J. Peter May 215) - Since ̃ K̃\(S 2 \) ∼ = Z with generator b, the theorem implies that multiplication by the “Bott element” b specifies an isomorphism [X, BU × Z] = ∼ = K ∼ ̃ K̃\(X\) −→ K ̃ K̃\(Σ 2 X\) ∼ = ∼ = [X, Ω 2 \(BU × Z\)] (J. Peter May 215) - Bott’s map β can also be proved to be a homotopy equivalence using only basic algebraic topology\. Since BU and ΩSU are simply connected spaces of the homotopy types of CW complexes, a relative version of the Hurewicz theorem called the Whitehead theorem shows that β will be a weak equivalence and therefore a homotopy equivalence if it induces an isomorphism on integral homology\. (J. Peter May 215) - A purely algebraic dualization argument proves that, as a ring, H∗ \(BU \) ∼ = ∼ Z[γi |i ≥ 1], = where γi is the image of a generator of H2i \(CP ∞ \) under the map induced by the inclusion of CP ∞ = BU \(1\) in BU (J. Peter May 216) - In any case, it should now be clear that we have a periodic Ω-prespectrum and therefore a generalized cohomology theory represented by it\. Definition\. The K-theory Ω-prespectrum KU has spaces KU2i = BU ×Z and KU2i+1 = U for all i ≥ 0\. The structure maps are given by the canonical homotopy equivalence U ≃ ΩBU = Ω\(BU × Z\) and the Bott equivalence BU × Z ≃ ΩU \. (J. Peter May 216) - We have a resulting reduced cohomology theory on based spaces such that ̃ K̃ 2i \(X\) = K̃\(X\) and K̃ 2i+1 \(X\) = K̃\(ΣX\) for all integers i\. This theory has products that are induced by tensor products of bundles over compact spaces and that are induced by suitable maps φ : KUi ∧ KUj −→ KUi+j in general, just as for the cup product in ordinary cohomology\. It is standard to view this simply as a Z2 -graded theory with groups ̃ K̃ 0 \(X\) and K̃ 1 \(X\)\. (J. Peter May 216) - Definition\. Let E0 be the zero section of E\. Define the projective bundle π : P \(E\) −→ X by letting the non-zero complex numbers act on E − E0 by scalar multiplication on fibers and taking the orbit space under this action\. Equivalently, the fiber π −1 \(x\) ⊂ P \(E\) is the complex projective space of lines through the origin in the fiber ξ −1 \(x\) ⊂ E\. Define the canonical line bundle L\(E\) over P \(E\) to be the subbundle of the pullback π ∗ E of ξ along π whose points are the pairs consisting of a line in a fiber of E and a point on that line\. Let Q\(E\) be the quotient bundle π ∗ E/L\(E\) and let H\(E\) denote the dual of L\(E\)\. (J. Peter May 216) - Regard K\(P \(E\)\) as a K\(X\)algebra via π ∗ : K\(X\) −→ K\(P \(E\)\)\. Theorem \(Bott periodicity\)\. Let L be a line bundle over X and let H = H\(L ⊕ ε\)\. Then the K\(X\)-algebra K\(P \(L ⊕ ε\)\) is generated by the single element [H] subject to the single relation \([H] − 1\)\([L][H] − 1\) = 0\. (J. Peter May 216) - It will be based on a generalization to projective bundles of the calculation of H ∗ \(CP n \)\. The proofs of both results are intertwined with the proof of the following “splitting principle,” which allows the deduction of explicit formulas about general bundles from formulas about sums of line bundles\. (J. Peter May 217) - Theorem \(Splitting principle\)\. There is a compact space F \(E\) and a map p : F \(E\) −→ X such that p∗ E is a sum of line bundles over F \(E\) and both p∗ : H ∗ \(X; Z\) −→ H ∗ \(F \(E\); Z\) and p∗ : K\(X\) −→ K\(F \(E\)\) are monomorphisms\. (J. Peter May 217) - Lemma \(Splitting lemma\)\. Both π ∗ : H ∗ \(X; Z\) −→ H ∗ \(P \(E\); Z\) and π ∗ : K\(X\) −→ K\(P \(E\)\) are monomorphisms\. (J. Peter May 217) - Theorem\. Let x = c1 \(L\(E\)\) ∈ H 2 \(P \(E\); Z\)\. Then H ∗ \(P \(E\); Z\) is the free Theorem\. Let x = c1 \(L\(E\)\) ∈ H 2 \(P \(E\); Z\)\. Then H ∗ \(P \(E\); Z\) is the free H \(X; Z\)-module on the basis 1, x, \. \. \., xn−1 , and the Chern classes of ξ are char∗ acterized by c0 \(ξ\) = 1 and the formula e fo n k=0 \(−1\) k ck \(E\)x n−k = 0\. (J. Peter May 217) - One first a line bundle E, L\(E\) = E and c1 \(E\) = c′1 \(E\) by the definition of x\. One first verifies by direct calculation that if E = L1 ⊕ · · · ⊕ Ln is a sum of line bundles, then ′ 1≤k≤n \(x − c1 \(Lk \)\) = 0\. This implies that ck \(E\) is the kth elementary symmetric 1≤k≤n ′ \(x − c1 \(Lk \)\) = 0\. This implies that ck k \(E\) is the kth elementary symmetric Q polynomial in the c1 \(Lk \)\. By the Whitney sum formula for the Chern classes, this implies that c′k k \(E\) = ck \(E\) in this case (J. Peter May 217) - The following analogue in K-theory of the previous theorem holds\. Observe that, since they are continuous operations on complex vector spaces, the exterior powers λk can be applied fiberwise to give natural operations on vector bundles\. Theorem\. Let H = H\(E\)\. Then K\(P \(E\)\) is the free K\(X\)-module on the Theorem\. Let H = H\(E\)\. Then K\(P \(E\)\) is the fre basis 1, [H], \. \. \., [H]n−1 , and the following formula holds: d t n k=0 \(−1\) k [H] k [λ k E] = 0\. (J. Peter May 218) - Projective bundles are closely related to Thom spaces (J. Peter May 218) - Theorem \(Thom isomorphism theorem\)\. Define Φ : K\(X\) −→ K ̃ K̃\(T \(ξ\)\) by Φ\(x\) = x · λE \. Then Φ is an isomorphism\. (J. Peter May 219) - We have seen above that ordinary cohomology and K-theory enjoy similar properties\. The splitting theorem implies a direct connection between (J. Peter May 219) - The example we are interested in is the “Chern character,” which gives rise to (J. Peter May 219) - Example\. Taking R = Q, define the Chern character ch\(E\) ∈ H ∗∗ \(X; Q\) by ch\(E\) = f ˆ mple\. Taking R = Q, define the fˆ\(E\), where f \(t\) = et = ti /i!\. (J. Peter May 219) - For line bundles L and L′ , we have c1 \(L ⊗ L′ \) = c1 \(L\) + c1 \(L′ \)\. One way to see this is to recall that BU \(1\) ≃ K\(Z, 2\) and that line bundles are classified by their Chern classes regarded as elements of [X+ , BU \(1\)] ∼ = ∼ = H 2 \(X; Z\)\. (J. Peter May 219) - Lemma\. The Chern character specifies a ring homomorphism ch : K\(X\) −→ H ∗∗ \(X; Q\)\. (J. Peter May 220) - Lemma\. For n ≥ 1, the Chern character maps ̃ K̃\(S 2n \) isomorphically onto the image of H 2n \(S 2n ; Z\) in H 2n \(S 2n ; Q\)\. Therefore cn : K̃\(S 2n \) −→ H 2n \(S 2n ; Z\) is a monomorphism with cokernel Z\(n−1\)! \. (J. Peter May 220) - Together with some of the facts given in Chapter 23 §7, this has a remarkable application to the study of almost complex structures on spheres\. Recall that a smooth manifold of even dimension admits an almost complex structure if its tangent bundle is the underlying real vector bundle of a complex bundle\. (J. Peter May 220) - Theorem\. S 2 and S 6 are the only spheres that admit an almost complex structure\. (J. Peter May 220) - It is classical that S 2 and S 6 admit almost complex structures and that S 4 does not\. (J. Peter May 220) - We then have the following basic result, which actually holds for general compact spaces X provided that we replace singular cohomology by Č Čech cohomology\. Theorem\. For any finite based CW complex X, ch induces an isomorphism ̃ K̃ ∗ \(X\) ⊗ Q −→ H ̃ H̃ ∗∗ \(X; Q\)\. (J. Peter May 220) - Visibly, this is a morphism of monoids, Λ\(ξ ⊕ η\) = Λ\(ξ\)Λ\(η\)\. It therefore extend to a homomorphism of groups Λ : K\(X\) −→ G, (J. Peter May 221) - There are natural operations in K-theory, called the Adams operations, that are somewhat analogous to the Steenrod operations in mod 2 cohomology\. In fact, the analogy can be given content by establishing a precise relationship between the Adams and Steenrod operations, but (J. Peter May 221) - Theorem\. For each non-zero integer k, there is a natural homomorphism of rings ψ k : K\(X\) −→ K\(X\)\. These operations satisfy the following properties\. \(1\) ψ 1 = id and ψ −1 is induced by complex conjugation of bundles\. \(2\) ψ k ψ ℓ = ψ kℓ = ψ ℓ ψ k \. \(3\) ψ p \(x\) ≡ xp mod p for any prime p\. \(4\) ψ k \(ξ\) = ξ k if ξ is a line bundle\. \(5\) ψ k \(x\) = k n x if x ∈ K ̃ K̃\(S 2n \)\. (J. Peter May 221) - Recall that the subring of symmetric polynomials in the polynomial algebra Z[x1 , \. \. \., xn ] is the polynomial algebra Z[σ1 , \. \. \., σn ], where σi = x1 x2 · · · xi + · · · is the ith elementary symmetric function\. (J. Peter May 221) - Remark\. The observant reader will have noticed that, by analogy with the definition of the Stiefel-Whitney classes, we can define characteristic classes in Ktheory by use of the Adams operations and the Thom isomorphism, setting ρk \(E\) = Φ−1 ψ k Φ\(1\) for n-plane bundles E (J. Peter May 223) - We give one of the most beautiful and impressive illustrations of the philosophy described in the first chapter\. We define a numerical invariant, called the “Hopf invariant,” of maps f : S 2n−1 −→ S n and show that it can only rarely take the value one\. (J. Peter May 223) - We then indicate several problems whose solution can be reduced to the question of when such maps f take the value one\. Adams’ original solution to the Hopf invariant one problem used secondary cohomology operations in ordinary cohomology and was a critical starting point of modern algebraic topology\. The later realization that a problem that required secondary operations in ordinary cohomology could be solved much more simply using primary operations in Ktheory had a profound impact on the further development of the subject\. (J. Peter May 223) - Definition\. Let X be the cofiber of a based map f : S 2n−1 −→ S n , where n ≥ 2\. Then X is a CW complex with a single vertex, a single n-cell i, and a single 2n-cell j\. (J. Peter May 223) - H ̃ H̃ ∗ \(X\) is free Abelian on generators x = [i] and y = [j]\. Define an integer h\(f \), the Hopf invariant of f , by x 2 = h\(f \)y (J. Peter May 223) - Theorem\. If h\(f \) = ±1, then n = 2, 4, or 8\. (J. Peter May 224) - Theorem\. If S n−1 is an H-space, then n = 1, 2, 4, or 8 (J. Peter May 224) - The determination of which spheres are H-spaces has the following implications\. Theorem\. Let ω : Rn × Rn −→ Rn be a map with a two-sided identity element e 6= 0 and no zero divisors\. Then n = 1, 2, 4, or 8\. (J. Peter May 226) - Note that ω need not be bilinear, just continuous\. Also, it need not have a strict unit; all that is required is that e be a two-sided unit up to homotopy for the restriction of ω to Rn − {0}\. Theorem\. If S n is parallelizable, then n = 0, 1, 3, or 7\. (J. Peter May 226) - suppose that S n is parallelizable, so that its tangent bundle τ is trivial\. (J. Peter May 226) - Cobordism theories were introduced shortly after K-theory, (J. Peter May 227) - We shall describe the cobordism of smooth closed manifolds, but this is in fact a particularly elementary example\. Other examples include smooth closed manifolds with extra structure on their stable normal bundles: orientation, complex structure, Spin structure, or symplectic structure for example\. All of these except the symplectic case have been computed completely\. (J. Peter May 227) - The area is pervaded by insights from algebraic topology that are quite mysterious geometrically\. For example, the complex cobordism groups turn out to be concentrated in even degrees: every smooth closed manifold of odd dimension with a complex structure on its stable normal bundle is the boundary of a compact manifold \(with compatible bundle information\)\. However, there is no geometric understanding of why this should be the case\. The analogue with “complex” replaced by “symplectic” is false\. (J. Peter May 227) - We consider the problem of classifying smooth closed n-manifolds M \. One’s first thought is to try to classify them up to diffeomorphism, but that problem is in principle unsolvable\. Thom’s discovery that one can classify such manifolds up to the weaker equivalence relation of “cobordism (J. Peter May 227) - We say that two smooth closed nmanifolds M and N are cobordant if there is a smooth compact manifold W whose boundary is the disjoint union of M and N , ∂W = M ∐ N \. We write Nn for the set of cobordism classes of smooth closed n-manifolds\. It is convenient to allow the empty set ∅ as an n-manifold for every n\. Disjoint union gives an addition on the set Nn \. (J. Peter May 227) - Nn is a vector space over Z2 \. Cartesian product of manifolds defines a multiplication Nm × Nn −→ Nm+n \. This operation is bilinear, associative, and commutative, and the zero dimensional manifold with a single point provides an identity element\. We conclude that N∗ is a graded Z2 -algebra\. (J. Peter May 227) - Theorem \(Thom\)\. N∗ is a polynomial algebra over Z2 on generators ui of dimension i for i > 1 and not of the form 2 r − 1\. (J. Peter May 227) - As already stated in our discussion of Stiefel-Whitney numbers, it follows from the proof of the theorem that a manifold is a boundary if and only if its normal Stiefel-Whitney numbers are zero (J. Peter May 228) - Theorem\. Two smooth closed n-manifolds are cobordant if and only if their normal Stiefel-Whitney numbers, or equivalently their tangential Stiefel-Whitney numbers, are equal\. (J. Peter May 228) - Explicit generators ui are known\. Write [M ] for the cobordism class of a manifold M \. Then we can take u2i = [RP 2i ] (J. Peter May 228) - The strategy for the proof of Thom’s theorem is to describe Nn as a homotopy group of a certain Thom space\. The homotopy group is a stable one, and it turns out to be computable by the methods of generalized homology theory (J. Peter May 228) - and we have the following translation of our problem in manifold theory to a problem in homotopy theory (J. Peter May 228) - shall sketch the proof in the next section, where we shall also explain the ring structure on π∗ \(T O\) that makes it a Z2 -algebra\. (J. Peter May 228) - Theorem \(Thom\)\. For sufficiently large q, Nn is isomorphic to πn+q \(T O\(q\)\)\. Therefore Nn ∼ = ∼ = πn \(T O\)\. Moreover, N∗ and π∗ \(T O\) are isomorphic as Z2 -algebras\. (J. Peter May 228) - Given a smooth closed n-manifold M , we may embed it in Rn+q for q sufficiently large, and we let ν be the normal bundle of the embedding\. \(By the Whitney embedding theorem, q = n suffices (J. Peter May 228) - Embed M as the zero section of the total space E\(ν\)\. Then a standard result in differential topology known as the tubular neighborhood theorem implies that the identity map of M extends to an embedding of E\(ν\) onto an open neighborhood U of M in R n+q \. (J. Peter May 228) - The “PontryaginThom construction” associates a map t : S n+q −→ T \(ν\) to our tubular neighborhood U (J. Peter May 229) - The Thom space was tailor made to allow this construction (J. Peter May 229) - By an implication of Sard’s theorem known as the transversality theorem (J. Peter May 229) - \. In fact, T O is a commutative and associative ring prespectrum in the sense of the following definition\. Definition\. Let T be a prespectrum\. Then T is a ring prespectrum if there are maps η : S 0 −→ T0 and φm,n : Tm ∧ Tn −→ Tm+n such that the following diagrams are homotopy commutative (J. Peter May 230) - For example, the Eilenberg-Mac Lane Ω-prespectrum of a commutative ring R is an associative and commutative ring prespectrum (J. Peter May 230) - 3\. It is denoted HR or sometimes, by abuse, K\(R, 0\)\. Similarly, the K-theory Ω-prespectrum is an associative and commutative ring prespectrum\. The sphere prespectrum, whose nth space is S n , is another example (J. Peter May 231) - Lemma\. If T is an associative ring prespectrum, then π∗ \(T \) is a graded ring\. If T is commutative, then π∗ \(T \) is commutative in the graded sense\. (J. Peter May 231) - Calculation of the homotopy groups π∗ \(T O\) proceeds by first computing the homology groups H∗ \(T O; Z2 \) and then showing that the stable Hurewicz homomorphism maps π∗ \(T O\) monomorphically onto an identifiable part of H∗ \(T O; Z2 \)\. (J. Peter May 231) - Just as we defined the homotopy groups of a prespectrum T by the formula πn \(T \) = colim πn+q \(Tq \), (J. Peter May 231) - we define the homology and cohomology groups of T with respect to a homology theory k∗ and cohomology theory k ∗ on spaces by the formulas kn \(T \) = colim k̃ k̃n+q \(Tq \), (J. Peter May 231) - In fact, this definition of cohomology is inappropriate in general, differing from the appropriate definition by a lim 1 error term\. However, the definition is correct when k ∗ is ordinary cohomology with coefficients in a field R and each H̃ n+q \(Tq ; R\) is a finite dimensional vector space over R\. (J. Peter May 232) - Observe that there is no cup product in H ∗ \(T ; R\): the maps in the limit system factor through the reduced cohomologies of suspensions, in which cup products are identically zero (J. Peter May 232) - The Hurewicz homomorphisms πn+q \(Tq \) −→ ̃ H̃n+q \(Tq ; Z\) pass to colimits to give the stable Hurewicz homomorphism h : πn \(T \) −→ Hn \(T ; Z\)\. (J. Peter May 232) - If T is an associative and commutative ring prespectrum, then h : π∗ \(T \) −→ H∗ \(T ; R\) is a map of graded commutative rings\. (J. Peter May 232) - Recall that we have Thom isomorphisms Φq : H n \(BO\(q\)\) −→ H ̃ H̃ n+q \(T O\(q\)\) obtained by cupping with the Thom class µq ∈ H ̃ H̃ q \(T O\(q\)\)\. (J. Peter May 232) - We therefore obtain a “stable Thom isomorphism” Φ : H n \(BO\) −→ H n \(T O\) on passage to limits\. We have dual homology Thom isomorphisms Φn : H ̃ H̃n+q \(T O\(q\)\) −→ Hn \(BO\(q\)\) (J. Peter May 232) - Theorem\. H∗ \(BO\) is the polynomial algebra Z2 [bi |i ≥ 1]\. Let ai ∈ Hi \(T O\) be the element characterized by Φ\(ai \) = bi \. Corollary\. H∗ \(T O\) is the polynomial algebra Z2 [ai |i ≥ 1]\. (J. Peter May 233) - Recall from Chapter 23 §6 that we have a homotopy equivalence j : RP ∞ −→ T O\(1\)\. (J. Peter May 233) - Since the Steenrod operations are stable and natural, they pass to limits to define natural operations Sq i : H n \(T \) −→ H n+i \(T \) for i ≥ 0 and prespectra T \. (J. Peter May 234) - The homology and cohomology of T O are built up from π∗ \(T O\) and Steenrod operations (J. Peter May 234) - Definition\. The mod 2 Steenrod algebra A is the quotient of the free associative Z2 -algebra generated by elements Sq i , i ≥ 1, by the ideal generated by the Adem relations \(which are stated in Chapter 22 §5\) (J. Peter May 234) - Lemma\. For spaces X, H ∗ \(X\) has a natural A-module structure\. Lemma\. For prespectra T , H ∗ \(T \) has a natural A-module structure (J. Peter May 234) - The elements of A are stable mod 2 cohomology operations, and our description of the cohomology of K\(Z2 , q\)s in Chapter 22 §5 implies that A is in fact the algebra of all stable mod 2 cohomology operations (J. Peter May 234) - Recall that HZ2 denotes the EilenbergMac Lane Ω-prespectrum {K\(Z2 , q\)}\. (J. Peter May 234) - Lemma\. As a vector space, A is isomorphic to H ∗ \(HZ2 \)\. (J. Peter May 234) - Theorem\. A has a basis consisting of the operations Sq I = Sq i1 · · · Sq ij , where I runs over the sequences {i1 , \. \. \., ij } of positive integers such that ir ≥ 2ir+1 for 1 ≤ r < j\. (J. Peter May 234) - What is still more important to us is that A not only has the composition product A ⊗ A −→ A, it also has a coproduct ψ : A −→ A ⊗ A\. (J. Peter May 234) - Algebraic structures like this, with compatible products and coproducts, are called “Hopf algebras\.” (J. Peter May 234) - Theorem\. Let N∗ be the algebra defined abstractly by N∗ = Z2 [ui |i > 1 and i 6= 2 r − 1], where deg ui = i\. Define a homomorphism of algebras f : H∗ \(T O\) −→ N∗ by f \(ai \) = homomorphism of algebras f : H∗ \(T ui if i is not of the form 2r − 1 0 if i = 2r − 1\. Then the composite g : H∗ \(T O\) γ − → A∗ ⊗ H∗ \(T O\) id ⊗f −−−→ A∗ ⊗ N∗ (J. Peter May 236) - is an isomorphism of both A-comodules and Z2 -algebras\. (J. Peter May 236) - Now consider the Hurewicz homomorphism h : π∗ \(T \) −→ H∗ \(T \) of a prespectrum T \. (J. Peter May 236) - Theorem\. h : π∗ \(T O\) −→ H∗ \(T O\) is a monomorphism and g◦h maps π∗ \(T O\) isomorphically onto N∗ \. (J. Peter May 236) - We shall prove that a smooth closed n-manifold M is a boundary if and only if all of its normal Stiefel-Whitney numbers are zero\. (J. Peter May 236) - Polynomials in the StiefelWhitney classes are elements of H ∗ \(BO\) (J. Peter May 236) - follows that cobordant manifolds have the same normal Stiefel-Whitney numbers\. (J. Peter May 236) - -Whitney numbers\. The assignment of Stiefel-Whitney numbers to corbordism classes of n-manifolds specifies a homomorphism # : H n \(BO\) ⊗ Nn −→ Z2 (J. Peter May 236) - To say that all normal Stiefel-Whitney numbers of M are zero is to say that w#[M ] = 0 for all w ∈ H n \(BO\) (J. Peter May 237) - this implies that [M ] = 0 and thus that M is a boundary\. (J. Peter May 237) - We think of prespectra as “stable objects” that have associated homotopy, homology, and cohomology groups\. Imagine that we have a good category of stable objects, analogous to the category of based spaces, that is equipped with all of the constructions that we have on based spaces: wedges \(= coproducts\), colimits, products, limits, suspensions, loops, homotopies, cofiber sequences, fiber sequences, smash products, function objects, and so forth\. Let us call the stable objects in our imagined category “spectra” and call the category of such objects S \. We have in mind an analogy with the notions of presheaf and sheaf\. (J. Peter May 238) - The homology and cohomology groups of Σ∞ X are the \(reduced\) homology and cohomology groups of X; the homotopy groups of Σ ∞ X are the stable homotopy groups of X\. (J. Peter May 238) - Continuing our thought exercise, we can form the homotopy category hS of spectra and can define homotopy groups in terms of homotopy classes of maps from sphere spectra to spectra (J. Peter May 239) - Reflection on the periodic nature of K-theory suggests that we should define sphere spectra of negative dimension and define homotopy groups πq \(X\) for all integers q (J. Peter May 239) - That is, we develop a theory of CW spectra using sphere spectra as the domains of attaching maps (J. Peter May 239) - The Whitehead and cellular approximation theorems hold, and every spectrum X admits a CW approximation ΓX −→ X\. We define the set [X, Y ] of morphisms X −→ Y in h̄ h̄S to be the set of homotopy classes of maps ΓX −→ ΓY \. This is a stable category in the sense that the functor Σ : h̄ h̄S −→ h h̄ h̄S is an equivalence of categories\. More explicitly, the natural maps X −→ ΩΣX and ΣΩX −→ X are isomorphisms in h h̄ h̄S \. (J. Peter May 239) - In particular, up to isomorphism, every object in the category h̄ h̄S is a suspension, hence a double suspension (J. Peter May 239) - This implies that each [X, Y ] is an Abelian group and composition is bilinear\. (J. Peter May 239) - so that cofiber sequences and fiber sequences are equivalent\. Therefore cofiber sequences give rise to long exact sequences of homotopy groups\. (J. Peter May 239) - The homotopy groups of wedges and products of spectra are given by topy π∗ \( i roups of Xi \) = i edges and product π∗ \(Xi \) and π∗ \( i of spectr Xi \) = i π∗ \(Xi \)\. (J. Peter May 239) - Therefore, if only Theref map i e, if only Q Xi −→ fi i nitely many πq \(Xi \) a Xi is an isomorphism (J. Peter May 239) - A spectrum E represents a homology theory E∗ and a cohomology theory E ∗ specified in terms of smash products and function spectra by Eq \(X\) = πq \(X ∧ E\) and E q \(X\) = π−q F \(X, E\) ∼ = ∼ = [X, Σ q E]\. (J. Peter May 239) - Moreover, every homology or cohomology theory on h h̄ h̄S is so represented by some spectrum E\. (J. Peter May 239) - Hπ for the “Eilenberg-Mac Lane spectrum” that represents ordinary cohomology with coefficients in (J. Peter May 239) - Its only non-zero homotopy group is π0 \(Hπ\) = π, and the Hurewicz homomorphism maps this group isomorphically onto H0 \(Hπ; Z\)\. When π = Z2 , the natural map H0 \(HZ2 ; Z\) −→ H0 \(HZ2 ; Z2 \) is also an isomorphism (J. Peter May 239) - Returning to our motivating example, we write M O for the “Thom spectrum” that arises from the Thom prespectrum T O\. The reader may sympathize with a student who claimed that M O stands for “Mythical Object\.” (J. Peter May 239) - However, these early constructions were far more primitive than our outline suggests\. While they gave a satisfactory stable category, the underlying category of spectra did not have products, limits, and function objects, and its smash product was not associative, commutative, or unital\. In fact, a fully satisfactory category of spectra was not constructed until 1995\. (J. Peter May 240) - Definition\. A spectrum E is a prespectrum such that the adjoints σ̃ : En −→ ΩEn+1 of the structure maps σ : ΣEn −→ En+1 are homeomorphisms\. A map f : T −→ T ′ of prespectra is a sequence of maps fn : Tn −→ Tn′ such that σ ′ n′ ◦ Σfn = fn+1 ◦ σn for all n\. A map f : E −→ E ′ of spectra is a map between E and E ′ regarded as prespectra\. (J. Peter May 240) - We define wedges and colimits of spectra by first performing the construction on the prespectrum level and then applying the functor L\. If we start with spectra and construct products or limits spacewise, then the result is again a spectrum; that is, limits of spectra are the limits of their underlying prespectra\. Thus the category S is complete and cocomplete\. (J. Peter May 240) - We now have cylinders E ∧ I+ and thus can define homotopies between maps of spectra\. Similarly we have cones CE = E ∧ I \(where I has basepoint 1\), suspensions ΣE = E ∧ S 1 , path spectra F \(I, E\) \(where I has (J. Peter May 240) - Similarly, we define the smash product T ∧X and function prespectrum F \(X, T \) of a based space X and a prespectrum T spacewise\. For a spectrum E, we define E ∧ X by applying L to the prespectrum level construction; the prespectrum F \(X, E\) is already a spectrum\. (J. Peter May 240) - basepoint 0\), and loop spectra ΩE = F \(S 1 , E\)\. The development of cofiber and fiber sequences proceeds exactly as for based spaces (J. Peter May 241) - The essential point is that homotopy and homology commute with colimits\. (J. Peter May 241) - It is not true that cohomology converts colimits to limits in general, because of lim 1 error terms, and this is one reason that our definition of the cohomology of prespectra via limits is inappropriate except under restrictions that guarantee the vanishing of lim 1 terms (J. Peter May 241) - We define QX = ∪Ωq Σq X, and we find that the nth space of Σ∞ X is QΣn X\. It should be apparent that the homotopy groups of the space QX are the stable homotopy groups of X\. (J. Peter May 241) - For example, the homotopy groups of the K-theory spectrum are Z for every even integer and zero for every odd integer (J. Peter May 241) - Thus, if we have a prespectrum T whose invariants we are interested in, such as an Eilenberg-Mac Lane Ω-prespectrum or the K-theory Ω-prespectrum, then we can construct a spectrum LKT that has the same invariants (J. Peter May 241) - The real work involves the smash product of spectra (J. Peter May 241) - there is also considerable payoff in explicit concrete calculations, as the computation of π∗ \(M O\) (J. Peter May 241) - The subject in its earlier days was blessed with some of the finest expositors of mathematics, for example Steenrod, Serre, Milnor, and Adams (J. Peter May 243) - Two introductions to algebraic topology starting from de Rham cohomology: R\. Bott and L\.W\. Tu\. Differential forms in algebraic topology\. Springer-Verlag\. 1982\. I\. Madsen and J\. Tornehave\. From calculus to cohomology\. de Rham cohomology and characteristic classes\. Cambridge University Press\. 1997\. (J. Peter May 244) - The classic reference on Morse theory, with an exposition of the Bott periodicity theorem: J\. Milnor\. Morse theory\. Annals of Math\. Studies No\. 51\. Princeton University Press\. 1963\. (J. Peter May 244) - Two good basic references on equivariant algebraic topology, classically called the theory of transformation groups \(see also §§16, 21 below\): G\. Bredon\. Introduction to compact transformation groups\. Academic Press\. 1972\. T\. tom Dieck\. Transformation groups\. Walter de Gruyter\. 1987\. A more advanced book, a precursor to much recent work in the area: T\. tom Dieck\. Transformation groups and representation theory\. Lecture Notes in Mathematics Vol\. 766\. Springer-Verlag\. 1979 (J. Peter May 245) - Two classical treatments and a good modern treatment of homological algebra: H\. Cartan and S\. Eilenberg\. Homological algebra\. Princeton University Press\. 1956\. S\. MacLane\. Homology\. Springer-Verlag\. 1963\. C\.A\. Weibel\. An introduction to homological algebra\. Cambridge University Press\. 1994\. (J. Peter May 245) - P\.G\. Goerss and J\.F\. Jardine\. Simplicial homotopy theory\. Birkhäuser\. To appear\. (J. Peter May 245) - Two classic papers of Serre: J\.-P\. Serre\. Homologie singuliére des espaces fibrés\. Applications\. Annals of Math\. \(2\)54\(1951\), 425–505\. J\.-P\. Serre\. Groupes d’homotopie et classes de groupes abéliens\. Annals of Math\. \(2\)58\(1953\), 198–232\. A nice exposition of some basic homotopy theory and of Serre’s work: S\.-T\. Hu\. Homotopy theory\. Academic Press\. 1959\. (J. Peter May 245) - There are other important spectral sequences in the context of fibrations, mainly due to Eilenberg and Moore\. Three references: S\. Eilenberg and J\.C\. Moore\. Homology and fibrations, I\. Comm\. Math\. Helv\. 40\(1966\), 199–236\. (J. Peter May 245) - L\. Smith\. Homological algebra and the Eilenberg-Moore spectral sequences\. Trans\. Amer\. Math\. Soc\. 129\(1967\), 58–93\. V\.K\.A\.M\. Gugenheim and J\.P\. May\. On the theory and applications of differential torsion products\. Memoirs Amer\. Math\. Soc\. No\. 142\. 1974\. There is a useful guidebook to spectral sequences: J\. McCleary\. User’s guide to spectral sequences\. Publish or Perish\. 1985\. (J. Peter May 246) - A compendium of the work of Steenrod and others on the construction and analysis of the Steenrod operations: N\.E\. Steenrod and D\.B\.A\. Epstein\. Cohomology operations\. Annals of Math\. Studies No\. 50\. Princeton University Press\. 1962\. (J. Peter May 246) - A general treatment of Steenrod-like operations: J\.P\. May\. A general algebraic approach to Steenrod operations\. In Lecture Notes in Mathematics Vol\. 168, 153–231\. Springer-Verlag\. 1970\. A nice book on mod 2 Steenrod operations and the Adams spectral sequence: R\. Mosher and M\. Tangora\. Cohomology operations and applications in homotopy theory\. Harper and Row\. 1968\. (J. Peter May 246) - A classic and a more recent standard treatment that includes K-theory: N\.E\. Steenrod\. Topology of fibre bundles\. Princeton University Press\. 1951\. Fifth printing, 1965\. D\. Husemoller\. Fibre bundles\. Springer-Verlag\. 1966\. Third edition, 1994\. A general treatment of classification theorems for bundles and fibrations: J\.P\. May\. Classifying spaces and fibrations\. Memoirs Amer\. Math\. Soc\. No\. 155\. 1975\. (J. Peter May 246) - A good reference for the basic calculations of characteristic classes: A\. Borel\. Topology of Lie groups and characteristic classes\. Bull\. Amer\. Math\. Soc\. 61\(1955\), 297–432\. (J. Peter May 246) - Two proofs of the Bott periodicity theorem that only use standard techniques of algebraic topology, starting from characteristic class calculations: H\. Cartan et al\. Périodicité des groupes d’homotopie stables des groupes classiques, d’après Bott\. Séminaire Henri Cartan, 1959/60\. Ecole Normale Supérieure\. Paris\. (J. Peter May 246) - E\. Dyer and R\.K\. Lashof\. A topological proof of the Bott periodicity theorems\. Ann\. Mat\. Pure Appl\. \(4\)54\(1961\), 231–254\. (J. Peter May 247) - Two classical lecture notes on K-theory: R\. Bott\. Lectures on K\(X\)\. W\.A\. Benjamin\. 1969\. This includes a reprint of perhaps the most accessible proof of the complex case of the Bott periodicity theorem, namely: M\.F\. Atiyah and R\. Bott\. On the periodicity theorem for complex vector bundles\. Acta Math\. 112\(1994\), 229–247\. (J. Peter May 247) - J\.F\. Adams\. Vector fields on spheres\. Annals of Math\. 75\(1962\), 603–632\. (J. Peter May 247) - The basic source for the structure theory of \(connected\) Hopf algebras: J\. Milnor and J\.C\. Moore\. On the structure of Hopf algebras\. Annals of Math\. 81\(1965\), 211–264\. (J. Peter May 247) - The classic analysis of the structure of the Steenrod algebra as a Hopf algebra: J\. Milnor\. The Steenrod algebra and its dual\. Annals of Math\. 67\(1958\), 150–171\. (J. Peter May 247) - Two classic papers of Adams; the first constructs the Adams spectral sequence relating the Steenrod algebra to stable homotopy groups and the second uses secondary cohomology operations to solve the Hopf invariant one problem: J\.F\. Adams\. On the structure and applications of the Steenrod algebra\. Comm\. Math\. Helv\. 32\(1958\), 180–214\. (J. Peter May 247) - J\.F\. Adams\. On the non-existence of elements of Hopf invariant one\. Annals of Math\. 72\(1960\), 20–104\. (J. Peter May 248) - The beautiful classic paper of Thom is still highly recommended: R\. Thom\. Quelques propriétés globals des variétés différentiables\. Comm\. Math\. Helv\. 28\(1954\), 17–86\. Thom computed unoriented cobordism\. Oriented and complex cobordism came later\. In simplest form, the calculations use the Adams spectral sequence: J\. Milnor\. On the cobordism ring Ω∗ and a complex analogue\. Amer\. J\. Math\. 82\(1960\), 505–521\. (J. Peter May 248) - Two classical references, the second of which also gives detailed information about complex cobordism that is of fundamental importance to the subject\. G\.W\. Whitehead\. Generalized homology theories\. Trans\. Amer\. Math\. Soc\. 102\(1962\), 227–283\. J\.F\. Adams\. Stable homotopy and generalised homology\. Chicago Lectures in Mathematics\. University of Chicago Press\. 1974\. Reprinted in 1995\. (J. Peter May 248) - Foundations for equivariant stable homotopy theory are established in: L\.G\. Lewis, Jr\., J\.P\. May, and M\.Steinberger \(with contributions by J\.E\. McClure\)\. Equivariant stable homotopy theory\. Lecture Notes in Mathematics Vol\. 1213\. Springer-Verlag\. 1986\. (J. Peter May 248) - \. Localization and completion; rational homotopy theory Since the early 1970s, it has been standard practice in algebraic topology to localize and complete topological spaces, and not just their algebraic invariants, at sets of primes and then to study the subject one prime at a time, or rationally\. Two of the basic original references are: D\. Sullivan\. The genetics of homotopy theory and the Adams conjecture\. Annals of Math\. 100\(1974\), 1–79\. A\.K\. Bousfield and D\.M\. Kan\. Homotopy limits, completions, and localizations\. Lecture Notes in Mathematics Vol\. 304\. Springer-Verlag\. 1972\. A more accessible introduction to localization and a readable recent paper on completion are: P\. Hilton, G\. Mislin, and J\. Roitberg\. Localization of nilpotent groups and spaces\. North-Holland\. 1975\. (J. Peter May 249) - When spaces are rationalized, there is a completely algebraic description of the result\. The main original reference and a more accessible source are: D\. Sullivan\. Infinitesimal computations in topology\. Publ\. Math\. IHES 47\(1978\), 269–332\. A\.K\. Bousfield and V\.K\.A\.M\. Gugenheim\. On PL de Rham theory and rational homotopy type\. Memoirs Amer\. Math\. Soc\. No\. 179\. 1976 (J. Peter May 249) - Another area well established by the mid-1970s\. The following book is a delightful read, with capsule introductions of many topics other than infinite loop space theory, a very pleasant starting place for learning modern algebraic topology: J\.F\. Adams\. Infinite loop spaces\. Annals of Math\. Studies No\. 90\. Princeton University Press\. 1978 (J. Peter May 249) - The following survey article is less easy going, but gives an indication of the applications to high dimensional geometric topology and to algebraic K-theory: J\.P\. May\. Infinite loop space theory\. Bull\. Amer\. Math\. Soc\. 83\(1977\), 456–494\. (J. Peter May 249) - Adams’ book cited in §16 gives a spectral sequence for the computation of stable homotopy groups in terms of generalized cohomology theories\. Starting from complex cobordism and related theories, its use has been central to two waves of major developments in stable homotopy theory (J. Peter May 250) - A good exposition for the first wave: D\.C\. Ravenel\. Complex cobordism and stable homotopy groups of spheres\. Academic Press\. 1986\. (J. Peter May 250) - The essential original paper and a very nice survey article on the second wave: E\. Devinatz, M\.J\. Hopkins, and J\.H\. Smith\. Nilpotence and stable homotopy theory\. Annals of Math\. 128\(1988\), 207–242\. (J. Peter May 250) - The cited Proceedings contain good introductory survey articles on several other topics in algebraic topology\. A larger scale exposition of the second wave is: D\.C\. Ravenel\. Nilpotence and periodicity in stable homotopy theory\. Annals of Math\. Studies No\. 128\. Princeton University Press\. 1992\. (J. Peter May 250) - There is a leap from the level of this introductory book to that of the most recent work in the subject\. One recent book that helps fill the gap is: P\. Selick\. Introduction to homotopy theory\. Fields Institute Monographs No\. 9\. American Mathematical Society\. 1997 (J. Peter May 250) - There is a recent expository book for the reader who would like to jump right in and see the current state of algebraic topology; although it focuses on equivariant theory, it contains introductions and discussions of many non-equivariant topics: J\.P\. May et al\. Equivariant homotopy and cohomology theory\. NSF-CBMS Regional Conference Monograph\. 1996\. (J. Peter May 250) - For the reader of the last section of this book whose appetite has been whetted for more stable homotopy theory, there is an expository article that motivates and explains the properties that a satisfactory category of spectra should have: J\.P\. May\. Stable algebraic topology and stable topological algebra\. Bulletin London Math\. Soc\. 30\(1998\), 225–234 (J. Peter May 250)