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we may compose with the map H ∗(X ⊗ X ′; A ⊗ A) −→ H ∗(X ⊗ X ′; A) induced by the multiplication of A to obtain a map H ∗(X; A) ⊗ H ∗(X ′; A) −→ H ∗(X ⊗ X ′; A). We are especially interested in the case when R = Z and A is either Z or a ﬁeld. CHAPTER 18 Axiomatic and cellular cohomology theory We give a treatment of cohomology that is precisely parallel to our treatment of homology. 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. This additional structure ties together the cohomology groups in diﬀerent degrees and is fundamentally important to most of the applications. 1. Axioms for cohomology Fix an Abelian group π and consider pairs of spaces (X, A). We shall see that π determines a “cohomology theory on pairs (X, A).” Theorem. For integers q, there exist contravariant functors H q(X, A; π) from the homotopy category of pairs of spaces to the category of Abelian groups together with natural transformations δ : H q(A; π) −→ H q+1(X, A; π), where H q(X; π) is deﬁned to be H q(X, ∅; π). These functors and natural transformations satisfy and are characterized by the following axioms. • DIMENSION If X is a point, then H 0(X; π) = π and H q(X; π) = 0 for all other integers. • EXACTNESS The following sequence is exact, where the unlabeled arrows are induced by the inclusions A −→ X and (X, ∅) −→ (X, A): · · · −→ H q(X, A; π) −→ H q(X; π) −→ H q(A; π) δ−→ H q+1(X, A; π) −→ · · ·.
• EXCISION If (X; A, B) is an excisive triad, so that X is the union of the interiors of A and B, then the inclusion (A, A ∩ B) −→ (X, B) induces an isomorphism H ∗(X, B; π) −→ H ∗(A, A ∩ B; π). • ADDITIVITY If (X, A) is the disjoint union of a set of pairs (Xi, Ai), then the inclusions (Xi, Ai) −→ (X, A) induce an isomorphism H ∗(X, A; π) −→ i H ∗(Xi, Ai; π). • WEAK EQUIVALENCE If f : (X, A) −→ (Y, B) is a weak equivalence, Q then f ∗ : H ∗(Y, B; π) −→ H ∗(X, A; π) is an isomorphism. We write f ∗ instead of H ∗(f ) or H q(f ). As in homology, our approximation theorems for spaces, pairs, maps, homotopies, and excisive triads directly imply that such a theory determines and is determined by an appropriate theory deﬁned on CW pairs, as spelled out in the following CW version of the theorem. 137 138 AXIOMATIC AND CELLULAR COHOMOLOGY THEORY Theorem. For integers q, there exist functors H q(X, A; π) from the homotopy category of pairs of CW complexes to the category of Abelian groups together with natural transformations δ : H q(A) −→ H q+1(X, A; π), where H q(X; π) is deﬁned to be H q(X, ∅; π). These functors and natural transformations satisfy and are characterized by the following axioms. • DIMENSION If X is a point, then H 0(X; π) = π and H q(X; π) = 0 for all other integers. • EXACTNESS The following sequence is exact, where the unlabeled arrows are induced by the inclusions A −→ X and (X, ∅) −→ (X, A): · · ·
−→ H q(X, A; π) −→ H q(X; π) −→ H q(A; π) δ −→ H q+1(X, A; π) −→ · · ·. • EXCISION If X is the union of subcomplexes A and B, then the inclusion (A, A ∩ B) −→ (X, B) induces an isomorphism H ∗(X, B; π) −→ H ∗(A, A ∩ B; π). • ADDITIVITY If (X, A) is the disjoint union of a set of pairs (Xi, Ai), then the inclusions (Xi, Ai) −→ (X, A) induce an isomorphism H ∗(X, A; π) −→ i H ∗(Xi, Ai; π). Such a theory determines and is determined by a theory as in the previous theorem. Q 2. Cellular and singular cohomology We deﬁne the cellular cochains of a CW pair (X, A) with coeﬃcients in an Abelian group π to be C∗(X, A; π) = Hom(C∗(X, A), π). We then deﬁne the cellular cohomology groups to be H ∗(X, A; π) = H ∗(C∗(X, A; π)). If M is a module over a commutative ring R, we have a natural identiﬁcation C∗(X, A; M ) ∼= HomR(C∗(X, A) ⊗ R, M ) which allows us to do homological algebra over R rather than over Z when convenient. In particular, if R is a ﬁeld, then H ∗(X, A; M ) ∼= HomR(H∗(X, A; R), M ). In general, with R = Z, we have a natural and splittable short exact sequence Z(Hn−1(X, A), π) −→ H n(X, A; π) −→ Hom(Hn(X, A), π) −→ 0. 0 −→ Ext1 The veriﬁcation of the axi
oms listed in the previous section is immediate, as in homology. The fact that cellularly homotopic maps induce the same map on cohomology uses our observations relating homotopies of chain complexes with homotopies of cochain complexes. For exactness, the fact that our chain complexes are free over Z implies that we have a short exact sequence of cochain complexes 0 −→ C∗(X, A; π) −→ C∗(X; π) −→ C∗(A; π) −→ 0. The required natural long exact sequence follows. The rest is the same as in homology. For general spaces X, we can use ΓX = \|S∗X\| as a canonical CW approximation functor. We deﬁne the singular cochains of X to be the cellular cochains 3. CUP PRODUCTS IN COHOMOLOGY 139 of ΓX. Then our passage from the cohomology of CW complexes to the cohomology of general spaces can be realized by taking the cohomology of singular cochain complexes. 3. Cup products in cohomology If X and Y are CW complexes, we have an isomorphism C∗(X × Y ) ∼= C∗(X) ⊗ C∗(Y ) of chain complexes and therefore, for any Abelian groups π and π′, an isomorphism of cochain complexes C∗(X × Y ; π ⊗ π′) ∼= Hom(C∗(X) ⊗ C∗(Y ), π ⊗ π′). By our observations about cochain complexes, there results a natural homomorphism H ∗(X; π) ⊗ H ∗(Y ; π′) −→ H ∗(X × Y ; π ⊗ π′). If X = Y and if π = π′ = R is a commutative ring, we can use the diagonal map ∆ : X −→ X × X and the product R ⊗ R −→ R to obtain a “cup product” ∪ : H ∗(X; R) ⊗R H ∗(X; R) −→ H ∗(X; R). More precisely, for p ≥ 0 and q ≥ 0, we have a product �
� : H p(X; R) ⊗R H q(X; R) −→ H p+q(X; R). We have noted that we can use C∗(X; R) instead of C∗(X) and so justify tensoring over R rather than Z. This product makes H ∗(X; R) into a graded unital, associative, and “commutative” R-algebra. Here commutativity is understood in the appropriate graded sense, namely xy = (−1)pqyx if deg x = p and deg y = q. The image of 1 ∈ R = H 0(∗; R) under the map π∗ : H 0(∗; R) −→ H 0(X; R) induced by the unique map π : X −→ {∗} is the unit (= identity element) for the product. In fact, the diagrams that say that H ∗(X; R) is unital, associative, and commutative result by passing to cohomology from the evident commutative diagrams tttttttttt tttttttttt X KKKKKKKKKK KKKKKKKKKK ∆ X × ∗ X × X id ×π π×id / ∗ × X, X ∆ ∆ X × X ∆×id X × X id ×∆ / X × X × X, and ∆ {wwwwwwwww X × X X t ∆ #HHHHHHHHH / X × X. o o / / / / { # / 140 AXIOMATIC AND CELLULAR COHOMOLOGY THEORY Here t : X × Y −→ Y × X is the transposition, t(x, y) = (y, x). The following diagrams commute in homology and cohomology with coﬃcients in R: and H∗(X) ⊗R H∗(Y ) α / H∗(X × Y ) τ t∗ H∗(Y ) ⊗R H∗(X) α / H∗(Y × X) H ∗(X) ⊗R H ∗(Y ) τ H ∗(Y ) ⊗R H ∗(X) α α / H ∗(X × Y ) ∗ t
/ H ∗(Y × X). In both diagrams, τ (x ⊗ y) = (−1)pqy ⊗ 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 Sp+q = Sp ∧ Sq −→ Sq ∧ Sp = Sp+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. 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. 4. An example: RP n and the Borsuk-Ulam theorem Remember that RP n is a CW complex with one q-cell for each q ≤ n. The diﬀerential on Cq(RP n) ∼= Z is zero if q is odd and multiplication by 2 if q is even. When we dualize to C∗(RP n), we ﬁnd that the diﬀerential on Cq(RP n) is multiplication by 2 if q is odd and zero if q is even. We read oﬀ that Z if q = 0 Z2 if 0 < q ≤ n and q is even Z if q = n is odd otherwise. 0 H q(RP n; Z) =    If we work mod 2, taking Z2 as coeﬃcient group, then the answer takes a nicer form, namely H q(RP n; Z2) = Z2 if 0 ≤ q ≤ n if q > n. 0 ( The reader may ﬁnd it instructive to compare with the calculations in homology, checking the correctness of the calculation by comparison with the universal coefﬁcient theorem. We shall later use Poincar´e 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. That is, for 1 ≤ q ≤ n, the unique non-zero element of H q(RP n; Z2) is the qth power of x. This means that the elements are so tightly bound together / / / / 4. AN EXAMPLE: RP n AND THE BORSUK-ULAM THEOREM 141 that knowledge of the cohomological behavior of a map f : RP m −→ RP n on cohomology in degree one determines its behavior on cohomology in all higher degrees. We assume that m ≥ 1 and n ≥ 1 to avoid triviality. Proposition. Let f : RP m −→ RP n be a map such that f∗ : π1(RP m) −→ π1(RP n) is non-zero. Then m ≤ n. Proof. Since π1(RP 1) = Z and π1(RP m) = Z2 if m ≥ 2, the result is certainly true if n = 1. Thus assume that n > 1 and assume for a contradiction that m > n. By the naturality of the Hurewicz isomorphism, f∗ : H1(RP m; Z) −→ H1(RP n; Z) is non-zero. By our universal coeﬃcient theorems, the same is true for mod 2 homology and for mod 2 cohomology. That is, if x is the non-zero element of H 1(RP n; Z2), then f ∗(x) is the non-zero element of H 1(RP m; Z2). By the naturality of cup products (f ∗(x))m = f ∗(xm). However, the left side is non-zero in H m(RP m; Z2) and the right side is zero since xm = 0 by our assumption that m > n. The contradiction establishes the conclu- sion. We use this fact together with covering space theory to prove a celebrated result known as the Borsuk-Ulam theorem. A map g : Sm −→ Sn 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: Sm pm RP m g f Sn pn / RP n, where pm and pn are the canonical coverings. Theorem. If m > n ≥ 1, then there exist no antipodal maps Sm −→ Sn. Proof. Suppose given an antipodal map g : Sm −→ Sn. According to the proposition, f∗ : π1(RP m) −→ π1(RP n) is zero. According to the fundamental theorem of covering space theory, there is a map ˜f : RP m −→ Sn such that pn ◦ ˜f = f. Let s ∈ Sm. Then ˜f (pm(s)) = ˜f (pm(−s)) must be either g(s) or g(−s), since these are the only two points in p−1 n (f (pm(s))). Thus either t = s or t = −s satisﬁes ˜f (pm(t)) = g(t). Therefore, by the fundamental theorem of covering space theory, the maps ˜f ◦ pm and g must be equal since they agree on a point. This is absurd: ˜f ◦ pm takes antipodal points to the same point, while g was assumed to be antipodal. Theorem (Borsuk-Ulam). For any continuous map f : Sn −→ Rn, there exists x ∈ Sn such that f (x) = f (−x). Proof. Suppose for a contradiction that f (x) 6= f (−x) for all x. We could then deﬁne a continuous antipodal map g : Sn −→ Sn−1 by letting g(x) be the point at which the vector from 0 through f (x) − f (−x) intersects Sn−1. / / / 142 AXIOMATIC AND CELLULAR COHOMOLOGY THEORY 5. Obstruction theory 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. It also explains why the letter π is so often
used to denote coeﬃcient groups. 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. Let (X, A) be a relative CW complex with relative skeleta X n and let Y be an n-simple space. The assumption on Y has the eﬀect that we need not worry about basepoints. Let f : X n −→ Y be a map. We ask when f can be extended to a map X n+1 −→ Y that restricts to the given map on A. If we compose the attaching maps Sn → X of cells of X \ A with f, we obtain elements of πn(Y ). These elements specify a well deﬁned “obstruction cocycle” cf ∈ Cn+1(X, A; πn(Y )). Clearly, by 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. If f and f ′ are maps X n → Y and h is a homotopy rel A of the restrictions of f and f ′ to X n−1, then f, f ′, and h together deﬁne a map h(f, f ′) : (X × I)n −→ Y. Applying ch(f,f ′) to cells j × I, we obtain a “deformation cochain” df,f ′,h ∈ Cn(X, A; πn(Y )) such that δdf,f ′,h = cf − cf ′. Moreover, given f and d, there exists f ′ that coincides with f on X n−1 and satisﬁes df,f ′ = d, where the constant homotopy h is understood. This gives the following result. Theorem. For f : X n −→ Y, the restriction of f to X n−1 extends to a map X
n+1 → Y if and only if [cf ] = 0 in H n+1(X, A; πn(Y )). It is natural to ask further when such extensions are unique up to homotopy, and a similar argument gives the answer. 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. PROBLEMS The ﬁrst few problems here are parallel to those at the end of Chapter 16. (1) Let X be a space that satisﬁes the hypotheses used to construct a universal cover ˜X and let A be an Abelian group. Using cellular or singular chains, show that C∗(X; A) ∼= HomZ[π](C∗( ˜X), A). 5. OBSTRUCTION THEORY 143 (2) Show that there is an isomorphism H ∗(K(π, 1); A) ∼= Ext∗ Z[π](Z, A). When A is a commutative ring, the Ext groups have algebraically deﬁned products, constructed as follows. The evident isomorphism Z ∼= Z ⊗ Z is covered by a map of free Z[π]-resolutions P −→ P ⊗ P, where Z[π] acts diagonally on tensor products, α(x ⊗ y) = αx ⊗ αy. This chain map is unique up to chain homotopy. It induces a map of chain complexes HomZ[π](P, A) ⊗ HomZ[π](P, A) −→ HomZ[π](P, A) and therefore an induced product on Ext∗ Z[π](Z, A). Convince yourself that the isomorphism above preserves products and explain the intuition (don’t worry about technical exactitude). (3) * Now use homological algebra to determine H ∗(RP ∞; Z2) as a ring. (4) Use the previous problem to deduce the ring structure on H ∗(RP n; Z
2) for each n ≥ 1. (5) Let p : Y −→ X be a covering space with ﬁnite ﬁbers, say of cardinality n. Construct a “transfer homomorphism” t : H ∗(Y ; A) −→ H ∗(X; A) and show that t ◦ p∗ : H ∗(X; A) −→ H ∗(X; A) is multiplication by n. (6) Let X and Y be CW complexes. Show that the interchange map t : X × Y −→ Y × X satisﬁes t∗([i] ⊗ [j]) = (−1)pq[j] ⊗ [i] for a p-cell of X and a q-cell of Y. Deduce that the cohomology ring H ∗(X) is commutative in the graded sense: x ∪ y = (−1)pqy ∪ x if deg x = p and deg y = q. An “H-space” is a space X with a basepoint e and a product φ : X × X −→ X such that the maps λ : X −→ X and ρ : X −→ X given by left and right multiplication by e are each homotopic to the identity map. Note that λ and ρ specify a map X ∨ X −→ X that is homotopic to the codiagonal or folding map ▽, which restricts to the identity on each wedge summand. The following two problems are optional review exercises. 7. If e is a nondegenerate basepoint for X, then φ is homotopic to a product φ′ such that left and right multiplication by e under the product φ′ are both identity maps. 8. Show that the product on π1(X, e) induced by the based map φ′ : X × X −→ X agrees with the multiplication given by composition of paths and that both products are commutative. 9. For an H-space X, the following diagram is commutative: X × X ∆×∆ / X × X × X × X id ×t×id / ×φ / X × X (Check it: it is too trivial to write down.) Let X be (n − 1)-connected, n ≥ 2
, and let x ∈ H n(X). (a) Show that φ∗(x) = x ⊗ 1 + 1 ⊗ x. / / / 144 AXIOMATIC AND CELLULAR COHOMOLOGY THEORY (b) Show that (∆ × ∆)∗(id × t × id)∗(φ × φ)∗(x ⊗ x) = x2 ⊗ 1 + (1 + (−1)n)(x ⊗ x) + 1 ⊗ x2. (c) Prove that, if n is even, then either 2(x ⊗ x) = 0 in H ∗(X × X) or x2 6= 0. Deduce that Sn cannot be an H-space if n is even. CHAPTER 19 Derivations of properties from the axioms Returning to the axiomatic approach to cohomology, we assume given a theory on pairs of spaces and give some deductions from the axioms. This may be viewed as a dualized review of what we did in homology, and we generally omit the proofs. The only signiﬁcant diﬀerence that we will encounter is in the computation of the cohomology of colimits. In a ﬁnal section, we show the uniqueness of (ordinary) cohomology with coeﬃcients in π. Prior to that section, we make no use of the dimension axiom in this chapter. A “generalized cohomology theory” E∗ is deﬁned to be a system of functors Eq(X, A) and natural transformations δ : Eq(A) −→ Eq+1(X, A) that satisfy all of our axioms except for the dimension axiom. Similarly, we have the notion of a generalized cohomology theory on CW pairs, and the following result holds. Theorem. A cohomology theory E∗ on pairs of spaces determines and is determined by its restriction to a cohomology theory E∗ on pairs of CW complexes. 1. Reduced cohomology groups and their properties For a based space X, we deﬁne the reduced cohomology of X to be ˜Eq(X) = Eq(X, ∗). There results a direct sum
decomposition E∗(X) ∼= ˜E∗(X) ⊕ E∗(∗) that is natural with respect to based maps. For ∗ ∈ A ⊂ X, the summand E∗(∗) maps isomorphically under the map E∗(X) −→ E∗(A), and the exactness axiom implies that there is a reduced long exact sequence · · · −→ ˜Eq−1(A) δ−→ Eq(X, A) −→ ˜Eq(X) −→ ˜Eq(A) −→ · · ·. The unreduced cohomology groups are recovered as the special cases E∗(X) = ˜E∗(X+) of reduced ones, and similarly for maps. Relative cohomology groups are also special cases of reduced ones. 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∗(X, A). We may replace any inclusion i : A −→ X by the canonical coﬁbration A −→ M i and then apply the result just given to obtain an isomorphism E∗(X, A) ∼= ˜E∗(Ci). 145 146 DERIVATIONS OF PROPERTIES FROM THE AXIOMS Theorem. For a nondegenerately based space X, there is a natural isomor- phism Σ : ˜Eq(X) ∼= ˜Eq+1(ΣX). Corollary. Let ∗ ∈ A ⊂ X, where i : A −→ X is a coﬁbration between nondegenerately based spaces. In the long exact sequence · · · −→ ˜Eq−1(A) δ−→ ˜Eq(X/A) −→ ˜Eq(X) −→ ˜Eq(A) −→ · · · of the pair (X, A), the connecting homomorphism δ is the composite ˜Eq−1(A) �
�−→ ˜Eq(ΣA) ∂ −→ ˜Eq(X/A). ∗ Corollary. For any n and q, ˜Eq(Sn) ∼= ˜Eq−n(∗). 2. Axioms for reduced cohomology Definition. A reduced cohomology theory ˜E∗ consists of functors ˜Eq from the homotopy category of nondegenerately based spaces to the category of Abelian groups that satisfy the following axioms. • EXACTNESS If i : A −→ X is a coﬁbration, then the sequence ˜Eq(X/A) −→ ˜Eq(X) −→ ˜Eq(A) is exact. • SUSPENSION For each integer q, there is a natural isomorphism Σ : ˜Eq(X) ∼= ˜Eq+1(ΣX). • ADDITIVITY If X is the wedge of a set of nondegenerately based spaces Xi, then the inclusions Xi −→ X induce an isomorphism ˜E∗(X) −→ ˜E∗(Xi). i • WEAK EQUIVALENCE If f : X −→ Y is a weak equivalence, then Q f ∗ : ˜E∗(Y ) −→ ˜E∗(X) is an isomorphism. The reduced form of the dimension axiom would read ˜H 0(S0) = π and ˜H q(S0) = 0 for q 6= 0. Theorem. A cohomology theory E∗ on pairs of spaces determines and is de- termined by a reduced cohomology theory ˜E∗ on nondegenerately based spaces. Definition. A reduced cohomology theory ˜E∗ on based CW complexes consists of functors ˜Eq from the homotopy category of based CW complexes to the category of Abelian groups that satisfy the following axioms. • EXACTNESS If A is a subcomplex of X, then the sequence ˜Eq(X/A) −→ ˜Eq(X) −→ ˜Eq(A) is exact. • SUSPENSION For each integer q, there is a natural isomorphism Σ :
˜Eq(X) ∼= ˜Eq+1(ΣX). 3. MAYER-VIETORIS SEQUENCES IN COHOMOLOGY 147 • ADDITIVITY If X is the wedge of a set of based CW complexes Xi, then the inclusions Xi −→ X induce an isomorphism ˜E∗(X) −→ ˜E∗(Xi). i Theorem. A reduced cohomology theory ˜E∗ on nondegenerately based spaces Q determines and is determined by its restriction to a reduced cohomology theory on based CW complexes. Theorem. A cohomology theory E∗ on CW pairs determines and is determined by a reduced cohomology theory ˜E∗ on based CW complexes. 3. Mayer-Vietoris sequences in cohomology We have Mayer-Vietoris sequences in cohomology just like those in homology. The proofs are the same. Poincar´e duality between the homology and cohomology of manifolds will be proved by an inductive comparison of homology and cohomology Mayer-Vietoris sequences. We record two preliminaries. Proposition. For a triple (X, A, B), the following sequence is exact: · · · Eq−1(A, B) δ−→ Eq(X, A) j −→ Eq(X, B) i −→ Eq(A, B) −→ · · ·. ∗ ∗ Here i : (A, B) −→ (X, B) and j : (X, B) −→ (X, A) are inclusions and δ is the composite Eq−1(A, B) −→ Eq−1(A) δ−→ Eq(X, A). Now let (X; A, B) be an excisive triad and set C = A ∩ B. Lemma. The map E∗(X, C) −→ E∗(A, C) ⊕ E∗(B, C) induced by the inclusions of (A, C) and (B, C) in (X, C) is an isomorphism. Theorem (Mayer-Vietoris sequence). Let (X; A, B) be an excisive triad and set C = A ∩ B. The
following sequence is exact: · · · −→ Eq−1(C) ∆ −−→ Eq(X) ∗ ∗ φ −→ Eq(A) ⊕ Eq(B) ∗ ψ −−→ Eq(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 Eq−1(C) δ−→ Eq(A, C) ∼= Eq(X, B) −→ Eq(X). For the relative version, let X be contained in some ambient space Y. Theorem (Relative Mayer-Vietoris sequence). The following sequence is exact: ∗ ∗ ∗ · · · −→ Eq−1(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 −→ Eq(Y, A) ⊕ Eq(Y, B) ψ −−→ Eq(Y, C) −→ · · ·. −−→ Eq(Y, X) φ∗(χ) = (k∗(χ), ℓ∗(χ)) and ψ∗(α, β) = i∗(α) − j∗(β) and ∆∗ is the composite Eq−1(Y, C) −→ Eq−1(A, C) ∼= Eq−1(X, B) δ−→ Eq(Y, X). 148 DERIVATIONS OF PROPERTIES FROM THE AXIOMS Corollary. The absolute and relative Mayer-Vietoris sequences are related by the following commutative diagram: Eq
−1(C) ∗ ∆ Eq(X) ∗ φ Eq(A) ⊕ Eq(B) ∗ ψ δ δ δ+δ Eq(C) δ Eq(Y, C) ∗ / / Eq+1(Y, X) ∆ ∗ φ / Eq+1(Y, A) ⊕ Eq+1(Y, B) ∗ / / Eq+1(Y, C). ψ 4. Lim1 and the cohomology of colimits In this section, we let X be the union of an expanding sequence of subspaces Xi, i ≥ 0. 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. Rather than go into the general theory, we simply display how the “ﬁrst right derived functor” lim1 of an inverse sequence of Abelian groups can be computed. Lemma. Let fi : Ai+1 −→ Ai, i ≥ 1, be a sequence of homomorphisms of Abelian groups. Then there is an exact sequence 0 −→ lim Ai α−→ where α is the diﬀerence of the identity map and the map with coordinates fi and β is the map whose projection to Ai is the canonical map given by the deﬁnition of a limit. iAi −→ lim1 Ai −→ 0, iAi Q Q β −→ That is, we may as well deﬁne lim1 Ai to be the displayed cokernel. We then have the following result. Theorem. For each q, there is a natural short exact sequence 0 −→ lim1 Eq−1(Xi) −→ Eq(X) π−→ lim Eq(Xi) −→ 0, where π is induced by the inclusions Xi −→ X. Proof. We use the notations and constructions in the proof that homology commutes with colimits and consider the excisive triad (tel Xi; A, B) with C = A ∩ B constructed there. By the
additivity axiom, i E∗(X2i), E∗(B) = i E∗(X2i+1), and E∗(C) = i E∗(Xi). E∗(A) = We construct the following commutative diagram, whose top row is the cohomology Mayer-Vietoris sequence of the triad (tel Xi; A, B) and whose bottom row is an exact sequence of the sort displayed in the previous lemma. Q Q Q / Eq(tel Xi) Eq(A) ⊕ Eq(B) Eq(C) Eq+1(tel Xi) / · · · ∼= / Eq(X) ′ π ′ β β ∼= Q Eq(Xi) Q(−1)i / Qi Eq(Xi) ′ α α ∼= ∼= Qi Eq(Xi) / Eq+1(X) / · · · Qi(−1)i / Qi Eq(Xi) / lim1 Eq(Xi) / 0. 0 / lim Eq(Xi) The commutativity of the bottom middle square is a comparison based on the sign used in the Mayer-Vietoris sequence. Here the map π′ diﬀers by alternating signs · · · · · · / / / / / / / / / / / / / / / / / / / / / / / / / / / 5. THE UNIQUENESS OF THE COHOMOLOGY OF CW COMPLEXES 149 from the canonical map π, but this does not aﬀect the conclusion. A chase of the diagram implies the result. The lim1 “error terms” are a nuisance, and it is important to know when they vanish. 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. Lem
ma. If the inverse sequence fi : Ai+1 −→ Ai satisﬁes the Mittag-Leﬄer condition, then lim1 Ai = 0. For example, for q < n, the inclusion X n −→ X n+1 of skeleta in a CW complex induces an isomorphism H q(X n+1; π) −→ H q(X n; π) and we conclude that the canonical map H q(X; π) −→ H q(X n; π) is an isomorphism for q < n. This is needed in the proof of the uniqueness of ordinary cohomology. 5. The uniqueness of the cohomology of CW complexes As with homology, one reason for deﬁning ordinary cohomology with coeﬃcients in an Abelian group π in terms of cellular cochains is the inevitability of the deﬁnition. 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 with diﬀerential the composite Cn(X; π) = H n(X n, X n−1; π), d : Cn(X; π) −→ Cn+1(X; π) H n(X n, X n−1; π) −→ H n(X n) 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). We deﬁne the reduced cochains ˜C∗(X; π) of a based space X to be the kernel of the map C∗(X; π) −→ C∗(∗; π) induced by the inclusion {∗} −→ X. For a CW pair (X, A), we deﬁne C∗(X, A; π) to be the kernel of the epimorphism C∗(X; π) −→ C∗(A; π) induced by the inclusion A
−→ X. The analogue of the previous result for reduced and relative cochains follows directly. This leads to a uniqueness theorem for cohomology just like that for homology. Theorem. There is a natural isomorphism H ∗(X, A; π) ∼= H ∗(C∗(X, A; π)) under which the natural transformation δ agrees with the natural transformation induced by the connecting homomorphisms associated to the short exact sequences 0 −→ C∗(X, A; π) −→ C∗(X; π) −→ C∗(A; π) −→ 0. 150 DERIVATIONS OF PROPERTIES FROM THE AXIOMS Proof. It suﬃces to obtain a natural isomorphism of reduced theories on based CW complexes X. We have seen that ˜H q(X; π) ∼= ˜H q(X n; π) for q < n, and we obtain a diagram dual to that used in the proof of the analogue in homology by arrow reversal. We leave further details as an exercise for the reader. PROBLEMS (1) Complete the proof of the uniqueness theorem for cohomology. In the following sequence of problems, we take cohomology with coeﬃcients in a commutative ring R and we write ⊗ for ⊗R. 2. Let A and B be subspaces of a space X. Construct a relative cup product H p(X, A) ⊗ H q(X, B) −→ H p+q(X, A ∪ B) and show that the following diagram is commutative: H p(X, A) ⊗ H q(X, B) H p+q(X, A ∪ B) H p(X) ⊗ H q(X) / H p+q(X). The horizontal arrows are cup products; the vertical arrows are induced from X −→ (X, A), and so forth. 3. Let X have a basepoint ∗ ∈ A ∩ B. Deduce a commutative diagram H p(X, A) ⊗ H q(X, B) H p+q(X, A ∪ B) ˜H p(X) ⊗ ˜H q(X) /
˜H p+q(X). 4. Let X = A ∪ B, where A and B are contractible and A ∩ B 6= ∅. Deduce that the cup product ˜H p(X) ⊗ ˜H q(X) −→ ˜H p+q(X) is the zero homomorphism. 5. Let X = ΣY = Y ∧ S1. Deduce that the cup product ˜H p(X) ⊗ ˜H q(X) −→ ˜H p+q(X) is the zero homomorphism. Commentary: Additively, cohomology groups are “stable,” in the sense that ˜H p(Y ) ∼= ˜H p+1(ΣY ). 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. / / / / / / CHAPTER 20 The Poincar´e duality theorem The crucial starting point for applications of algebraic topology to geometric topology is the Poincar´e duality theorem. It gives a tight algebraic constraint on the homology and cohomology groups of compact manifolds. 1. Statement of the theorem It is apparent that there is a kind of duality relating the construction of hoIn its simplest form, this is reﬂected by the fact that mology and cohomology. evaluation of cochains on chains gives a natural homomorphism Cp(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. Taking π = ρ to be a commutative ring R and using its product, there results a pairing H p(X; R) ⊗R Hp(X; R) −→ R. It is usually written hα, xi for α ∈ H p(X; R) and x ∈ Hp(X
; R). 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). That is, the cohomology groups of X are the vector space duals of the homology groups of X. Now let M be a compact manifold of dimension n. 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. We shall not go into the proof but shall take the result as known. We have the cup product H p(M ; R) ⊗ H n−p(M ; R) −→ H n(M ; R). 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. One version of the Poincar´e 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). 151 152 THE POINCAR ´E DUALITY THEOREM In fact, Poincar´e duality does not require the commutative ring R to be a ﬁeld, and it is useful to allow R-modules π as coeﬃcients in our homology and cohomology groups. We shall gradually make sense of and prove the following theorem. Theorem (Poincar´e 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). We shall deﬁne the notion of an R-orientation and an R-fundamental class of a manifold in §3, and we shall later prove the following result. 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). The isomorphism D is given by the adjoint of the cup product pairing determined by z, but it is more convenient to describe it in terms of the “cap product.” For any space X and R-module π, there is a cap product ∩ : H p(X; π) ⊗R Hn(X; R) −→ Hn−p(X; π). We shall deﬁne it in the next section. The isomorphism D is speciﬁed by D(α) = α ∩ z. When π = R, we shall prove that the cap product, cup product, and evaluation pairing are related by the fundamental identity hα ∪ β, xi = hβ, α ∩ xi. Taking x = z, this shows that in this case D is adjoint to the cup product pairing determined by z. We explain a few consequences before beginning to ﬁll in the details and proofs. Let M be a connected compact oriented (= Z-oriented) n-manifold. Taking integer coeﬃcients, we have D : H p(M ) ∼= Hn−p(M ). With p = 0, this shows that Hn(M ) ∼= Z with generator the fundamental class z. With p = n, it shows that H n(M ) ∼= Z with generator ζ dual to z, hζ, zi = 1. The relation between ∪ and ∩ has the following consequence. Corollary. Let Tp ⊂ H p(M ) be the torsion subgroup. The cup product pair- ing α ⊗ β −→ hαβ, zi induces a nonsingular pairing H p(M )/Tp ⊗ H n−p(M )/Tn−p
−→ Z. Proof. If α ∈ Tp, say rα = 0, and β ∈ H n−p(M ), then r(α∪β) = 0 and therefore α ∪ β = 0 since H n(M ) = Z. Thus the pairing vanishes on torsion elements. Since Ext1 Z(H∗(M ), Z) is a torsion group. By the universal coeﬃcient theorem, this implies that Z(Zr, Z) = Zr and each Hp(M ) is ﬁnitely generated, Ext1 H p(M )/Tp = Hom(Hp(M ), Z). Thus, if α ∈ H p(M ) projects to a generator of the free Abelian group H p(M )/Tp, then there exists a ∈ Hp(M ) such that hα, ai = 1. By Poincar´e duality, there exists β ∈ H n−p(M ) such that β ∩ z = a. Then hβ ∪ α, zi = hα, β ∩ zi = 1. We shall see that any simply connected manifold, such as CP n, is orientable. The previous result allows us to compute the cup products of CP n. 2. THE DEFINITION OF THE CAP PRODUCT 153 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. We know that CP n is a CW complex with one 2q-cell for each q, 0 ≤ q ≤ n. Therefore the conclusion is correct additively: H 2q(CP n) is a free Abelian group on one generator for 0 ≤ q ≤ n. Moreover CP n−1 is the (2n − 1)skeleton of CP n, and the inclusion CP n−1 −→ CP n therefore induces an isomorphism H 2q(CP n) −→ H 2q(CP n−1) for q < n. We proceed by induction on n, the conclusion being obvious for CP 1 ∼= S2. The induction hypothesis implies that if α generates H 2(CP n), then α
q generates H 2q(CP n) for q < n. By the previous result, there exists β ∈ H 2n−2(CP n) such that hα ∪ β, zi = 1. Clearly β must be a generator, so that β = ±αn−1, and therefore αn must generate H 2n(CP n). In the presence of torsion in the cohomology of M, it is convenient to work with coeﬃcients in a ﬁeld. We shall see that an oriented manifold is R-oriented for any commutative ring R. The same argument as for integer coeﬃcients gives the following more convenient nonsingular pairing result. Corollary. Let M be a connected compact R-oriented n-manifold, where R is 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. Corollary. As a graded ring, H ∗(RP n; Z2) is the truncated polynomial algebra Z2[α]/(αn+1), where deg α = 1. That is, αq is the non-zero element of H q(RP n; Z2) for 1 ≤ q ≤ n. 2. The deﬁnition of the cap product To deﬁne the cap product, we may as well assume that X is a CW complex, by CW approximation. The diagonal map ∆ : X −→ X × X is not cellular, but it is homotopic to a cellular map ∆′. Thus we have a chain map ∆′ ∗ : C∗(X) −→ C∗(X × X) ∼= C∗(X) ⊗ C∗(X). It carries Cn(X) to and using that R ∼= R ⊗R R, we obtain P Cp(X) ⊗ Cn−p(
X). Tensoring over a commutative ring R ∆′ ∗ : C∗(X; R) −→ C∗(X × X; R) ∼= C∗(X; R) ⊗R C∗(X; R). For an R-module π, we deﬁne ∩ : C∗(X; π) ⊗R C∗(X; R) −→ C∗(X; π) 154 THE POINCAR ´E DUALITY THEOREM to be the composite C∗(X; π) ⊗R C∗(X; R) id ⊗∆ ′ ∗ C∗(X; π) ⊗R C∗(X; R) ⊗R C∗(X; R) ε⊗id π ⊗R C∗(X; R) ∼= C∗(X; π). Here ε evaluates cochains on chains. Precisely, it must be interpreted as zero on Cp(X; π) ⊗R Cq(X; R) if p 6= q and the evident evaluation map HomR(Cp(X; R), π) ⊗R Cp(X; R) −→ π if p = q. Therefore the cap product is given degreewise by maps ∩ : Cp(X; π) ⊗R Cn(X; R) −→ Cn−p(X; π). To understand this, it makes sense to think in terms of C∗(X; π) regraded by negative degrees and so thought of as a chain complex rather than a cochain complex. Our convention on cochains that (dα)(x) = (−1)p+1α(dx) for α ∈ Cp(X; π) and x ∈ Cp+1(X; R) means that ε ◦ d = 0, where d is the tensor product diﬀerential on the chain complex C∗(X; π) ⊗R C∗(X; R). That is, ε is a map of chain complexes, where π is thought of as a chain complex concentrated in degree
zero, with zero diﬀerential. It follows that ∩ is a chain map. That is, d(α ∩ x) = (dα) ∩ x + (−1)deg αα ∩ dx. 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). To relate the cap and cup products, recall that the latter is induced by ∆′∗ : C∗(X; R) ⊗R C∗(X; R) ∼= C∗(X × X; R) −→ C∗(X; R). It is trivial that the following diagram commutes: C∗(X × X; R) ⊗R C∗(X; R) id ⊗∆ ′ ∗ C∗(X × X; R) ⊗R C∗(X × X; R) ′∗ ∆ ⊗id C∗(X; R) ⊗R C∗(X; R) ε ε / R. We may identify the chains and cochains of X × X on the top row with tensor products of chains and cochains of X. After this identiﬁcation, the right-hand map / / / 3. ORIENTATIONS AND FUNDAMENTAL CLASSES 155 ε becomes the composite C∗(X; R) ⊗R C∗(X; R) ⊗R C∗(X; R) ⊗R C∗(X; R) id ⊗t⊗id C∗(X; R) ⊗R C∗(X; R) ⊗R C∗(X; R) ⊗R C∗(X; R) ε⊗ε R ⊗R R ∼= R. Noting the agreement of signs introduced by the two maps t, we see that this composite is the same as the composite C∗(X; R) ⊗R C∗(X; R) ⊗R
C∗(X; R) ⊗R C∗(X; R) t⊗id ⊗ id C∗(X; R) ⊗R C∗(X; R) ⊗R C∗(X; R) ⊗R C∗(X; R) id ⊗ε⊗id C∗(X; R) ⊗R C∗(X; R) ε R. Inspecting deﬁnitions, we see that, on elements, these observations prove the fundamental identity hα ∪ β, xi = hβ, α ∩ xi. For use in the proof of the Poincar´e duality theorem, we observe that the cap product generalizes to relative cap products ∩ : H p(X, A; π) ⊗R Hn(X, A; R) −→ Hn−p(X; π) and ∩ : H p(X; π) ⊗R Hn(X, A; R) −→ Hn−p(X, A; π) for pairs (X, A). Indeed, we may assume that (X, A) is a CW pair and that ∆′ restricts to a map A −→ A × A that is homotopic to the diagonal of A. Via the quotient map X −→ X/A, ∆′ induces relative diagonal approximations ∆′ ∗ : C∗(X, A; R) −→ C∗(X, A; R) ⊗ C∗(X; R) and ∆′ ∗ : C∗(X, A; R) −→ C∗(X; R) ⊗ C∗(X, A; R). These combine with the evident evaluation maps to give the required relative cap products. 3. Orientations and fundamental classes Let M be an n-manifold, not necessarily compact; the extra generality will be crucial to our proof of the Poincar´e duality theorem. For x ∈ M, we can choose a coordinate chart U ∼= Rn with x ∈ U. By excision, exactness, and homotopy invariance, we have isomorphisms Hi(M, M − x) ∼= Hi(U
, U − x) ∼= ˜Hi−1(U − x) ∼= ˜Hi−1(Sn−1). 156 THE POINCAR ´E DUALITY THEOREM This holds with any coeﬃcient group, but we agree to take coeﬃcients in a given commutative ring R. Thus Hi(M, M − x) = 0 if i 6= n and Hn(M, M − x) ∼= R. 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). We say that M is R-orientable if it admits an R-orientation. When R = Z, we refer to orientations and orientability. There are various equivalent ways of formulating these notions. We leave it as an exercise for the reader to reconcile the present deﬁnition of orientability with any other deﬁnition he or she may have seen. Clearly an R-fundamental class z determines an R-orientation: given any open cover {Ui}, we take zi to be the image of z in Hn(M, M − Ui). The converse holds when M is compact. To show this, we need the following vanishing theorem, which we shall prove in the next section. Theorem (Vanishing). Let
M be an n-manifold. For any coeﬃcient group π, Hi(M ; π) = 0 if i > n, and ˜Hn(M ; π) = 0 if M is connected and is not compact. We can use this together with Mayer-Vietoris sequences to construct R-fundamental classes at compact subspaces from R-orientations. To avoid trivialities, we tacitly assume that n > 0. (The trivial case n = 0 forced the use of reduced homology in the statement; where arguments use reduced homology below, it is only to ensure that what we write is correct in dimension zero.) 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. Proof. First assume that K is contained in a coordinate chart U ∼= Rn. By excision and exactness, we then have Hi(M, M − K; π) ∼= Hi(U, U − K; π) ∼= ˜Hi−1(U − K; π). Since U − K is open in U, the vanishing theorem implies that ˜Hi−1(U − K; π) = 0 for i > n. In fact, a lemma used in the proof of the vanishing theorem will prove this directly. In this case, an R-fundamental class in Hn(M, M − U ) maps to an R-fundamental class in Hn(M, M − K). A general compact subset K of M can be written as the union of ﬁnitely many compact subsets, each of which is contained in a coordinate chart. By induction, it suﬃces to prove the result for K ∪ L under the assumption that it holds for K, L, and K ∩ L. With any coeﬃcients, we have 3. ORIENTATIONS AND FUNDAMENTAL CLASSES 157 the Mayer-Vietoris sequence · · · −→ Hi+1(M, M − K ∩ L) ∆−
→ Hi(M, M − K ∪ L) ψ −→ Hi(M, M − K) ⊕ Hi(M, M − L) φ −→ Hi(M, M − K ∩ L) −→ · · ·. The vanishing of Hi(M, M − K ∪ L; π) for i > n follows directly. Now take i = n and take coeﬃcients in R. Then ψ is a monomorphism. The R-fundamental classes zK ∈ Hn(M, M − K) and zL ∈ Hn(M, M − L) determined by a given R-orientation both map to the R-fundamental class zK∩L ∈ Hn(M, M − K ∩ L) determined by the given R-orientation. Therefore and there exists a unique zK∪L ∈ Hn(M, M − K ∪ L) such that φ(zK, zL) = zK∩L − zK∩L = 0 ψ(zK∪L) = (zK, zL). Clearly zK∪L is an R-fundamental class of M at K ∪ L. The vanishing theorem also implies the following dichotomy, which we have already noticed in our examples of explicit calculations. 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. Proof. Since M − x is connected and not compact, Hn(M − x; π) = 0 and thus Hn(M ; π) −→ Hn(M, M − x; π) ∼= π is a monomorphism for all coeﬃcient groups π. coeﬃcient theorem, In particular, by the universal Hn(M ; Z) ⊗ Zq −→ Hn(M, M − x; Z) ⊗ Zq ∼= Zq is a monomorphism for all positive integers q. If Hn(M ; Z) 6=
0, then Hn(M ; Z) ∼= Z with generator mapped to some multiple of a generator of Hn(M, M − x; Z). By the mod q monomorphism, the coeﬃcient must be ±1. As an aside, the corollary leads to a striking example of the failure of the naturality of the splitting in the universal coeﬃcient theorem. Consider a connected, compact, non-orientable n-manifold M. Let x ∈ M and write Mx for the pair (M, M − x). Since M is Z2-orientable, the middle vertical arrow in the following diagram is an isomorphism between copies of Z2: 0 0 / Hn(M ) ⊗ Z2 Hn(M ; Z2) Z 1 (Hn−1(M ), Z2) Tor 0 ∼= 0 / Hn(Mx) ⊗ Z2 / Hn(Mx; Z2) Z 1 (Hn−1(Mx), Z2) / Tor 0 / 0. Clearly Hn−1(M, M − x) = 0, and the corollary gives that Hn(M ) = 0. 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. / / / / / / / / / / / 158 THE POINCAR ´E DUALITY THEOREM 4. The proof of the vanishing theorem Let M be an n-manifold, n > 0. Take all homology groups with coeﬃcients in a given Abelian group π in this section. We must prove the intuitively obvious statement that Hi(M ) = 0 for i > n and the much more subtle statement that Hn(M ) = 0 if M is connected and is not compact. The last statement is perhaps the technical heart of our proof of the Poincar´e duality theorem. 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. Proof. Let γ : Y −→ X be a
CW approximation of X and let x = γ∗(y). If y is represented by a cycle z ∈ Cq(Y ), then z, as a ﬁnite linear combination of q-cells, is an element of Cq(L) for some ﬁnite subcomplex L of Y. Let K = γ(L) and let k be the image of the homology class represented by z. Then K is compact and k maps to x. We need two lemmas about open subsets of Rn to prove the vanishing theorem, the ﬁrst of which is just a special case. Lemma. If U is open in Rn, then Hi(U ) = 0 for i ≥ n. Proof. Let s ∈ Hi(U ), i ≥ n. There is a compact subspace K of U and an element k ∈ Hi(K) that maps to s. We may decompose Rn as a CW complex whose n-cells are small n-cubes in such a way that there is a ﬁnite subcomplex L of Rn with K ⊂ L ⊂ U. (To be precise, use a cubical grid with small enough mesh.) For i > 0, the connecting homomorphisms ∂ are isomorphisms in the commutative diagram Hi+1(Rn, L) Hi+1(Rn, U ) ∂ Hi(L) ∂ / Hi(U ). Since (Rn, L) has no relative q-cells for q > n, the groups on the left are zero for i ≥ n. Since s is in the image of Hi(L), s = 0. Lemma. Let U be open in Rn. Suppose that t ∈ Hn(Rn, U ) maps to zero in Hn(Rn, Rn − x) for all x ∈ Rn − U. Then t = 0. Proof. We prove the equivalent statement that if s ∈ ˜Hn−1(U ) maps to zero in ˜Hn−1(Rn − x) for all x ∈ Rn − U, then s = 0. Choose a compact subspace K of U such that s is in the image of ˜Hn−1(K). Then K is contained in an open subset V whose closure ¯V is compact and contained in U, hence
s is the image of an element r ∈ ˜Hn−1(V ). We claim that r maps to zero in ˜Hn−1(U ), so that s = 0. Of course, r maps to zero in ˜Hn−1(Rn − x) if x 6∈ U. Let T be an open contractible subset of Rn such that ¯V ⊂ T and ¯T is compact. For example, T could be a large enough open cube. Let L = T − (T ∩ U ). For each x ∈ ¯L, choose a closed cube D that contains x and is disjoint from V. A ﬁnite set {D1,..., Dq} of these cubes covers ¯L. Let Ci = Di ∩ T and observe that (Rn − Di) ∩ T = T − Ci. We see by / / / 4. THE PROOF OF THE VANISHING THEOREM 159 induction on p that r maps to zero in ˜Hn−1(T − (C1 ∪ · · · ∪ Cp)) for 0 ≤ p ≤ q. This is clear if p = 0. For the inductive step, observe that T − (C1 ∪ · · · ∪ Cp) = (T − (C1 ∪ · · · ∪ Cp−1)) ∩ (Rn − Dp) and that Hn((T − (C1 ∪ · · · ∪ Cp−1)) ∪ (Rn − Dp)) = 0 by the previous lemma. Therefore the map ˜Hn−1(T − (C1 ∪ · · · ∪ Cp)) −→ ˜Hn−1(T − (C1 ∪ · · · ∪ Cp−1)) ⊕ ˜Hn−1(Rn − Dp) in the Mayer-Vietoris sequence is a monomorphism. Since r ∈ ˜Hn−1(V ) maps to zero in the two right-hand terms, by the induction hypothesis and the contractibility of Dp to a point x 6∈ U, it maps to zero in the left-hand term. Since V ⊂ T − (C1 ∪ · · · ∪ Cq) ⊂ T ∩ U
⊂ U, this implies our claim that r maps to zero in ˜Hn−1(U ). Proof of the vanishing theorem. Let s ∈ Hi(M ). We must prove that s = 0 if i > n and if i = n when M is connected and not compact. Choose a compact subspace K of M such that s is in the image of Hi(K). Then K is contained in some ﬁnite union U1 ∪ · · · ∪ Uq of coordinate charts, and it suﬃces to prove that Hi(U1 ∪ · · · ∪ Uq) = 0 for the speciﬁed values of i. Inductively, using that Hi(U ) = 0 for i ≥ n when U is an open subset of a coordinate chart, it suﬃces to prove that Hi(U ∪ V ) = 0 for the speciﬁed values of i when U is a coordinate chart and V is an open subspace of M such that Hi(V ) = 0 for the speciﬁed values of i. We have the Mayer-Vietoris sequence Hi(U ) ⊕ Hi(V ) −→ Hi(U ∪ V ) −→ ˜Hi−1(U ∩ V ) −→ ˜Hi−1(U ) ⊕ ˜Hi−1(V ). If i > n, the vanishing of Hi(U ∪ V ) follows immediately. Thus assume that M is connected and not compact and consider the case i = n. We have Hn(U ) = 0, Hn(V ) = 0, and ˜Hn−1(U ) = 0. It follows that Hn(U ∪ V ) = 0 if and only if i∗ : ˜Hn−1(U ∩ V ) −→ ˜Hn−1(V ) is a monomorphism, where i : U ∩ V −→ V is the inclusion. We claim ﬁrst that Hn(M ) −→ Hn(M, M − y) is the zero homomorphism for any y ∈ M. If x ∈ M and L is a path in M connecting x to y, then the diagram Hn(M ) / Hn(M, M − L) Hn(M, M − x
) ∼= 6lllllllllllll (RRRRRRRRRRRRR ∼= Hn(M, M − y) shows that if s ∈ Hn(M ) maps to zero in Hn(M, M − x), then it maps to zero in Hn(M, M − y). If s is in the image of Hn(K) where K is compact, we may choose a point x ∈ M − K. Then the map K −→ M −→ (M, M − x) factors through (M − x, M − x) and therefore s maps to zero in Hn(M, M − x). This proves our claim. / 6 ( 160 THE POINCAR ´E DUALITY THEOREM Now consider the following diagram, where y ∈ U − U ∩ V : Hn(U ∪ V ) vmmmmmmmmmmmmm Hn(V, U ∩ V ) / Hn(U ∪ V, U ∩ V ) ∂ vlllllllllllll ∂ ˜Hn−1(U ∩ V ) ∂ Hn(U, U ∩ V ) Hn(M ) 0 / Hn(M, M − y) ∼= Hn(U, U − y) i∗ ˜Hn−1(V ). Let r ∈ ker i∗. Since ˜Hn−1(U ) = 0, the bottom map ∂ is an epimorphism and there exists s ∈ Hn(U, U ∩ V ) such that ∂(s) = r. We claim that s maps to zero in Hn(U, U − y) for every y ∈ U − (U ∩ V ). By the previous lemma, this will imply that s = 0 and thus r = 0, so that i∗ is indeed a monomorphism. Since i∗(r) = 0, there exists t ∈ Hn(V, U ∩ V ) such that ∂(t) = r. Let s′ and t′ be the images of s and t in Hn(U ∪ V, U ∩ V ). Then ∂(s′ − t′) = 0, hence there exists w ∈ Hn(U ∪ V ) that maps to s′ − t′. Since
w maps to zero in Hn(M, M − y), so does s′ − t′. Since the map (V, U ∩ V ) −→ (M, M − y) factors through (M − y, M − y), t and thus also t′ maps to zero in Hn(M, M − y). Therefore s′ maps to zero in Hn(M, M − y) and thus s maps to zero in Hn(U, U − y), as claimed. 5. The proof of the Poincar´e duality theorem Let M be an R-oriented n-manifold, not necessarily compact. Unless otherwise speciﬁed, we take homology and cohomology with coeﬃcients in a given R-module π in this section. Remember that homology is a covariant functor with compact supports. Cohomology is a contravariant functor, and it does not have compact supports. We would like to prove the Poincar´e 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. 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 c (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) induced by the inclusions (M, M − L) ⊂ (M, M − K) for K ⊂ L. This is the cohomology of M with compact supports. Intuitively, thinking in terms of singular cohomology, its elements are represented by cocycles that vanish oﬀ some compact subspace. THE PROOF OF THE POINCAR ´E DUALITY THEOREM 161 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. For such f, we obtain an induced homomorphism f ∗ : H ∗ c (M ) in an evident way. However, we shall make no use of this contravariant functoriality. c (N ) −→ H ∗ What we shall use is a kind of covariant functoriality that will allow us to compare long exact sequences in homology and cohomology. Explicitly, for an open subspace U of M, we obtain a homomorphism H q c (M ) by passage to colimits from the excision isomorphisms c (U ) −→ H q H q(U, U − K) −→ H q(M, M − K) for compact subspaces K of U. For each compact subspace K of M, the R-orientation of M determines a fundamental class zK ∈ Hn(M, M − K; R). Taking the relative cap product with zK, we obtain a duality homomorphism DK : H p(M, M − K) −→ Hn−p(M ). If K ⊂ L, the following diagram commutes: H p(M, M − K) H p(M, M − L) (PPPPPPPPPPPP DK wnnnnnnnnnnnn DL Hn−p(M ). We may therefore pass to colimits to obtain a duality homomorphism D : H p c (M ) −→ Hn−p(M ). If U is open in M and is given the induced R-orientation, then the following naturality diagram commutes: H p c (U ) D / Hn−p(U ) H p c (M ) D / Hn−p(M ). If M itself is compact, then M is coﬁnal among the compact subspaces of M. Therefore H p c (M ) = H p(M ), and the present duality map D coincides with that of the Poincar´e duality theorem as originally stated. We shall prove a generalization to not necessarily compact manifolds. Theorem (Poincar´e duality). Let M be an R-oriented n-manifold. Then
D : H p c (M ) −→ Hn−p(M ) is an isomorphism. Proof. We shall prove that D : H p c (U ) −→ Hn−p(U ) is an isomorphism for every open subspace U of M. The proof proceeds in ﬁve steps. Step 1. The result holds for any coordinate chart U. We may take U = M = Rn. The compact cubes K are coﬁnal among the compact subspaces of Rn. For such K and for x ∈ K, H p(Rn, Rn − K) ∼= H p(Rn, Rn − x) ∼= ˜H p−1(Sn−1) ∼= ˜H p(Sn). c (Rn) are clearly isomorphisms. By the The maps of the colimit system deﬁning H p deﬁnition of the cap product, we see that D : H n(Rn, Rn − x) −→ H0(Rn) is an / / ( w / / 162 THE POINCAR ´E DUALITY THEOREM isomorphism. Therefore DK is an isomorphism for every compact cube K and so D : H n c (Rn) −→ H0(Rn) is an isomorphism. Step 2. If the result holds for open subspaces U and V and their intersection, then it holds for their union. Let W = U ∩ V and Z = U ∪ V. The compact subspaces of Z that are unions of a compact subspace K of U and a compact subspace L of V are coﬁnal among all of the compact subspaces of Z. For such K and L, we have the following commutative diagram with exact rows. We let J = K ∩ L and N = K ∪ L, and we write UK = (U, U − K), and so on, to abbreviate notation. / H p(ZJ ) H p(ZK ) ⊕ H p(ZL) H p(ZN ) / H p+1(ZJ ) ∼= ∼= ∼= / H p(WJ ) H p(UK ) ⊕ H p(VL) H p(ZN ) H p+1(WJ
) D D⊕D D D / Hn−p(W ) / Hn−p(U ) ⊕ Hn−p(V ) / Hn−p(Z) / Hn−p−1(W ) The top row is the relative Mayer-Vietoris sequence of the triad (Z; Z − K, Z − L). The middle row results from the top row by excision isomorphisms. The bottom row is the absolute Mayer-Vietoris sequence of the triad (Z; U, V ). The left two squares commute by naturality. The right square commutes by a diagram chase from the deﬁnition of the cap product. The entire diagram is natural with respect to pairs (K, L). We obtain a commutative diagram with exact rows on passage to colimits, and the conclusion follows by the ﬁve lemma. Step 3. If the result holds for each Ui in a totally ordered set of open subspaces {Ui}, then it holds for the union U of the Ui. Any compact subspace K of U is contained in a ﬁnite union of the Ui and therefore in one of the Ui. Since homology is compactly supported, it follows that colim Hn−p(Ui) ∼= Hn−p(U ). On the cohomology side, we have colimi H p c (Ui) = colimi colim{K\|K ⊂ Ui} H p(Ui, Ui − K) ∼= colim{K ⊂ U} colim{i\|K ⊂ Ui} H p(Ui, Ui − K) ∼= colim{K ⊂ U} H p(U, U − K) = H p c (U ). Here the ﬁrst isomorphism is an (algebraic) interchange of colimits isomorphism: both composite colimits are isomorphic to colim H p c (Ui, Ui − K), where the colimit runs over the pairs (K, i) such that K ⊂ Ui. The second isomorphism holds since colim{i\|K ⊂ Ui} H p(Ui, Ui − K) ∼= H p(U, U − K) because the
colimit is taken over a system of inverses of excision isomorphisms. The conclusion follows since a colimit of isomorphisms is an isomorphism. Step 4. The result holds if U is an open subset of a coordinate neighborhood. We may take M = Rn. If U is a convex subset of Rn, then U is homeomorphic to Rn and Step 1 applies. Since the intersection of two convex sets is convex, it follows by induction from Step 2 that the conclusion holds for any ﬁnite union of convex open subsets of Rn. Any open subset U of Rn is the union of countably many convex open subsets. By ordering them and letting Ui be the union of the ﬁrst i, we see that the conclusion for U follows from Step 3. THE ORIENTATION COVER 163 Step 5. The result holds for any open subset U of M. We may as well take M = U. By Step 3, we may apply Zorn’s lemma to conclude that there is a maximal open subset V of M for which the conclusion holds. If V is not all of M, say x 6∈ V, we may choose a coordinate chart U such that x ∈ U. By Steps 2 and 4, the result holds for U ∪ V, contradicting the maximality of V. This completes the proof of the Poincar´e duality theorem. 6. The orientation cover There is an orientation cover of a manifold that helps illuminate the notion of orientability. For the moment, we relax the requirement that the total space of a cover be connected. Here we take homology with integer coeﬃcients. Proposition. Let M be a connected n-manifold. Then there is a 2-fold cover p : ˜M −→ M such that ˜M is connected if and only if M is not orientable. Proof. Deﬁne ˜M to be the set of pairs (x, α), where x ∈ M and where α ∈ Hn(M, M − x) ∼= Z is a generator. Deﬁne p(x, α) = x. If U ⊂ M is open and β ∈ Hn(M, M − U ) is a fundamental class of M at U, deﬁne hU, βi = {(x
, α)\|x ∈ U and β maps to α}. The sets hU, βi form a base for a topology on ˜M. In fact, if (x, α) ∈ hU, βi ∩ hV, γi, we can choose a coordinate neighborhood W ⊂ U ∩ V such that x ∈ W. There is a unique class α′ ∈ Hn(M, M − W ) that maps to α, and both β and γ map to α′. Therefore hW, α′i ⊂ hU, βi ∩ hV, γi. Clearly p maps hU, βi homeomorphically onto U and p−1(U ) = hU, βi ∪ hU, −βi. Therefore ˜M is an n-manifold and p is a 2-fold cover. Moreover, ˜M is oriented. Indeed, if U is a coordinate chart and (x, α) ∈ hU, βi, then the following maps all induce isomorphisms on passage to homology: ( ˜M, ˜M − hU, βi) (M, M − U ) ( ˜M, ˜M − (x, α)) (M, M − x) (hU, βi, hU, βi − (x, α)) p ∼= / (U, U − x). Via the diagram, β ∈ Hn(M, M − U ) speciﬁes an element ˜β ∈ Hn( ˜M, ˜M − hU, βi), and ˜β is independent of the choice of (x, α). These classes are easily seen to specify an orientation of ˜M. Essentially by deﬁnition, an orientation of M is a cross section s : M −→ ˜M : if s(U ) = hU, βi, then these β specify an orientation. Given one section s, changing the signs of the β gives a second section −s such that ˜M = im(s) ∐ im(−s), showing that ˜M is not connected if M is oriented. The theory of covering spaces gives the following consequence. O O / O O 164 THE POINCAR ´E DUALITY THEOREM Corollary. If M is
simply connected, or if π1(M ) contains no subgroup of If M is orientable, then M admits exactly two index 2, then M is orientable. orientations. Proof. If M is not orientable, then p∗(π1( ˜M )) is a subgroup of π1(M ) of index 2. This implies the ﬁrst statement, and the second statement is clear. 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 Z2oriented. 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. PROBLEMS (1) Prove: there is no homotopy equivalence f : CP 2n −→ CP 2n that reverses orientation (induces multiplication by −1 on H4n(CP 2n)). In the problems below, M is assumed to be a compact connected n-manifold (without boundary), where n ≥ 2. 2. Prove that if M is a Lie group, then M is orientable. 3. Prove that if M is orientable, then Hn−1(M ; Z) is a free Abelian group. 4. Prove that if M is not orientable, then the torsion subgroup of Hn−1(M ; Z) is cyclic of order 2 and Hn(M ; Zq) is zero if q is odd and is cyclic of order 2 if q is even. (Hint: use universal coeﬃcients and the transfer homomorphism of the orientation cover.) 5. Let M be oriented with fundamental class z. Let f : Sn −→ M be a map such that f∗(in) = qz, where in ∈ Hn(Sn; Z) is the fundamental class and q 6= 0. (a) Show that f∗ : H∗(Sn; Zp) −→ H
∗(M ; Zp) is an isomorphism if p is a prime that does not divide q. 6. (b) Show that multiplication by q annihilates Hi(M ; Z) if 1 ≤ i ≤ n − 1. (a) Let M be a compact n-manifold. Suppose that M is homotopy equivalent to ΣY for some connected based space Y. Deduce that M has the same integral homology groups as Sn. (Hint: use the vanishing of cup products on ˜H ∗(ΣY ) and Poincar´e duality, treating the cases M orientable and M non-orientable separately.) (b) Deduce that M is homotopy equivalent to Sn. Does it follow that Y is homotopy equivalent to Sn−1? 7. * Essay: The singular cohomology H ∗(M ; R) is isomorphic to the de Rham cohomology of M. Why is this plausible? Sketch proof? CHAPTER 21 The index of manifolds; manifolds with boundary The Poincar´e duality theorem imposes strong constraints on the Euler characteristic of a manifold. It also leads to new invariants, most notably the index. Moreover, there is a relative version of Poincar´e 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. In particular, the index of a compact oriented 4n-manifold M is zero if M is a boundary. We shall later outline the theory of cobordism, which leads to necessary and suﬃcient algebraic conditions for a manifold to be a boundary. 1. The Euler characteristic of compact manifolds The Euler characteristic χ(X) of a space with ﬁnitely generated homology is deﬁned by χ(X) = i(−1)i rank Hi(X; Z). The universal coeﬃcient theorem implies that P χ(X) = i(−1)i dim Hi(X; F ) for any ﬁeld of coeﬃcients F. Examination of the relevant short exact sequences shows that P χ(X) = i(−1)i rank Ci(X; Z) for any decomposition of
X as a ﬁnite CW complex. The veriﬁcations of these statements are immediate from earlier exercises. P Now consider a compact oriented n-manifold. Recall that we take it for granted that M can be decomposed as a ﬁnite CW complex, so that each Hi(M ; Z) is ﬁnitely generated. By the universal coeﬃcient theorem and Poincar´e duality, we have Hi(M ; F ) ∼= H i(M ; F ) ∼= Hn−i(M ; F ) for any ﬁeld F. 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. If n = 2m and M is oriented, then χ(M ) = m−1 i=0 (−1)i2 dim Hi(M ) + (−1)m dim Hm(M ) for any ﬁeld F of coeﬃcients. Let us take F = Q. Of course, we can replace homology by cohomology in the deﬁnition and formulas for χ(M ). The middle dimensional cohomology group H m(M ) plays a particularly important role. Recall that we have the cup product pairing P φ : H m(M ) ⊗ H m(M ) −→ Q 165 166 THE INDEX OF MANIFOLDS; MANIFOLDS WITH BOUNDARY speciﬁed by φ(α, β) = hα ∪ β, zi. This pairing is nonsingular. Since α ∪ β = (−1)mβ ∪ 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. Proof. It suﬃces to prove that dim H m(M ) is even, where n = 2m, and this is immediate from the following algebraic observation. 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. Proof. We proceed by induction on dim V, and we may assume that V 6= 0. Since φ(x, y) = −φ(y, x), φ(x, x) = 0 for all x ∈ V. Choose x1 6= 0. Certainly there exists y1 such that φ(x1, y1) = 1, and x1 and y1 are then linearly independent. Deﬁne W = {x\|φ(x, x1) = 0 and φ(x, y1) = 0} ⊂ V. That is, W is the kernel of the homomorphism ψ : V −→ F × F speciﬁed by ψ(x) = (φ(x, x1), φ(x, y1)). Since ψ(x1) = (0, 1) and ψ(y1) = (−1, 0), ψ is an epimorphism. Thus dim W = dim V −2. Since φ restricts to a nonsingular skew symmetric bilinear form on W, the conclusion follows from the induction hypothesis. 2. The index of compact oriented manifolds To study manifolds of dimension 4k, we consider an analogue for symmetric bilinear forms of the previous algebraic lemma. Since we will need to take square roots, we will work over R. 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 φ. Proof. We proceed by induction on dim V, and we may assume that V 6= 0. Clearly q(rx) = r2q(x). Since we can take square roots in R, we can choose x1 ∈ V such that q(x1) = ±1. Deﬁne ψ : V −→ R by ψ(x) = φ(x, x1) and let W = ker ψ. Since ψ(x1) = ±1, ψ is an epimorphism and dim W = dim V − 1. Since φ restricts to a nonsingular symmetric bilinear form on W, the existence of a basis as speciﬁed follows directly from the induction hypothesis. Invariance means that the integer r − s is independent of the choice of basis on which q takes values ±1, and we leave the veriﬁcation to the reader. 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. The Euler characteristic and index are related by the following congruence. Proposition. For any compact oriented n-manifold, χ(M ) ≡ I(M ) mod 2. 2. THE INDEX OF COMPACT ORIENTED MANIFOLDS 167 Proof. If n is odd, then χ(M ) = 0 and I(M ) = 0. If n ≡ 2 mod 4, then If n = 4k, then I(M ) = r − s, where r + s = χ(M ) is even and I(M ) = 0. dim H 2k(M ; R) ≡ χ(M ) mod 2. Observe that the index of M changes sign if the orientation of M is reversed.
We write −M for M with the reversed orientation, and then I(−M ) = −I(M ). We also have the following algebraic identities. Write H ∗(M ) = H ∗(M ; R). 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 ′. Proof. There is nothing to prove unless n = 4k, in which case H 2k(M ∐ M ′) = H 2k(M ) × H 2k(M ′). Clearly the cup product of an element of H ∗(M ) with an element of H ∗(M ′) is zero, and the cup product form on H 2k(M ∐ M ′) is given by φ((x, x′), (y, y′)) = φ(x, y) + φ(x′, y′) for x, y ∈ H 2k(M ) and x′, y′ ∈ H 2k(M ′). The conclusion follows since the signature of a sum of forms is the sum of the signatures. 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. Proof. We must ﬁrst make sense of the induced orientation on M × N. For CW pairs (X, A) and (Y, B), we have an identiﬁcation of CW complexes (X × Y )/(X × B ∪ A × Y ) ∼= (X/A) ∧ (Y /B) and therefore an isomorphism C∗(X × Y, X × B ∪ A × Y ) ∼= C∗(X, A) ⊗ C∗(Y, B). This implies a relative K¨unneth theorem for arbitrary pairs (X, A) and (Y, B). For subspaces K ⊂ M and L ⊂ N, (M × N, M × N − K × L)
= (M × N, M × (N − L) ∪ (M − K) × N ). In particular, for points x ∈ M and y ∈ Y, (M × N, M × N − (x, y)) = (M × N, M × (N − y) ∪ (M − x) × N ). Therefore fundamental classes zK of M at K and zL of N at L determine a fundamental class zK×L of M × N at K × L. In particular, the image under Hm(M ) ⊗ Hn(N ) −→ Hm+n(M × N ) of the tensor product of fundamental classes of M and N is a fundamental class of M × N. Turning to the claimed product formula, we see that there is nothing to prove unless m + n = 4k, in which case H 2k(M × N ) = H i(M ) ⊗ H j(N ). The cup product form is given by Xi+j=2k φ(x ⊗ y, x′ ⊗ y′) = (−1)(deg y)(deg x ′ )+mnhx ∪ x′, zM ihy ∪ y′, zN i 168 THE INDEX OF MANIFOLDS; MANIFOLDS WITH BOUNDARY for x, x′ ∈ H ∗(M ) and y, y′ ∈ H ∗(N ). If m and n are odd, then the signature of this form is zero. If m and n are even, then this form is the sum of the tensor product of the cup product forms on the middle dimensional cohomology groups of M and N and a form whose signature is zero. Here, if m and n are congruent to 2 mod 4, the signature is zero since the lemma of the previous section implies that the signature of the tensor product of two skew symmetric forms is zero. When m and n are congruent to 0 mod 4, the conclusion holds since the signature of the tensor product of two symmetric forms is the product of their signatures. We leave the detailed veriﬁcations of these algebraic statements as exercises for the reader. 3. Manifolds with boundary 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 Hn. 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. 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. It is a fundamental question in topology to determine which closed manifolds are boundaries. The question makes sense with varying kinds of extra structure. For example, we can ask whether or not a smooth (= diﬀerentiable) closed manifold is the boundary of a smooth manifold (with the induced smooth structure). Numerical invariants in algebraic topology give criteria. One such criterion is given by the following consequence of the Poincar´e duality theorem. Remember that χ(M ) = 0 if M is a closed manifold of odd dimension. Proposition. If M = ∂W, where W is a compact (2m + 1)-manifold, then χ(M ) = 2χ(W ). Proof. The product W × I is a (2m + 2)-manifold with ∂(W × I) = (W × {0}) ∪ (M × I) ∪ (W × {1}). Let U = ∂(W × I) − (W × {1}) and V = ∂(W × I) − (W × {0}). Then U and V are open subsets of ∂(W × I). Clearly U and V are both homotopy equivalent to W and U ∩ V is homotopy equivalent to M. We have the Mayer-Vietoris sequence Hi+1(U �
� V ) / Hi(U ∩ V ) Hi(U ) ⊕ Hi(V ) Hi(U ∪ V ) ∼= ∼= Hi+1(∂(W × I)) / Hi(M ) / Hi(W ) ⊕ Hi(W ) / Hi(∂(W × I)). Therefore 2χ(W ) = χ(M )+χ(∂(W ×I)). However, χ(∂(W ×I)) = 0 since ∂(W ×I) is a closed manifold of odd dimension. Corollary. If M = ∂W for a compact manifold W, then χ(M ) is even. POINCAR ´E DUALITY FOR MANIFOLDS WITH BOUNDARY 169 For example, since χ(RP 2m) = 1 and χ(CP n) = n+ 1, this criterion shows that RP 2m and CP 2m cannot be boundaries. Notice that we have proved that these are not boundaries of topological manifolds, let alone of smooth ones. 4. Poincar´e duality for manifolds with boundary 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. To prove this, we must ﬁrst obtain a relative form of the Poincar´e duality theorem applicable to manifolds with boundary. 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 ˚M = M − ∂M is R-orientable; similarly, an R-orientation of M is an R-orientation of its interior. To study these notions, we shall need the following result, which is intuitively clear but is somewhat technical to prove. 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. 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). It follows that the inclusion ˚M −→ M is a homotopy equivalence and the inclusion ∂M −→ M is a coﬁbration. We take homology with coeﬃcients in R in the next two results. Proposition. An R-orientation of M determines an R-orientation of ∂M. Proof. Consider a coordinate chart U of a point x ∈ ∂M. If dim M = n, then U is homeomorphic to an open half-disk in Hn. Let V = ∂U = U ∩ ∂M and let y ∈ ˚U = U − V. We have the following chain of isomorphisms: Hn( ˚M, ˚M − ˚U ) ∼= Hn( ˚M, ˚M − y) ∼= Hn(M, M − y) ∼= Hn(M, M − ˚U) ∂−→ Hn−1(M − ˚U, M − U ) ∼= Hn−1(M − ˚U, (M − ˚U) − x) ∼= Hn−1(∂M, ∂M − x) ∼= Hn−1(∂M, ∂M − V ). The ﬁrst and last isomorphisms are restrictions of the sort that enter into the deﬁnition of an R-orientation, and the third isomorphism is similar. We see by use of a small boundary collar that the inclusion ( ˚M, ˚M − y) −→ (M, M − y) is a homotopy equivalence, and that gives the second isomorphism. The connecting homomorphism is that of the triple (M, M − ˚U, M −U ) and is an isomorphism since H∗(M, M − U ) ∼= H∗(M, M ) = 0. The isomorphism that follows comes from the observation that the inclusion (M − ˚U ) − x −→ M − U is a homotopy equivalence, and the next to last isomorphism is given by excision of ˚M − ˚U. The conclusion
is an easy consequence of these isomorphisms. 170 THE INDEX OF MANIFOLDS; MANIFOLDS WITH BOUNDARY 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. Proof. Since ˚M is a non-compact manifold without boundary and ˚M −→ M is a homotopy equivalence, Hn(M ) ∼= Hn( ˚M ) = 0 by the vanishing theorem. Therefore ∂ : Hn(M, ∂M ) −→ Hn−1(∂M ) is a monomorphism. Let V be a boundary collar and let N = M − V. Then N is a closed subspace and a deformation retract of the R-oriented open manifold ˚M, and we have Hn( ˚M, ˚M − N ) ∼= Hn(M, M − ˚M ) = Hn(M, ∂M ). Since M is compact, N is a compact subspace of ˚M. Therefore the R-orientation of ˚M determines a fundamental class in Hn( ˚M, ˚M − N ). Let z be its image in Hn(M, ∂M ). Then z restricts to a generator of Hn(M, M − y) ∼= Hn( ˚M, ˚M − y) for every y ∈ ˚M. Via naturality diagrams and the chain of isomorphisms in the previous proof, we see that ∂z restricts to a generator of Hn−1(∂M, ∂M − x) for all x ∈ ∂M and is the fundamental class determined by the R-orientation of ∂M. Theorem (Relative Poincar´e 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 D : H p(M, ∂M ) −→ Hn−p(M ) and D : H p(M ) −→ Hn−p(M, ∂M ). Proof. The following diagram commutes by inspection of deﬁnitions: H p−1(∂M ) H p(M, ∂M ) H p(M ) H p(∂M ) D D D D Hn−p(∂M ) / Hn−p(M ) / Hn−p(M, ∂M ) / Hn−p−1(∂M ). Here D for ∂M is obtained by capping with ∂z and is an isomorphism. By the ﬁve lemma, it suﬃces to prove that D : H p(M ) −→ Hn−p(M, ∂M ) is an isomorphism. To this end, let N = M ∪∂M M be the “double” of M and let M1 and M2 be the two copies of M in N. Clearly N is a compact manifold without boundary, and it is easy to see that N inherits an R-orientation from the orientation on M1 and the negative of the orientation on M2. Of course, ∂M = M1 ∩ M2. If U is the union of M1 and a boundary collar in M2 and V is the union of M2 and a boundary collar in M1, then we have a Mayer-Vietoris sequence for the triad (N ; U, V ). Using the evident equivalences of U with M1, V with M2, and U ∩ V with ∂M, this gives the exact sequence in the top row of the following commutative diagram. The bottom row is the exact sequence of the pair (N, ∂M ), and the isomorphism results from the homeomorphism N/∂M ∼= (M1/∂M ) ∨ (M2/∂M ); we abbreviate N1 = (M1, ∂. THE INDEX OF MANIFOLDS THAT ARE BOUNDARIES 171 and N2 = (M2, ∂M ): H p(N ) D H
p(M1) ⊕ H p(M2) ψ H p(∂M ) ∆ / H p+1(N ) D⊕D D D Hn−p(N ) / Hn−p(N1) ⊕ Hn−p(N2) Hn−p−1(∂M ) / Hn−p−1(N ) ∼= Hn−p(N ) / Hn−p(N, ∂M ) / Hn−p−1(∂M ) / Hn−p−1(N ). 1(x) − i∗ The top left square commutes by naturality. In the top middle square, we have ψ(x, y) = i∗ 2(y), where i1 : ∂M −→ M1 and i2 : ∂M −→ M2 are the inclusions. Since D for M2 is the negative of D for M1 under the identiﬁcations with M, the commutativity of this square follows from the relation D ◦ i∗ = ∂ ◦ D : H p(M ) −→ Hn−p−1(∂M ), i : ∂M −→ M, which holds by inspection of deﬁnitions. For the top right square, ∆ is the the top composite in the diagram H p(∂M ) δ H p+1(M1, ∂M ) ∼= H p+1(N, M2) / H p+1(N ) D D D Hn−p−1(∂M ) i1 ∗ / Hn−p−1(M1) / Hn−p−1(N ). The right square commutes by naturality, and D ◦ δ = i1∗ ◦ D by inspection of deﬁnitions. By the ﬁve lemma, since the duality maps D for N and ∂M are isomorphisms, both maps D between direct summands must be isomorphisms. The conclusion follows. 5. The index of manifolds that are boundaries We shall prove the following theorem. Theorem. If M is the boundary of a compact oriented (4k + 1)-manifold, then I(M ) = 0. We �
�rst 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. Proof. Let r and s be as in the deﬁnition of the signature. Then r + s = 2n and we must show that r = s. We prove that r ≥ n. Applied to the form −φ, this will also give that s ≥ n, implying the conclusion. We proceed by induction on n. Let {x1,..., xn, z1,..., zn} be a basis for V, where {x1,..., xn} is a basis for W. Deﬁne θ : V −→ Rn and ψ : V −→ Rn by θ(x) = (φ(x, x1),..., φ(x, xn)) and ψ(x) = (φ(x, z1),..., φ(x, zn)). Since φ is nonsingular, ker θ ∩ ker ψ = 0. Since ker θ and ker ψ each have dimension at least n, neither can have dimension more than n and θ and ψ must both be epimorphisms. Choose y1 such that θ(y1) = (1, 0,..., 0). Let q(x) = φ(x, x) and / / / / / / / / / / / / / / / / / 172 THE INDEX OF MANIFOLDS; MANIFOLDS WITH BOUNDARY note that q(x) = 0 if x ∈ W. Since q(x1) = 0 and φ(x1, y1) = 1, q(ax1 + y1) = 2a + q(y1) for a ∈ R. Taking a = (1 − q(y1))/2, we ﬁnd
q(ax1 + y1) = 1. If n = 1, this gives r ≥ 1 and completes the proof. If n > 1, deﬁne ω : V −→ R2 by ω(x) = (φ(x, x1), φ(x, y1)). Since ω(x1) = (0, 1) and ω(y1) = (1, q(y1)), ω is an epimorphism. Let V ′ = ker ω and let W ′ ⊂ V ′ be the span of {x2,..., xn}. The restriction of φ to V ′ satisﬁes the hypothesis of the lemma, and the induction hypothesis together with the construction just given imply that r ≥ n. Take homology and cohomology with coeﬃcients in R. Lemma. Let M = ∂W, where W is a compact oriented (4k + 1)-manifold, and let i : M −→ W be the inclusion. Let φ : H 2k(M ) ⊗ H 2k(M ) −→ R be the cup product form. Then the image of i∗ : H 2k(W ) −→ H 2k(M ) is a subspace of half the dimension of H 2k(M ) on which φ is identically zero. Proof. Let z ∈ H4k+1(W, M ) be the fundamental class. For α, β ∈ H 2k(W ), φ(i∗(α), i∗(β)) = hi∗(α ∪ β), ∂zi = hα ∪ β, i∗∂zi = 0 since i∗∂ = 0 by the long exact sequence of the pair (W, M ). Thus φ is identically zero on im i∗. The commutative diagram with exact rows H 2k(W ) D H2k+1(W, M ) ∗ i ∂ H 2k(M ) δ / H 2k+1(W, M ) D D / H2k(M ) / H2k(W ) i∗ implies that H 2k(M ) ∼= im i∗ ⊕ im δ ∼= im i∗
⊕ im i∗. Since i∗ and i∗ are dual homomorphisms, im i∗ and im i∗ are dual vector spaces and thus have the same dimension. PROBLEMS Let M be a compact connected n-manifold with boundary ∂M, where n ≥ 2. (1) Prove: ∂M is not a retract of M. (2) Prove: if M is contractible, then ∂M has the homology of a sphere. (3) Assume that M is orientable. Let n = 2m + 1 and let K be the kernel of the homomorphism Hm(∂M ) −→ Hm(M ) induced by the inclusion, where homology is taken with coeﬃcients in a ﬁeld. Prove: dim Hm(∂M ) = 2 dim K. Let n = 3 in the rest of the problems. 4. Prove: if M is orientable, ∂M is empty, and H1(M ; Z) = 0, then M has the same homology groups as a 3-sphere. 5. Prove: if M is nonorientable and ∂M is empty, then H1(M ; Z) is inﬁnite. (Hint for the last three problems: use the standard classiﬁcation of closed 2- manifolds and think about ﬁrst homology groups.) 6. Prove: if M is orientable and H1(M ; Z) = 0, then ∂M is a disjoint union of 2-spheres. 7. Prove: if M is orientable, ∂M 6= φ, and ∂M contains no 2-spheres, then H1(M ; Z) is inﬁnite. / / / / / 5. THE INDEX OF MANIFOLDS THAT ARE BOUNDARIES 173 8. Prove: if M is nonorientable and ∂M contains no 2-spheres and no pro- jective planes, then H1(M ; Z) is inﬁnite. CHAPTER 22 Homology, cohomology, and K(π, n)s We have given an axiomatic deﬁnition of ordinary homology and cohomology, and we have shown how
to realize the axioms by means of either cellular or singular chain and cochain complexes. 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. 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. Finally, we present the important notion of cohomology operations and relate them to the cohomology of Eilenberg-Mac Lane spaces. 1. K(π, n)s and homology Recall that a reduced homology theory on based CW complexes is a sequence of functors ˜Eq from the homotopy category of based CW complexes to the category of Abelian groups. Each ˜Eq must satisfy the exactness and additivity axioms, and there must be a natural suspension isomorphism. Up to isomorphism, ordinary reduced homology with coeﬃcients in π is characterized as the unique such theory that satisﬁes the dimension axiom: ˜E0(S0) = π and ˜Eq(S0) = 0 if q 6= 0. We proceed to construct such a theory homotopically. For based spaces X and Y, we let [X, Y ] denote the set of based homotopy classes of based maps X −→ Y. Recall that we require Eilenberg-Mac Lane spaces K(π, n) to have the homotopy types of CW complexes and that, up to homotopy equivalence, there is a unique such space for each n and π. 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 ˜σ : K(π, n) −→ ΩK(π, n + 1).
This map is the adjoint of a map σ : ΣK(π, n) −→ K(π, n + 1). We may take the smash product of the map σ with a based CW complex X and use the suspension homomorphism on homotopy groups to obtain maps Σ−→ πq+n+1(Σ(X ∧ K(π, n))) πq+n(X ∧ K(π, n)) = πq+n+1(X ∧ ΣK(π, n)) (id ∧σ)∗ −−−−−→ πq+n+1(X ∧ K(π, n + 1)). 175 176 HOMOLOGY, COHOMOLOGY, AND K(π, n)S Theorem. For CW complexes X, Abelian groups π and integers n ≥ 0, there are natural isomorphisms ˜Hq(X; π) ∼= colimn πq+n(X ∧ K(π, n)). It suﬃces to verify the axioms, and the dimension axiom is clear. If X = S0, then X ∧ K(π, n) = K(π, n). Here the homotopy groups in the colimit system are zero if q 6= 0, and, if q = 0, the colimit runs over a sequence of isomorphisms between copies of π. The veriﬁcations of the rest of the axioms are exercises in the use of the homotopy excision and Freudenthal suspension theorems, and it is worthwhile to carry out these exercises in greater generality. Definition. A prespectrum is a sequence of based spaces Tn, n ≥ 0, and based maps σ : ΣTn −→ Tn+1. The example at hand is the Eilenberg-Mac Lane prespectrum {K(π, n)}. Another example is the “suspension prespectrum” {ΣnX} of a based space X; the required maps Σ(ΣnX) −→ Σn+1X are the evident identiﬁcations. When X = S0, this is called the sphere prespectrum. 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 ˜Eq(X) = colimn πq+n(X ∧ Tn), where the colimit is taken over the maps πq+n(X ∧ Tn) Σ−→ πq+n+1(Σ(X ∧ Tn)) ∼= πq+n+1(X ∧ ΣTn) id ∧σ−−−→ πq+n+1(X ∧ Tn+1). Then the functors ˜Eq deﬁne a reduced homology theory on based CW complexes. Proof. Certainly the ˜E are well deﬁned functors from the homotopy category of based CW complexes to the category of Abelian groups. We must verify the exactness, additivity, and suspension axioms. Without loss of generality, we may take the Tn to be CW complexes with one vertex and no other cells of dimension less than n. Then X ∧ Tn is a quotient complex of X × Tn, and it too has one vertex and no other cells of dimension less than n. In particular, it is (n − 1)-connected. If A is a subcomplex of X, then the homotopy excision theorem implies that the quotient map (X ∧ Tn, A ∧ Tn) −→ ((X ∧ Tn)/(A ∧ Tn), ∗) ∼= ((X/A) ∧ Tn, ∗) is a (2n − 1)-equivalence. We may restrict to terms with n > q − 1 in calculating ˜Eq(X), and, for such q, the long exact sequence of homotopy groups of the pair (X ∧ Tn, A ∧ Tn) gives that the sequence πq+n(A ∧ Tn) −→ πq+n(X ∧ Tn) −→ πq+n((X/A) ∧ Tn) is exact. Since passage to colimits preserves exact sequences, this proves the exactness axiom. We need some preliminaries to prove the additivity axiom. Definition. Deﬁne the weak product Yi of
a set of based spaces Yi to i Yi consisting of those points all but ﬁnitely many of whose w i Q be the subspace of coordinates are basepoints. Q Q 2. K(π, n)S AND COHOMOLOGY 177 Lemma. For a set of based spaces {Yi}, the canonical map iπq(Yi) −→ πq( w i Yi) is an isomorphism. P Q Proof. The homotopy groups of w i Yi are the colimits of the homotopy groups of the ﬁnite subproducts of the Yi, and the conclusion follows. Lemma. If {Yi} is a set of based CW complexes, then w i Yi is a CW complex whose cells are the cells of the ﬁnite subproducts of the Yi. If each Yi has a single w i Yi coincides vertex and no q-cells for q < n, then the (2n − 1)-skeleton of w with the (2n − 1)-skeleton of i Yi is a (2n − 1)-equivalence. i Yi. Therefore the inclusion i Yi −→ Q Q W W Q Returning to the proof of the additivity axiom, suppose given based CW com- plexes Xi and consider the natural map ( iXi) ∧ Tn ∼= i(Xi ∧ Tn) −→ w i (Xi ∧ Tn). W W Q It induces isomorphisms on πq+n for q < n − 1, and the additivity axiom follows. Finally, we must prove the suspension axiom. We have the suspension map πq+n(X ∧ Tn) Σ−→ πq+n+1(Σ(X ∧ Tn)) ∼= πq+n+1((ΣX) ∧ Tn). By the Freudenthal suspension theorem, it is an isomorphism for q < n−1. Keeping track of suspension coordinates and their permutation, we easily check that these maps commute with the maps deﬁning the colimit systems for X and for ΣX. Therefore they induce a natural suspension isomorphism ˜Eq(X) ∼= ˜Eq+1(ΣX). This completes the proof of the theorem. Example. Applying the theorem to the sphere prespectrum, we
ﬁnd that the q (X) give the values of a reduced homology theory; it is stable homotopy groups πs called “stable homotopy theory.” 2. K(π, n)s and cohomology The homotopical description of ordinary cohomology theories is both simpler and more important to the applications than the homotopical description of ordinary homology theories. Theorem. For CW complexes X, Abelian groups π, and integers n ≥ 0, there are natural isomorphisms ˜H n(X; π) ∼= [X, K(π, n)]. The dimension axiom is built into the deﬁnition of K(π, n), as we see by taking X = S0. As in homology, it is worthwhile to carry out the veriﬁcation of the remaining axioms in greater generality. We ﬁrst state some properties of the functor [−, Z] on based CW complexes that is “represented” by a based space Z. Lemma. For any based space Z, the functor [X, Z] from based CW complexes X to pointed sets satisﬁes the following properties. • HOMOTOPY If f ≃ g : X −→ Y, then f ∗ = g∗ : [Y, Z] −→ [X, Z]. 178 HOMOLOGY, COHOMOLOGY, AND K(π, n)S • EXACTNESS If A is a subcomplex of X, then the sequence [X/A, Z] −→ [X, Z] −→ [A, Z] is exact. • ADDITIVITY If X is the wedge of a set of based CW complexes Xi, then the inclusions Xi −→ X induce an isomorphism [X, Z] −→ [Xi, Z]. If Z has a multiplication φ : Z × Z −→ Z such that the basepoint ∗ of Z is Q 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]. The trivial map X −→ Z acts as zero. If Z is homotopy associative, in the sense that there is
a homotopy between the maps given on elements by (xy)z and by x(yz), then the addition is associative. If, further, Z is homotopy commutative, in the sense that there is a homotopy between the maps given on elements by xy and by yx, then this addition is commutative. 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. Lemma. If Z is a grouplike homotopy associative and commutative H-space, then the functor [X, Z] takes values in Abelian groups. 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 H- space and Ω2Y is a grouplike homotopy associative and commutative H-space. Recall too that we have [ΣX, Y ] ∼= [X, ΩY ] for any based spaces X and Y. Definition. An Ω-prespectrum is a sequence of based spaces Tn and weak homotopy equivalences ˜σ : Tn −→ ΩTn+1. It is usual, but unnecessary, to require the Tn to have the homotopy types of CW complexes, in which case the ˜σ are homotopy equivalences. Specialization of the observations above leads to the following fundamental fact. Theorem. Let {Tn} be an Ω-prespectrum. Deﬁne ˜Eq(X) = [X, Tq] [X, �
�−qT0] Then the functors ˜Eq deﬁne a reduced cohomology theory on based CW complexes. Proof. We have already veriﬁed the exactness and additivity axioms, and the if q ≥ 0 if q < 0. weak equivalences ˜σ induce the suspension isomorphisms: ˜Eq(X) = [X, Tq] −→ [X, ΩTq+1] ∼= [ΣX, Tq+1] = ˜Eq+1(ΣX). 3. CUP AND CAP PRODUCTS 179 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. 3. Cup and cap products Changing notations, let A and B be Abelian groups and X and Y be based spaces. We have an external product ˜H p(X; A) ⊗ ˜H q(Y ; B) −→ ˜H p+q(X ∧ Y ; A ⊗ B). Indeed, if X and Y are based CW complexes, then we have an isomorphism of cellular chain complexes ˜C∗(X) ⊗ ˜C∗(Y ) ∼= ˜C∗(X ∧ Y ). On passage to cochains with coeﬃcients in A, B, and A ⊗ B, this induces a homomorphism of cochain complexes ˜C∗(X; A) ⊗ ˜C∗(Y ; B) −→ ˜C∗(X ∧ Y ; A ⊗ B). In turn, this induces the cited product on passage to cohomology. With X = Y, we can apply the diagonal ∆ : X −→ X ∧ X and any homomorphism A ⊗ B −→ C to obtain a cup product ˜H p(X; A) ⊗ ˜H q(X; B) −→ ˜H p+q(X; C). When X = X ′ duced cohomology of X ′. + for an unbased space X ′, this gives the cup product on the unre- We can obtain these external products and therefore their induced cup products homotopically
. The smash product of maps gives a pairing [X, K(A, p)] ⊗ [Y, K(B, q)] −→ [X ∧ Y, K(A, p) ∧ K(B, q)]. Therefore, to obtain an external product, we need only obtain a suitable map φp,q : K(A, p) ∧ K(B, q) −→ K(A ⊗ B, p + q). Such a map may be interpreted as an element of ˜H p+q(K(A, p) ∧ K(B, q); A ⊗ B). Since the space K(A, p) ∧ K(B, q) is (p + q − 1)-connected, the universal coeﬃcient, K¨unneth, and Hurewicz theorems give isomorphisms ˜H p+q(K(A, p) ∧ K(B, q); A ⊗ B) ∼= Hom( ˜Hp+q(K(A, p) ∧ K(B, q)), A ⊗ B) ∼= Hom( ˜Hp(K(A, p)) ⊗ ˜Hq(K(B, q)), A ⊗ B) ∼= Hom(πp(K(A, p)) ⊗ πq(K(B, q)), A ⊗ B) = Hom(A ⊗ B, A ⊗ B). Therefore the identity homomorphism on the group A ⊗ B gives rise to the required map φp,q. Arguing similarly, it is easy to check that the system of maps {φp,q} is associative, commutative, and unital in the sense that the following diagrams are homotopy commutative. Indeed, translating back along isomorphisms of the form just displayed, each of the diagrams translates to an elementary algebraic identity. K(A, p) ∧ K(B, q) ∧ K(C, r) φ∧id / K(A ⊗ B, p + q) ∧ K(C, r) id ∧φ φ K(A, p) ∧ K(B ⊗ C, q + r) / K(A �
� B ⊗ C, p + q + r), φ / / 180 HOMOLOGY, COHOMOLOGY, AND K(π, n)S and K(A, p) ∧ K(B, q) t K(B, q) ∧ K(A, p) φ φ / K(A ⊗ B, p + q) K(t,p+q) / K(B ⊗ A, p + q), S0 ∧ K(A, p) K(A, p) i∧id K(Z, 0) ∧ K(A, p) φ / K(Z ⊗ A, p), where i : S0 −→ Z = K(Z, 0) takes 0 to 0 and 1 to 1. The associativity, graded commutativity, and unital properties of the cup product follow. The cup products on cohomology deﬁned in terms of cellular cochains and in terms of the homotopical representation of cohomology agree. To see this, observe that the identity homomorphism of A speciﬁes a fundamental class via the isomorphisms ιp ∈ ˜H p(K(A, p), A) Hom(A, A) ∼= Hom(πp(K(A, p)), A) ∼= Hom( ˜Hp(K(A, p)), A) ∼= ˜H p(K(A, p); A). A moment’s thought shows that the two cup products will agree on arbitrary pairs of cohomology classes if they agree when applied to ιp ⊗ ιq for all p and q. We may take our Eilenberg-Mac Lane spaces to be CW complexes and give their smash product the induced CW structure. Considering representative cycles for generators of our groups as images under the Hurewicz homomorphism of representative maps Sp −→ K(A, p), we ﬁnd that the required agreement follows from the canonical identiﬁcations Sp ∧ Sq ∼= Sp+q. We can also construct cap products homotopically. To do so, it is convenient to bring function spaces into play, using the obvious isomorphisms and evaluation maps [X, Y ] ∼= π0F (X, Y ) ε : F (X
, Y ) ∧ X −→ Y. We wish to construct the cap product ˜H p(X; A) ⊗ ˜Hn(X; B) −→ ˜Hn−p(X; A ⊗ B), and it is equivalent to construct π0(F (X, K(A, p)))⊗colimq πn+q(X ∧K(B, q)) −→ colimr πn−p+r(X ∧K(A⊗B, r)). Changing the variable of the second colimit by setting r = p + q and recalling the algebraic fact that tensor products commute with colimits, we can rewrite this as colimq(π0(F (X, K(A, p))) ⊗ πn+q(X ∧ K(B, q))) −→ colimq πn+q(X ∧ K(A ⊗ B, p + q)). Thus it suﬃces to deﬁne maps π0(F (X, K(A, p))) ⊗ πn+q(X ∧ K(B, q)) −→ πn+q(X ∧ K(A ⊗ B, p + q)). / / / 3. CUP AND CAP PRODUCTS 181 These are given by the following composites: π0(F (X, K(A, p))) ⊗ πn+q(X ∧ K(B, q)) ∧ πn+q(F (X, K(A, p)) ∧ X ∧ K(B, q)) (id ∧∆∧id)∗ πn+q(F (X, K(A, p)) ∧ X ∧ X ∧ K(B, q)) (ε∧id)∗ πn+q(K(A, p) ∧ X ∧ K(B, q)) (id ∧φp,q)∗(t∧id)∗ πn+q(X ∧ K(A ⊗ B, p + q)). Similarly, we can construct the evaluation pairing ˜H n(X; A) ⊗ ˜Hn(X
; B) −→ A ⊗ B homotopically. It is obtained by passage to colimits over q from the composites π0(F (X, K(A, n))) ⊗ πn+q(X ∧ K(B, q)) ∧ πn+q(F (X, K(A, n)) ∧ X ∧ K(B, q)) (ε∧id)∗ πn+q(K(A, n) ∧ K(B, q)) (φn,q)∗ πn+q(K(A ⊗ B, n + q)) = A ⊗ B. The following formula relating the cup and cap products to the evaluation pairing was central to our discussion of Poincar´e duality: hα ∪ β, xi = hβ, α ∩ xi ∈ R, where α ∈ ˜H p(X; R), β ∈ ˜H q(X; R), and x ∈ ˜Hp+q(X; R) for a commutative ring R. It is illuminating to rederive this from our homotopical descriptions of these products. In fact, a straightforward diagram chase shows that this formula is a direct consequence of the following elementary facts, where X, Y, and Z are based spaces. First, the following diagram commutes: 182 HOMOLOGY, COHOMOLOGY, AND K(π, n)S F (X, Y ) ∧ F (X, Z) ∧ X id ∧ id ∧∆ / / F (X, Y ) ∧ F (X, Z) ∧ X ∧ X (∧)∧id (∧)∧id ∧ id F (X ∧ X, Y ∧ Z) ∧ X id ∧∆ / F (X ∧ X, Y ∧ Z) ∧ X ∧ X F (∆,id)∧id F (X, Y ∧ Z) ∧ X ε ε / Y ∧ Z. Second, the right vertical composite in the diagram coincides with the common composite in the commutative diagram F (X, Y ) ∧ F (X, Z) ∧ X ∧ X t∧id / F (X,
Z) ∧ F (X, Y ) ∧ X ∧ X id ∧t∧id id ∧ε∧id F (X, Y ) ∧ X ∧ F (X, Z) ∧ X F (X, Z) ∧ Y ∧ X ε∧ε Y ∧ Z t Z ∧ Y ε∧id F (X, Z) ∧ X ∧ Y. id ∧t The observant reader will see a punch line here: everything in this section applies equally well to the homology and cohomology theories represented by Ωprespectra. A little more precisely, thinking of the case when A = B = C is a commutative ring in the discussion above, we see by use of the product on A that we have a well behaved system of product maps φp,q : K(A, p) ∧ K(A, q) −→ K(A, p + q). We have analogous cup and cap products and an evaluation pairing for the theories represented by any Ω-prespectrum {Tn} with such a system of product maps φp,q : Tp ∧ Tq −→ Tp+q. 4. Postnikov systems 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. 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. This decomposition answers a natural question: how close are the homotopy groups of a CW complex X to being a complete set of invariants for its homotopy n K(πn(X), n) has the same homotopy groups as X but is generally type? Since 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. Q 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, / / / o o o o 4. POSTNIKOV SYSTEMS 183 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 kn+2 : Xn −→ K(πn+1(X), n + 2), and αn induces an isomorphism πq(X) → πq(Xn) for q ≤ n. πq(Xn) = 0 for q > n. The system can be displayed diagrammatically as follows: It follows that... Xn+1 kn+3 / K(πn+2(X), n + 3) pn+1 X / Xn αn+1 ={{{{{{{{{ αn 00000000000000000 α1... X1 kn+2 K(πn+1(X), n + 2) k3 / K(π2(X), 3). Our requirement that Eilenberg-Mac Lane spaces have the homotopy types of CW complexes implies (by a result of Milnor) that each Xn has the homotopy type of 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” that specify the system are to be regarded as cohomology classes kn+2 kn+2 ∈ H n+2(Xn; πn+1(X)). These classes together with the homotopy groups πn(X) specify the weak homotopy type of X. We outline the proof of the following theorem. Theorem. A simple space X of the homotopy type of a CW complex has a Postnikov system. Proof. Assume inductively that αn : X → Xn has been constructed. A consequence
of the homotopy excision theorem shows that the coﬁber C(αn) is (n + 1)-connected and satisﬁes πn+2(C(αn)) = πn+1(X). More precisely, the canonical map η : F (αn) → ΩC(αn) induces an isomorphism on πq for q ≤ n + 1. We construct j : C(αn) → K(πn+1(X), n + 2) by inductively attaching cells to C(αn) to kill its higher homotopy groups. We take the composite of j and the inclusion Xn ⊂ C(αn) to be the k-invariant kn+2 : Xn −→ K(πn+1(X), n + 2). / = / / / / 184 HOMOLOGY, COHOMOLOGY, AND K(π, n)S By our deﬁnition of a Postnikov system, we must deﬁne Xn+1 to be the homotopy ﬁber of kn+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) = kn+2(x). The map pn+1 : Xn+1 → Xn is given by pn+1(ω, x) = x, and the map αn+1 : X → Xn+1 is given by αn+1(x) = (ω(x), αn(x)), where ω(x)(t) = j(x, 1 − t), (x, 1 − t) being a point on the cone CX ⊂ C(αn). Clearly pn+1 ◦ αn+1 = αn. It is evident that αn+1 induces an isomorphism on πq for q ≤ n, and a diagram chase shows that this also holds for q = n + 1. 5. Cohomology operations 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. The following simple observation actually applies to represented functors on arbitrary categories. We shall use it to describe cohomology operations, but it also applies to describe many other invariants in algebraic topology, such as the characteristic classes of vector bundles. Lemma (Yoneda). There is a canonical bijection between natural transforma- tions Φ : k −→ k′ and elements φ ∈ k′(Z). Proof. Given Φ, we deﬁne φ to be Φ(id), where id ∈ k(Z) = [Z, Z] is the identity map. Given φ, we deﬁne Φ : k(X) −→ k′(X) by the formula Φ(f ) = f ∗(φ). Here f is a map X −→ Z, and it induces f ∗ = k′(f ) : k′(Z) −→ k′(X). It is simple to check that these are inverse bijections. We are interested in the case when k′ is also represented, say k′(X) = [X, Z ′]. Corollary. There is a canonical bijection between natural transformations Φ : [−, Z] −→ [−, Z ′] and elements φ ∈ [Z, Z ′]. Definition. Suppose given cohomology theories ˜E∗ and ˜F ∗. A cohomology operation of type q and degree n is a natural transformation ˜Eq −→ ˜F q+n. A stable cohomology operation of degree n is a sequence {Φq} of cohomology operations of type q and degree n such that the following diagram commutes for each q and each based space X: ˜Eq(X) Φq ˜Eq+n(X) Σ Σ ˜Eq+1(ΣX) Φq+1 / / ˜Eq+1+n(ΣX). We generally abbreviate notation by setting Φq = Φ. In general, cohomology operations are only natural transformations of setvalued functors. However, stable operations are necessarily homomorphisms of cohomology groups, as the reader is encouraged to check. Theorem. Cohomology operations ˜H q(−; �
�) −→ ˜H q+n(−; ρ) are in canonical bijective correspondence with elements of ˜H q+n(K(π, q); ρ). Proof. Translate to the represented level, apply the previous corollary, and translate back. / / 5. COHOMOLOGY OPERATIONS 185 This seems very abstract, but it has very concrete consequences. To determine all cohomology operations, we need only compute the cohomology of all EilenbergMac Lane spaces. 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. We brieﬂy indicate a key example of how this works, without proofs. Theorem. For n ≥ 0, there are stable cohomology operations Sqn : H q(X; Z2) −→ H q+n(X; Z2), called the Steenrod operations. They satisfy the following properties. (i) Sq0 is the identity operation. (ii) Sqn(x) = x2 if n = deg x and Sqn(x) = 0 if n > deg x. (iii) The Cartan formula holds: Sqn(xy) = Sqi(x)Sqj (y). i+j=n X In fact, the Steenrod operations are uniquely characterized by the stated properties. There are also formulas, called the Adem relations, describing SqiSqj, as a linear combination of operations Sqi+j−kSqk, 2k ≤ i, when 0 < i < 2j; explicitly, SqiSqj = X0≤k≤[i/2] j − k − 1 i − 2k Sqi+j−kSqk. It turns out that the Steenrod operations generate all mod 2 cohomology operIn fact, the identity map of K(Z2, q) speciﬁes a fundamental class ιq ∈ ations. 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 SqI = S
qi1 · · · Sqij for a sequence of positive integers I = {i1,..., ij}, the generators are the SqI ιq for those sequences I such that ir ≥ 2ir+1 for 1 ≤ r < j and i1 < i2 + · · · + ij + q. PROBLEMS (1) For Abelian groups π and ρ, show that [K(π, n), K(ρ, n)] ∼= Hom(π, ρ). (Hint: use the natural isomorphism [X, K(ρ, n)] ∼= ˜H n(X; ρ) and universal coeﬃcients.) (a) Let f : π −→ ρ be a homomorphism of Abelian groups. Construct (2) cohomology operations f ∗ : H q(X; π) −→ H q(X; ρ) for all q. (b) Let 0 −→ π f −→ ρ g −→ σ −→ 0 be an exact sequence of Abelian groups. Construct cohomology operations β : H q(X; σ) −→ H q+1(X; π) for all q such that the following is a long exact sequence: ∗ ∗ · · · −→ H q(X; π) f −→ H q(X; ρ) g −→ H q(X; σ) β −→ H q+1(X; π) −→ · · ·. The β are called Bockstein operations. (3) Using the calculation of H ∗(K(Z, 2); Z2) stated in the text, prove that Sq1 : H q(X; Z2) −→ H q+1(X; Z2) coincides with the Bockstein operation associated to the short exact sequence 0 −→ Z2 −→ Z4 −→ Z2 −→ 0. 186 HOMOLOGY, COHOMOLOGY, AND K(π, n)S (4) Prove that each Φq of a stable cohomology operation {Φq} is a natural homomorphism. (5) Write H ∗(RP ∞; Z2) = Z2[α], deg α = 1. Compute Sqi(αj ) for all i
and j. CHAPTER 23 Characteristic classes of vector bundles Some of the most remarkable applications of algebraic topology result from the translation of problems in geometric topology into problems in homotopy theory. The essential intermediary in many of these translations is the theory of vector bundles. We here explain the classiﬁcation theorem for vector bundles and its relationship to the theory of characteristic classes. The reader is assumed to be familiar with the tangent and normal bundles of smooth manifolds and to be reasonably well acquainted with the deﬁnitions and elementary properties of vector bundles in general. 1. The classiﬁcation of vector bundles Let E be a (real) vector bundle over a base space B. Thus we are given a projection p : E −→ B such that, for each b ∈ B, the ﬁber p−1(b) is a copy of Rn for some n. In the case of non-connected base spaces, the ﬁbers over points in diﬀerent components may have diﬀerent dimension. We say that p is an n-plane bundle if all ﬁbers have dimension n. For each U in some open cover of B, there is a homeomorphism (a “coordinate chart”) φU : U × Rn −→ p−1(U ) over U that restricts to a linear isomorphism on each ﬁber. We shall require our open covers to be numerable, as can always be arranged when B is paracompact. For a second vector bundle q : D −→ A, a map (g, f ) : D −→ E of vector bundles is a pair of maps f : A −→ B and g : D −→ E such that p ◦ g = f ◦ q and g : q−1(a) −→ p−1(f (a)) is linear for all a ∈ A. This gives the category of vector bundles. 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. We are mainly interested in the subcategories of n-plane bundles and maps that are linear isomorphisms on ﬁbers. We say that two vector bundles over B are equivalent if they are isomorphic over B, so that there is an is
omorphism (g, id) between them. We let En(B) denote the set of equivalence classes of n-plane bundles over B. (That this really is a well deﬁned set will emerge shortly.) If f : A −→ B is a continuous map, then the pullback of f and a vector bundle p : E −→ B is a vector bundle f ∗E over A. Moreover, a bundle D over A is equivalent to f ∗E if and only if there is a map (g, f ) : D −→ E that is an isomorphism on ﬁbers. Thus we have a contravariant set-valued functor En(−) on spaces. 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. This is a crucial starting point for the translation of geometric problems to homotopical ones. 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 187 188 CHARACTERISTIC CLASSES OF VECTOR BUNDLES 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 homo- topic maps f0, f1 : A −→ B are equivalent. Sketch proof. Let h : A × I −→ B be a homotopy f0 ≃ f1. Then the restrictions of h∗E over A × {0} and A × {1} can be identiﬁed with f ∗ 0 E and f ∗ 1 E. Thus we change our point of view and consider a general n-plane bundle p : E −→ B × I. It suﬃces to show that the restrictions E0 and E1 of E over B × {0} and B × {1} are equivalent. Deﬁne r : B × I −→ B × I by r(b, t) = (b, 1). We claim that there is a map g : E −→ E such that (g, r) is a map
of vector bundles. It follows that E is equivalent to r∗E, and it is easy to see that r∗E is isomorphic to the bundle E1 × I. The restriction of g to E0 will be an equivalence to E1. To construct g, one ﬁrst proves, using the compactness of I, that there is a numerable open cover O of B such that the restriction of E to U × I is trivial for all U ∈ O. One then uses trivializations φU : U × I × Rn −→ p−1(U × I) together with functions λU : B −→ I such that λ−1 U (0, 1] = U to construct g by gradually pushing the bundle to the right along neighborhoods where it is trivial. 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)] for some space BO(n). It is far more useful to have an explicit concrete construction of the relevant “classifying space” BO(n). More precisely, we think of “BO(n)” as specifying a homotopy type of spaces, and we want an explicit representative of the homotopy type. Here [X, Y ] denotes unbased homotopy classes of maps. 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.” Let Vn(Rq) be the Stiefel variety of orthonormal n-frames in Rq. 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 (Sq−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 It too is a compact manifold. For example, V1(Rq) = Sq−1 space of Vn(Rq). and G1(Rq) = RP q−1. The standard inclusion of Rq in Rq+1 induces inclusions Vn(Rq) ⊂ Vn(Rq+1) and Gn(Rq) ⊂ Gn(Rq+1). We deﬁne Vn(R∞) and Gn(R∞) to be the unions of the Vn(Rq) and Gn(Rq), with the topology of the union. We deﬁne the classifying space BO(n) to be Gn(R∞). Let Eq n, so that γq n 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 Eq n n(x, v) = x. When n = 1, γq by γq 1 is called the “canonical line bundle” over RP q−1. We may let q go to inﬁnity. We let En = E∞ n : En −→ BO(n). This is our universal bundle, and it is not hard to verify that it is indeed an n-plane bundle. We must explain why it is universal. (Technically, it is n and let γn = γ∞ 2. CHARACTERISTIC CLASSES FOR VECTOR BUNDLES 189 usual to assume that base spaces are paracompact, but the restriction to numerable systems of coordinate charts in our deﬁnition of vector bundles allows the use of general base spaces.) 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. Sketch proof. To illustrate ideas, let M be a smooth compact n-manifold smoothly embedded in Rq and let τ (M ) be its tangent bundle. The tangent plane τx at a point x ∈ M ⊂ Rq is then embedded as an aﬃne plane through x in Rq. Translating to a plane through the origin by subtracting x from each vector, n)−1(f (x)). we obtain a point f (x) ∈ Gn(Rq) and an isomorphism gx : τx −→ (γq The gx glue together to give a map (g, f ) of bundles from E(τ (M )) to Eq n; it is called the Gauss map of the tangent bundle of M. Similarly, using the orthogonal complements of tangent planes, we obtain the Gauss map E(ν) −→ Eq q−n of the normal bundle ν of the embedding of M in Rq. For a general n-plane bundle p : E −→ B, we must construct a map (g, f ) : E −→ En of vector bundles that is an isomorphism on ﬁbers; it will follow that E is equivalent to f ∗En, thus showing that Φ is surjective. It suﬃces to construct a map ˆg : E −→ R∞ that is a linear monomorphism on ﬁbers, since we can then deﬁne f (e) to be the image under ˆg of the ﬁber through e and can deﬁne g(e) = (f (e), ˆg(e)). One ﬁrst shows that one can construct a countable numerable cover of coordinate charts from a general numerable cover of coordinate charts. Using trivializations φU : U × Rn −→ p−1(U ) and functions λU : B −→ I such that U = λ−1 U (0, 1], we deﬁne ˆgU : E −→ Rn by ˆgU (
e) = λU (p(e)) · p2(φ−1 U (e)) for e ∈ p−1(U ), where p2 : U × Rn −→ Rn is the projection, and ˆgU (e) = 0 for e 6 ∈ p−1(U ). Taking R∞ to be the sum of countably many copies of Rn, we then deﬁne ˆg = gU. P To show that Φ is injective, we must show further that the resulting classifying map f is unique up to homotopy, and for this it suﬃces to show that any two maps (g0, f0) and (g1, f1) of vector bundles from E to En that are isomorphisms on ﬁbers are bundle homotopic. These bundle maps are determined by their second coordinates ˆg0 and ˆg1, which are maps E −→ R∞. Provided that ˆg0(e) is not a negative multiple of ˆg1(e) for any e, we obtain a homotopy ˆh : ˆg0 ≃ ˆg1 by setting ˆh(e, t) = (1 − t)ˆg0(e) + tˆg1(e). The proviso ensures that ˆh is a monomorphism on ﬁbers, and ˆh determines the required bundle homotopy E × I −→ En. For the general case, let i0 and i1 be the linear isomorphisms from R∞ to itself that send the qth standard basis element eq to e2q and e2q−1, respectively. The composites i0 ◦ ˆg0 and i1 ◦ ˆg1 determine bundle maps k0 and k1 from E to En, and the construction just given applies to give bundle homotopies from g0 to k0, from k0 to k1, and from k1 to g1. 2. Characteristic classes for vector bundles 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) ∈ kq(B) to bundles ξ with base space B. 190 CHARACTERISTIC CLASSES OF VECTOR BUNDLES 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). Clearly c(ξ) = c(ξ′) if ξ is equivalent to ξ′. 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)). The formal similarity to the deﬁnition of cohomology operations is obvious, and we shall illustrate how to exploit this similarity in the following sections. Clearly calculation of k∗(BO(n)) determines all characteristic classes. 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. We are particularly interested in Whitney sums of bundles. We have the evident Cartesian product, or external sum, of an m-plane bundle over A and an n-plane bundle over B; it is an (m + n)-plane bundle over A × B. 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. 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∗ teristic class c on (n + 1)-plane bundles, then n(γn+1) = γn ⊕ ε. If we have a charac- i∗ nc(γn+1) = c(γn ⊕ ε), and this leads by naturality to a description of c(ξ⊕ε) for general
n-plane bundles ξ. To give an explicit description of in, we may think of BO(n + 1) as Gn+1(R∞ ⊕ R); precisely, we use an isomorphism between R∞ ⊕ R and R∞ to deﬁne a homeomorphism Gn+1(R∞ ⊕ R) ∼= Gn+1(R∞), and we check that the homotopy class of this homeomorphism is independent of the choice of isomorphism. We then deﬁne in on Gn(R∞) by sending an n-plane x in R∞ to the (n + 1)-plane x ⊕ R. Similarly, we have a classifying map pm,n : BO(m) × BO(n) −→ BO(m + n) that is characterized up to homotopy by p∗ characteristic class c on (m + n)-plane bundles, then m,n(γm+n) = γm × γn. If we have a p∗ m,nc(γm+n) = c(γm × γn), and this leads by naturality to a description of c(ζ × ξ) for general m-plane bundles ζ and n-plane bundles ξ. To give an explicit description of pm,n, we may think of BO(m + n) as Gm+n(R∞ ⊕ R∞); precisely, we use an isomorphism between R∞ ⊕ R∞ and R∞ to deﬁne a homeomorphism Gm+n(R∞ ⊕ R∞) ∼= Gm+n(R∞), and we check that the homotopy class of this homeomorphism is independent of the choice of isomorphism. We then deﬁne pm,n on Gm(R∞) × Gn(R∞) by sending (x, y) to x ⊕ y, where x is an m-plane in R∞ and y is an n-plane in R∞. We bring this down to earth by describing all characteristic classes in mod 2 cohomology. In fact, we have the following equivalent pair of theorems. The ﬁrst uses 3. STIEF
EL-WHITNEY CLASSES OF MANIFOLDS 191 the language of characteristic classes, while the second describes H ∗(BO(n); Z2) together with the induced maps i∗ m,n. 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. n and p∗ (1) w0(ξ) = 1 and wi(ξ) = 0 if i > dim ξ. (2) w1(γ1) 6= 0, where γ1 is the universal line bundle over RP ∞. (3) wi(ξ ⊕ ε) = wi(ξ). (4) wi(ζ ⊕ ξ) = i j=0 wj(ζ) ∪ wi−j (ξ). Every mod 2 characteristic class for n-plane bundles can be written uniquely as a polynomial in the Stiefel-Whitney classes {w1,..., wn}. P Theorem. For n ≥ 1, there are elements wi ∈ H i(BO(n); Z2), i ≥ 0, called the Stiefel-Whitney classes. They satisfy and are uniquely characterized by the following axioms. (1) w0 = 1 and wi = 0 if i > n. (2) w1 6= 0 when n = 1. (3) i∗ n(wi) = wi. (4) p∗ m,n(wi) = i j=0 wj ⊗ wi−j. The mod 2 cohomology H ∗(BO(n); Z2) is the polynomial algebra Z2[w1,..., wn]. P For the uniqueness, suppose given another collection of classes w′ that satisfy the stated properties. Since BO(1) = RP ∞, w1 = w′ non-zero element of H 1(RP ∞; Z2). Therefore wi = w′ assume that this is true for all m < n. Visibly i∗ less than n, and this implies that wi = w′ visible but easily checked that the p∗ 1,n
−1(w′ 1,n−1(wn) = p∗ p∗ i for all n ≥ 1 1 is the unique i for all i when n = 1, and we n−1 is an isomorphism in degrees i in H i(BO(n); Z2) for i < n. It is less m,n are all monomorphisms in all degrees. Since n), this implies that wn = w′ n. 3. Stiefel-Whitney classes of manifolds It is convenient to consider H ∗∗(X) = i H i(X) and to write its elements as formal sums xi, deg xi = i. In practice, we usually impose conditions that guarantee that the sum is ﬁnite. We deﬁne the total Stiefel-Whitney class w(ξ) of wi(ξ); here the sum is clearly ﬁnite. Note in particular a vector bundle ξ to be that w(εq) = 1, where εq is the trivial q-plane bundle. With this notation, we have the formula P P Q w(ζ ⊕ ξ) = w(ζ) ∪ w(ξ). It is usual to write wi(M ) = wi(τ (M )) and w(M ) = w(τ (M )) for a smooth compact manifold M. 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, 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 ). If M has dimension n, then ν has dimension q − n and must satisfy wi(ν) = 0 if i > q − n. Calculation of wi(ν) from the Whitney duality formula can lead to a contradiction if q is too small. 192 CHARACTERISTIC CLASSES OF VECTOR BUNDLES One calculation is immediate. Since the normal bundle of the standard embedding Sq −→ Rq+1 is trivial, w(Sq) = 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 = Sq. 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. (We called it γq+1 before.) The total space of ζq consists of pairs (x, v), where x is a line in Rq+1 and v is a point on that line. This is a subbundle of the trivial (q + 1)-plane bundle εq+1, and we write ζ⊥ q for the complementary bundle whose points are pairs (x, w) such that w is orthogonal to the line x. Thus 1 ζq ⊕ ζ⊥ q ∼= εq+1. Write H ∗(RP q; Z2) = Z2[α]/(αq+1), deg α = 1. Thus α = w1(ζq). Since ζq is a line bundle, wi(ζq) = 0 for i > 1. The formula w(ζq) ∪ w(ζ⊥ q ) = 1 implies that w(ζ⊥ q ) = 1 + α + · · · + αq. We can describe τ (RP q) in terms of ζq. Consider a point x ∈ Sq and write (x, v) for a typical vector in the tangent plane of Sq at x. Then x is orthogonal to v in Rq+1 and (x, v) and (−x, −v) have the same image in τ (RP q). If Lx is the line through x, then this image point determines and is determined by the linear map f : Lx −→ L⊥ x that sends x to v. Starting from this, it is easy to check that τ (RP q) is isomorphic to the bundle Hom(ζq, ζ⊥ q ). As for any line bundle, we have Hom(ζq, ζq) ∼= ε since the identity homomorphisms of the ﬁbers specify a cross
section. Again, as for any bundle over a smooth manifold, a choice of Euclidean metric determines an isomorphism Hom(ζq, ε) ∼= ζq. These facts give the following calculation of τ (RP q) ⊕ ε: τ (RP q) ⊕ ε ∼= Hom(ζq, ζ⊥ ∼= Hom(ζq, ζ⊥ ∼= (q + 1) Hom(ζq, ε) ∼= (q + 1)ζq. q ) ⊕ Hom(ζq, ζq) q ⊕ ζq) ∼= Hom(ζq, εq+1) Therefore w(RP q) = w((q + 1)ζq) = w(ζq)q+1 = (1 + α)q+1 = q + 1 i αi. X0≤i≤q Explicit computations are obtained by computing mod 2 binomial coeﬃcients. 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. If Rq+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. 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 Poincar´e duality and the Steenrod operations in H ∗(M ; Z2). In analogy with w(M ), we deﬁne the total Steenrod square of an element x by Sq
(x) = i Sqi(x). P 4. CHARACTERISTIC NUMBERS OF MANIFOLDS 193 Theorem (Wu formula). Let M be a smooth closed n-manifold with fundamental class z ∈ Hn(M ; Z2). Then the total Stiefel-Whitney class w(M ) is equal vi ∈ H ∗∗(M ; Z2) is the unique cohomology class such that to Sq(v), where v = hv ∪ x, zi = hSq(x), zi for all x ∈ H ∗(M ; Z2). Thus, for k ≥ 0, vk ∪ x = Sqk(x) for all x ∈ H n−k(M ; Z2), and P wk(M ) = Sqi(vj ). Xi+j=k Here the existence and uniqueness of v is an easy exercise from the Poincar´e duality theorem. The basic reason that such a formula holds is that the StiefelWhitney classes can be deﬁned in terms of the Steenrod operations, as we shall see shortly. The Wu formula implies that the Stiefel-Whitney classes are homotopy invariant: if f : M −→ M ′ is a homotopy equivalence between smooth closed nmanifolds, then f ∗ : H ∗(M ′; Z2) −→ H ∗(M ; Z2) satisﬁes f ∗(w(M ′)) = w(M ). In fact, the conclusion holds for any map f, not necessarily a homotopy equivalence, that induces an isomorphism in mod 2 cohomology. Since the tangent bundle of M depends on its smooth structure, this is rather surprising. 4. Characteristic numbers of manifolds Characteristic classes determine important numerical invariants of manifolds, called their characteristic numbers. 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.) 1 · · · wrn In particular, if ri are integers such that iri = n, then the monomial wr1 n 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. In fact, we have the following observation. P Lemma. If M is the boundary of a smooth compact (n + 1)-manifold W, then all tangential Stiefel-Whitney numbers of M are zero. Proof. Using a smooth tubular neighborhood, we see that there is an inward- pointing normal vector ﬁeld along M that spans a trivial bundle ε such that τ (W )\|M ∼= τ (M ) ⊕ ε. Therefore, if i : M −→ W is the inclusion, then i∗(wj (W )) = wj (M ). Let f be a polynomial in the wj of degree n. Recall that the fundamental class of M is ∂z, where z ∈ Hn+1(W, M ) is the fundamental class of the pair (W, M ). We have hf (M ), ∂zi = hi∗f (W ), ∂zi = hf (W ), i∗∂zi = 0 since i∗∂ = 0 by the long exact homology sequence of the pair. 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. 194 CHARACTERISTIC CLASSES OF VECTOR BUNDLES Proof. The Whitney duality formula implies that every wi(M ) is a polynomial in the wi(ν(M )) and every wi(ν(M )) is a polynomial in the wi(M ). We shall explain the following amazing result of Thom in the last chapter. 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. 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. In practice, it might be ﬁendishly diﬃcult to actually construct a manifold with boundary M geometrically. 5. Thom spaces and the Thom isomorphism theorem 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.” This construction will also be at the heart of the proof of Thom’s theorem in the last chapter. 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 Sn 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 ∞). Observe that this construction is functorial with respect to maps of vector bundles. 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). 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. If the bundle ξ is trivial, so that E = B ×
Rn, then Sph(E) = B × Sn. Quotienting out B amounts to the same thing as giving B a disjoint basepoint and then forming the smash product B+ ∧ Sn. That is, in this case the Thom complex is ΣnB+. Therefore, for any cohomology theory k∗, kq(B) = ˜kq(B+) ∼= ˜kn+q(T ξ). There is a conceptual way of realizing this isomorphism. For any n-plane bundle ξ : E −→ B, we have a projection ξ : Sph(E) −→ B and a quotient map π : Sph(E) −→ T ξ. We can compose their product with the diagonal map of Sph(E) to obtain a composite map Sph(E) −→ Sph(E) × Sph(E) −→ B × T ξ. This sends all points at ∞ to points of B × {∞}. Therefore it factors through a map ∆ : T ξ −→ B+ ∧ T ξ, 5. THOM SPACES AND THE THOM ISOMORPHISM THEOREM 195 which is called the “Thom diagonal.” For a commutative ring R, we can use ∆ to deﬁne a cup product H p(B; R) ⊗ ˜H q(T ξ; R) −→ ˜H p+q(T ξ; R). When the bundle ξ is trivial, we let µ ∈ ˜H n(B+ ∧ Sn; R) be the suspension of the identity element 1 ∈ H 0(B; R), and we ﬁnd that x −→ x∪µ speciﬁes the suspension isomorphism H q(B; R) ∼= ˜H n+q(B+ ∧ Sn; R) = ˜H n+q(T ξ; R). Now consider a general bundle ξ. On neighborhoods U of B over which ξ is trivial, we have H q(U ; R) ∼= ˜H n+q(T (ξ\|U ); R). The isomorphism depends on the trivialization φU : U × Rn −→ ξ−1(U ). It is natural to ask if
these isomorphisms patch together to give a global isomorphism H q(B+) −→ ˜H n+q(T ξ). 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. This leads to the following deﬁnition and theorem. For a point b ∈ B, let Sn b be the one-point compactiﬁcation of the ﬁber ξ−1(b); since Sn b is the Thom space of ξ\|b, we have a canonical map ib : Sn b −→ T ξ. Definition. Let ξ : E −→ B be an n-plane bundle. An R-orientation, or Thom class, of ξ is an element µ ∈ ˜H n(T ξ; R) such that, for every point b ∈ B, b(µ) is a generator of the free R-module ˜H n(Sn i∗ b ). 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 ). Theorem (Thom isomorphism theorem). Let µ ∈ ˜H n(T ξ; R) be a Thom class for an n-plane bundle ξ : E −→ B. Deﬁne Φ : H q(B; R) −→ ˜H n+q(T ξ; R) by Φ(x) = x ∪ µ. Then Φ is an isomorphism. Sketch Proof. When R is a ﬁeld, this can be proved by an inductive MayerVietoris sequence argument. To exploit inverse images of open subsets of B, it is convenient to observe that, by easy homotopy and excision arguments, ˜H ∗(T ξ) ∼= H ∗(Sph(E), B) ∼= H ∗(Sph(E), Sph(E)0) ∼= H ∗(E, E0), where E0 and Sph(E)0 are the subspaces of E and Sph(E) obtained by deleting {0} from each ﬁber. Use of a ﬁeld ensures that the
cohomology of the relevant direct limits is the inverse limit of the cohomologies. An alternative argument that works for general R can be obtained by ﬁrst showing that one can assume that B is a CW complex, by replacing ξ by its pullback along a CW approximation of B, and then proceeding by induction over the restrictions of ξ to the skeleta of B; one point is that the restriction of ξ to any cell is trivial and another is that the cohomology of B is the inverse limit of the cohomologies of its skeleta. 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). 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 ). In the present situation, the existence of a Thom class ensures that there is no deviation for the sphere bundle Sph(E) −→ B, so that H ∗(Sph(E); R) ∼= H ∗(B; R) ⊗ H ∗(Sn; R). 196 CHARACTERISTIC CLASSES OF VECTOR BUNDLES The section given by the points at ∞ induces an isomorphism of H ∗(B; R) ⊗ H 0(Sn; R) with H ∗(B; R), and the quotient map Sph(E) −→ T ξ induces an isomorphism of ˜H ∗(T ξ; R) with H ∗(B; R) ⊗ H n(Sn; R). 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 Mayer- Vietoris argument. 6. The construction of the Stiefel-Whitney classes We indicate two constructions of the Stiefel-Whitney classes. Each has distinct advantages over the other. 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−1SqiΦ(1) = Φ−1Sqiµ. Naturality is obvious. Axiom 1 is immediate from the relations Sq0 = id and Sqi(x) = 0 if i > deg x. For axiom 2, we use the following observation. Lemma. There is a homotopy equivalence j : RP ∞ −→ T γ1. Proof. T γ1 is homeomorphic to D(γ1)/S(γ1). Here S(γ1) is the inﬁnite sphere S∞, which is the universal cover of RP ∞ and is therefore contractible. The zero section RP ∞ −→ D(γ1) and the quotient map D(γ1) −→ T γ1 are homotopy equiv- alences, and their composite is the required homotopy equivalence j. Since Sq1(x) = x2 if deg x = 1, the lemma implies that Sq1 is non-zero on the Thom class of γ1, verifying axiom 2. For axiom 3, we easily check that T (ξ ⊕ ε) ∼= ΣT (ξ) for any vector bundle ξ and that the Thom class of ξ ⊕ ε is the suspension of the Thom class of ξ. Thus axiom 3 follows from the stability of the Steenrod operations. For axiom 4, we easily check that, for any vector bundles ζ and ξ, T (ζ × ξ) ∼= T ζ ∧ T ξ and the Thom class of ζ × ξ is the tensor product of the Thom classes of ζ and ξ. Interpreting the Cartan formula for the Steenrod operations externally in the cohomology of products and therefore of smash products, we see that it implies axiom 4. That is, the properties that axiomatize the Steenrod operations directly imply the properties that axiomatize the Stiefel-Whitney classes. We next take the classifying space point of view. As we shall explain in §8, passage from topological groups to their classifying spaces is a product-preserving functor, at
least up to homotopy. We may embed (Z2)n = O(1)n in O(n) as the subgroup of diagonal matrices. The classifying space BO(1) is RP ∞, and we obtain a map ω : (RP ∞)n ≃ B(O(1)n) −→ BO(n) upon passage to classifying spaces. The symmetric group Σn is contained in O(n) as the subgroup of permutation matrices, and the diagonal subgroup O(1)n is closed under conjugation by symmetric matrices. Application of the classifying space functor to conjugation by permutation matrices induces the corresponding permutation of the factors of BO(1)n, and it induces the identity map on BO(n). Indeed, up to homotopy, inner conjugation by an element of G induces the identity map on BG for any topological group G. 7. CHERN, PONTRYAGIN, AND EULER CLASSES 197 By the K¨unneth theorem, we see that H ∗((RP ∞)n; Z2) = ⊗n i=1H ∗(RP ∞; Z2) = Z2[α1,..., αn], where the generators αi are of degree one. The symmetric group Σn acts on this cohomology ring by permuting the variables αi. 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. Here αj1 · · · αji, 1 ≤ j1 < · · · < jn, σi = has degree i. The induced map ω∗ : H ∗(BO(n); Z2) −→ H ∗((RP ∞)n; Z2) takes values in H ∗((RP ∞)n; Z2)Σn. We shall give a general reason why this is so in §8. The resulting map P ω∗ : H ∗(BO(n); Z2) −→ H ∗((RP ∞)n; Z2)Σn is a ring homomorphism between polynomial al
gebras on generators of the same degrees. It turns out to be a monomorphism and therefore an isomorphism. We redeﬁne the Stiefel-Whitney classes by letting wi be the unique element such that ω∗(wi) = σi for 1 ≤ i ≤ n and deﬁning w0 = 1 and wi = 0 for i > n. Then axioms 1 and 2 for the Stiefel-Whitney classes are obvious, and we derive axioms 3 and 4 from algebraic properties of elementary symmetric functions. 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: i Sqi(wj ) = t= wi−twj+t. 7. Chern, Pontryagin, and Euler classes The theory of the previous sections extends appropriately to complex vector bundles and to oriented real vector bundles. 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. For oriented real n-plane bundles, we use the Grassmann varieties of oriented n-planes, the points of which are planes x together with a chosen orientation. In fact, the fundamental groups of the real Grassmann varieties are Z2, and their universal covers are their orientation covers. These covers are the oriented Grassmann varieties ˜Gn(Rq). We write BU (n) = Gn(C∞) and BSO(n) = ˜Gn(R∞), and we construct universal complex n-plane bundles γn : EUn −→ BU (n) and oriented n-plane bundles ˜γn : ˜En −→ BSO(n) as in the ﬁrst section. Let E Un(B) denote the set of equivalence classes of complex nplane bundles over B and let ˜En(B) denote the set of equivalence classes of oriented real n-plane bundles over B; it is required that bundle maps (
g, f ) be orientation preserving, in the sense that the induced map of Thom spaces carries the orientation of the target bundle to the orientation of the source bundle. The universal bundle ˜γn has a canonical orientation which determines an orientation on f ∗ ˜En for any map f : B −→ BSO(n). 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. 198 CHARACTERISTIC CLASSES OF VECTOR BUNDLES Theorem. The natural transformation Φ : [−, BSO(n)] −→ ˜En(−) obtained by sending the homotopy class of a map f : B −→ BSO(n) to the equivalence class of the oriented n-plane bundle f ∗ ˜En is a natural isomorphism of functors. The deﬁnition of characteristic classes for complex n-plane bundles and for oriented real n-plane bundles in a cohomology theory k∗ is the same as for real n-plane bundles, and the Yoneda lemma applies. 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 ˜γn speciﬁes a canonical bijection between characteristic classes of oriented n-plane bundles and elements of k∗(BSO(n)). Clearly we have a 2-fold cover πn : BSO(n) −→ BO(n). The mod 2 characteristic classes for oriented n-plane bundles are as one might expect from this. Continue to write wi for π∗(wi) ∈ H i(BSO(n); Z2); here w1 = 0 since BSO(n) is simply connected. Theorem. H ∗(BSO(n); Z2) ∼= Z2[w2,..., wn]. If we regard a complex n-plane bundle as a real 2n-plane bundle, then the complex structure induces a canonical orientation. By the Yoneda lemma, the resulting natural transformation r
: E Un(−) −→ ˜En(−) is represented by a map r : BU (n) −→ BSO(2n). Explicitly, ignoring its complex structure, we may identify C∞ with R∞ ⊕ R∞ ∼= R∞ and so regard a complex n-plane in C∞ as an oriented 2n-plane in R∞. Similarly, we may complexify real bundles ﬁberwise and so obtain a natural transformation c : En(−) −→ E Un(−). It is represented by a map c : BO(n) −→ BU (n). Explicitly, identifying C∞ with R∞ ⊗R C, we may complexify an n-plane in R∞ to obtain an n-plane in C∞. 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. Characteristic classes constructed in this fashion satisfy homotopy invariance properties that fail for general characteristic classes. In the complex case, with integral coeﬃcients, we have a parallel to our second approach to Stiefel-Whitney classes that leads to a description of H ∗(BU (n); Z) in terms of Chern classes. We may embed (S1)n = U (1)n in U (n) as the subgroup of diagonal matrices. The classifying space BU (1) is CP ∞, and we obtain a map ω : (CP ∞)n ≃ B(U (1)n) −→ BU (n) upon passage to classifying spaces. The symmetric group Σn is contained in U (n) as the subgroup of permutation matrices, and the diagonal subgroup U (1)n is closed under conjugation by symmetric matrices. Application of the classifying space functor to conjugation by permutation matrices induces the corresponding permutation of the factors of BU (1)n, and it induces the identity map on BU (n). By the
K¨unneth theorem, we see that H ∗((CP ∞)n; Z) = ⊗n i=1H ∗(CP ∞; Z) = Z[β1,..., βn], where the generators βi are of degree two. The symmetric group Σn acts on this cohomology ring by permuting the variables βi. The subring H ∗((CP ∞)n; Z)Σn of 7. CHERN, PONTRYAGIN, AND EULER CLASSES 199 elements invariant under the action is the polynomial algebra on the elementary symmetric functions σi, 1 ≤ i ≤ n, in the variables βi. Here βj1 · · · βji, 1 ≤ j1 < · · · < jn, σi = has degree 2i. The induced map ω∗ : H ∗(BU (n); Z) −→ H ∗((CP ∞)n; Z) takes values in H ∗((CP ∞)n; Z)Σn. The resulting map P ω∗ : H ∗(BU (n); Z) −→ H ∗((CP ∞)n; Z)Σn is a ring homomorphism between polynomial algebras on generators of the same degrees. It turns out to be a monomorphism and thus an isomorphism when tensored with any ﬁeld, and it is therefore an isomorphism. We deﬁne the Chern classes by letting ci, 1 ≤ i ≤ n, be the unique element such that ω∗(ci) = σi. 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∗ (4) p∗ n(ci) = ci. m,n(ci) = i j=0 cj ⊗ ci−j. The integral cohomology H ∗(BU (n); Z) is
the polynomial algebra Z[c1,..., cn]. P Here we take axiom 1 as a deﬁnition and we interpret axiom 2 as meaning that c1 corresponds to the identity map of CP ∞ under the canonical identiﬁcation of [CP ∞, CP ∞] with H 2(CP ∞; Z). Axioms 3 and 4 can be read oﬀ from algebraic properties of elementary symmetric functions. The theorem admits an immediate interpretation in terms of characteristic classes. Observe that, since H ∗(BU (n); Z) is a free Abelian group, the theorem remains true precisely as stated with Z replaced by any other commutative ring of coeﬃcients R. We continue to write ci for the image of ci in H ∗(BU (n); R) under the homomorphism induced by the unit Z −→ R of the ring R. The reader deserves to be warned about a basic inconsistency in the literature. Remark. With the discussion above, c1(γn+1 ) is the canonical generator of H 2(CP n; Z), where γn+1 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 γn+1 is the canonical generator of H 2(CP n; Z). With this convention, the nth Chern class would be (−1)ncn. It is often unclear in the literature which convention is being followed. 1 1 1 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)ic∗(c2i), c∗ : H 4i(BU (n); R) −→ H 4i(BO(n); R); also write pi for π∗ n(pi) ∈ H 4i(BSO(n); R). Definition. Deﬁne the Euler class e(

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