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of another product called cross product. However, to deﬁne the cross product from scratch takes some work, so we will proceed in the opposite order, ﬁrst giving an elementary deﬁnition of cup product by an explicit formula with simplices, then afterwards deﬁning cross product in terms of cup product. The other approach of deﬁning cup product via cross product is explained at the end of §3.B. To deﬁne the cup product we consider cohomology with coeﬃcients in a ring R, the most common choices being Z, Zn, and Q. For cochains ϕ ∈ C k(X; R) and ψ ∈ C ℓ(X; R), the cup product ϕ ` ψ ∈ C k+ℓ(X; R) is the cochain whose value on a singular simplex σ : k+ℓ→X is given by the formula (ϕ ` ψ)(σ ) = ϕ ∆ σ \|\| [v0, ···, vk] ψ σ \|\| [vk, ···, vk+ℓ] where the right-hand side is the product in R. To see that this cup product of cochains induces a cup product of cohomology classes we need a formula relating it to the coboundary map: Lemma 3.6. δ(ϕ`ψ) = δϕ`ψ+(−1)kϕ`δψ for ϕ ∈ C k(X; R) and ψ ∈ C ℓ(X; R). Proof: For σ : k+ℓ+1→X we have ∆ (δϕ ` ψ)(σ ) = k+1 (−1)iϕ Xi=0 k+ℓ+1 σ \|\|[v0, ···, vi, ···, vk+1] b ψ σ \|\|[vk+1, ···, vk+ℓ+1] (−1)k(ϕ ` δψ)(σ ) = (−1)iϕ Xi=k σ \|\|[v0, ···, vk] ψ σ \|\|[vk, ···, vi, ·
··, vk+ℓ+1] When we add these two expressions, the last term of the ﬁrst sum cancels the ﬁrst term of the second sum, and the remaining terms are exactly δ(ϕ ` ψ)(σ ) = (ϕ ` ψ)(∂σ ) ⊔⊓ since ∂σ = (−1)iσ \|\| [v0, ···, vi, ···, vk+ℓ+1]. k+ℓ+1 i=0 b P b Cup Product Section 3.2 207 From the formula δ(ϕ ` ψ) = δϕ ` ψ ± ϕ ` δψ it is apparent that the cup product of two cocycles is again a cocycle. Also, the cup product of a cocycle and a coboundary, in either order, is a coboundary since ϕ ` δψ = ±δ(ϕ ` ψ) if δϕ = 0, and δϕ ` ψ = δ(ϕ ` ψ) if δψ = 0. It follows that there is an induced cup product H k(X; R) × H ℓ(X; R) `-----------------→ H k+ℓ(X; R) This is associative and distributive since at the level of cochains the cup product obviously has these properties. If R has an identity element, then there is an identity element for cup product, the class 1 ∈ H 0(X; R) deﬁned by the 0 cocycle taking the value 1 on each singular 0 simplex. A cup product for simplicial cohomology can be deﬁned by the same formula as for singular cohomology, so the canonical isomorphism between simplicial and singu- lar cohomology respects cup products. Here are three examples of direct calculations of cup products using simplicial cohomology. Example 3.7. Let M be the closed orientable surface of genus g ≥ 1 with the complex structure shown in the ﬁgure for the case g = 2. The cup product of ∆ interest is H 1(M)× H 1(M)→H 2(M). Taking Z coefﬁcients, a basis for H1(M
) is formed by the edges ai and bi, as we showed in Example 2.36 when we computed the homology of M using cellular homology. We have H 1(M) ≈ Hom(H1(M), Z) by cellular cohomology or the universal coeﬃcient theorem. A basis for H1(M) determines a dual basis for Hom(H1(M), Z), so dual to ai is the cohomology class αi assigning the value 1 to ai and 0 to the other basis elements, and similarly we have cohomology classes βi dual to bi. To represent αi by a simplicial cocycle ϕi we need to choose values for ϕi on the edges radiating out from the central vertex in such a way that δϕi = 0. This is the ‘cocycle condition’ discussed in the introduction to this chapter, where we saw that it has a geometric interpretation in terms of curves transverse to the edges of M. With this interpretation in mind, consider the arc labeled αi in the ﬁgure, which represents a loop in M meeting ai in one point and disjoint from all the other basis elements aj and bj. We deﬁne ϕi to have the value 1 on edges meeting the arc αi and the value 0 on all other edges. Thus ϕi counts the number of intersections of each edge with the arc αi. In similar fashion we obtain a cocycle ψi counting intersections with the arc βi, and ψi represents the cohomology class βi dual to bi. Now we can compute cup products by applying the deﬁnition. Keeping in mind that the ordering of the vertices of each 2 simplex is compatible with the indicated ` ψ1 takes the value 0 on all orientations of its edges, we see for example that ϕ1 2 simplices except the one with outer edge b1 in the lower right part of the ﬁgure, 208 Chapter 3 Cohomology `ψ1 takes the value 1 on the 2 chain c formed by where it takes the value 1. Thus ϕ1 the sum of all the 2 simplices with the signs indicated in the center of the ﬁgure. It is an easy calculation that ∂c = 0. Since there are no 3 simplices, c is not a boundary
, so ` ψ1)(c) is a generator it represents a nonzero element of H2(M). The fact that (ϕ1 ` ψ1 of Z implies both that c represents a generator of H2(M) ≈ Z and that ϕ1 represents the dual generator γ of H 2(M) ≈ Hom(H2(M), Z) ≈ Z. Thus α1 ` β1 = γ. In similar fashion one computes: αi ` βj = γ, i = j 0, i ≠ j     = −(βi ` αj), αi ` αj = 0, βi ` βj = 0   These relations determine the cup product H 1(M)× H 1(M)→H 2(M) completely since cup product is distributive. Notice that cup product is not commutative in this exam` αi). We will show in Theorem 3.11 below that this is the ple since αi worst that can happen: Cup product is commutative up to a sign depending only on ` βi = −(βi dimension, assuming that the coeﬃcient ring itself is commutative. One can see in this example that nonzero cup products of distinct classes αi or βj occur precisely when the corresponding loops αi or βj intersect. This is also true for the cup product of αi or βi with itself if we allow ourselves to take two copies of the corresponding loop and deform one of them to be disjoint from the other. Example 3.8. The closed nonorientable surface N of genus g can be treated in similar fashion if we use Z2 coeﬃcients. Using the complex structure shown, the edges ai give a basis for H1(N; Z2), and the dual basis elements αi ∈ H 1(N; Z2) can be represented by cocycles with values given by counting intersections with the arcs labeled αi in the ﬁgure. Then ` αi is the nonzero element of one computes that αi H 2(N; Z2) ≈ Z2 and αi ` αj = 0 for i ≠ j. In particular, when g = 1 we have N = RP2, and the cup product of a generator of
H 1(RP2; Z2) with itself is a generator of H 2(RP2; Z2). ∆ The remarks in the paragraph preceding this example apply here also, but with the following diﬀerence: When one tries to deform a second copy of the loop αi in the present example to be disjoint from the original copy, the best one can do is make ` αi is now nonzero. it intersect the original in one point. This reﬂects the fact that αi Example 3.9. Let X be the 2 dimensional CW complex obtained by attaching a 2 cell to S 1 by the degree m map S 1→S 1, z ֏ zm. Using cellular cohomology, or cellular homology and the universal coeﬃcient theorem, we see that H n(X; Z) consists of a Z for n = 0 and a Zm for n = 2, so the cup product structure with Z coeﬃcients is uninteresting. However, with Zm coeﬃcients we have H i(X; Zm) ≈ Zm for i = 0, 1, 2, Cup Product Section 3.2 209 so there is the possibility that the cup product of two 1 dimensional classes can be nontrivial. To obtain a complex structure on X, take a regular m gon subdivided into m triangles Ti around a central vertex v, as shown in the ﬁgure for the case m = 4, then ∆ ∆ identify all the outer edges by rotations of the m gon. complex structure with 2 vertices, m+1 This gives X a edges, and m 2 simplices. A generator α of H 1(X; Zm) is represented by a cocycle ϕ assigning the value 1 to the edge e, which generates H1(X). The condition that ϕ be a cocycle means that ϕ(ei) + ϕ(e) = ϕ(ei+1) for all i, subscripts being taken mod m. So we may take ϕ(ei) = i ∈ Zm. Hence (ϕ ` ϕ)(Ti) = ϕ(ei)ϕ(e) = i. The map h : H 2(X; Zm)→Hom(H2(X; Zm), Zm) is an isomorphism since i Ti is
a generator of H2(X; Zm) and there are 2 cocycles taking the value 1 on i Ti, for example the P cocycle taking the value 1 on one Ti and 0 on all the others. The cocycle ϕ ` ϕ takes i Ti, hence represents 0 + 1 + ··· + (m − 1) times the value 0 + 1 + ··· + (m − 1) on a generator β of H 2(X; Zm). In Zm the sum 0 + 1 + ··· + (m − 1) is 0 if m is odd and k if m = 2k since the terms 1 and m − 1 cancel, 2 and m − 2 cancel, and so on. Thus, writing α2 for α ` α, we have α2 = 0 if m is odd and α2 = kβ if m = 2k. P P In particular, if m = 2, X is RP2 and α2 = β in H 2(RP2; Z2), as we showed already in Example 3.8. The cup product formula (ϕ ` ψ)(σ ) = ϕ also gives relative cup products H k(X; R) × H ℓ(X, A; R) H k(X, A; R) × H ℓ(X; R) H k(X, A; R) × H ℓ(X, A; R) σ \|\| [v0, ···, vk] `-----------------→ H k+ℓ(X, A; R) `-----------------→ H k+ℓ(X, A; R) `-----------------→ H k+ℓ(X, A; R) ψ σ \|\| [vk, ···, vk+ℓ] since if ϕ or ψ vanishes on chains in A then so does ϕ ` ψ. There is a more general relative cup product H k(X, A; R) × H ℓ(X, B; R) `-----------------→ H k+ℓ(X, A ∪ B; R) when A and B are open subsets of X or subcomplexes of the CW complex X. This is obtained in the following way. The absolute cup product restricts to a cup product C k(X, A; R)× C ℓ(
X, B; R)→C k+ℓ(X, A + B; R) where C n(X, A + B; R) is the subgroup of C n(X; R) consisting of cochains vanishing on sums of chains in A and chains in If A and B are open in X, the inclusions C n(X, A ∪ B; R) ֓ C n(X, A + B; R) B. induce isomorphisms on cohomology, via the ﬁve-lemma and the fact that the restriction maps C n(A ∪ B; R)→C n(A + B; R) induce isomorphisms on cohomology as we saw in the discussion of excision in the previous section. Therefore the cup product C k(X, A; R)× C ℓ(X, B; R)→C k+ℓ(X, A + B; R) induces the desired relative cup product 210 Chapter 3 Cohomology H k(X, A; R)× H ℓ(X, B; R)→H k+ℓ(X, A ∪ B; R). This holds also if X is a CW complex with A and B subcomplexes since here again the maps C n(A ∪ B; R)→C n(A + B; R) induce isomorphisms on cohomology, as we saw for homology in §2.2. Proposition 3.10. For a map f : X→Y, the induced maps f ∗ : H n(Y ; R)→H n(X; R) satisfy f ∗(α ` β) = f ∗(α) ` f ∗(β), and similarly in the relative case. Proof: This comes from the cochain formula f ♯(ϕ) ` f ♯(ψ) = f ♯(ϕ ` ψ) : (f ♯ϕ ` f ♯ψ)(σ ) = f ♯ϕ f ♯ψ σ \|\|[v0, ···, vk] f σ \|\|[v0, ···, vk] ψ = ϕ = (ϕ ` ψ)(f σ ) = f ♯(ϕ ` ψ)(σ ) σ \|\|[
vk, ···, vk+ℓ] f σ \|\|[vk, ···, vk+ℓ] ⊔⊓ The natural question of whether the cup product is commutative is answered by the following: Theorem 3.11. The identity α ` β = (−1)kℓβ ` α holds for all α ∈ H k(X, A; R) and β ∈ H ℓ(X, A; R), when R is commutative. Taking α = β, this implies in particular that if α is an element of H k(X, A; R) with k odd, then 2(α ` α) = 0 in H 2k(X, A; R), or more concisely, 2α2 = 0. Hence if H 2k(X, A; R) has no elements of order two, then α2 = 0. For example, if X is the 2 complex obtained by attaching a disk to S 1 by a map of degree m as in Example 3.9 above, then we can deduce that the square of a generator of H 1(X; Zm) is zero if m is odd, and is either zero or the unique element of H 2(X; Zm) ≈ Zm of order two if m is even. As we showed, the square is in fact nonzero when m is even. ∆ Proof: Consider ﬁrst the case A = ∅. For cochains ϕ ∈ C k(X; R) and ψ ∈ C ℓ(X; R) one can see from the deﬁnition that the cup products ϕ ` ψ and ψ ` ϕ diﬀer only by k+ℓ. The idea of the proof is to study a particularly a permutation of the vertices of nice permutation of vertices, namely the one that totally reverses their order. This has the convenient feature of also reversing the ordering of vertices in any face. For a singular n simplex σ : [v0, ···, vn]→X, let σ be the singular n simplex obtained by preceding σ by the linear homeomorphism of [v0, ···, vn] reversing the order of the vertices. Thus σ (vi) = σ (
vn−i). This reversal of vertices is the product of n + (n − 1) + ··· + 1 = n(n + 1)/2 transpositions of adjacent vertices, each of which reverses orientation of the n simplex since it is a reﬂection across an (n − 1) dimensional hyperplane. So to take orientations into account we would expect that a sign εn = (−1)n(n+1)/2 ought to be inserted. Hence we deﬁne a homomorphism ρ : Cn(X)→Cn(X) by ρ(σ ) = εnσ. We will show that ρ is a chain map, chain homotopic to the identity, so it induces the identity on cohomology. From this the theorem quickly follows. Namely, the Cup Product Section 3.2 211 formulas (ρ∗ϕ ` ρ∗ψ)(σ ) = ϕ ρ∗(ψ ` ϕ)(σ ) = εk+ℓψ ψ εkσ \|\|[vk, ···, v0] εℓσ \|\|[vk+ℓ, ···, vk] σ \|\|[vk, ···, v0] σ \|\|[vk+ℓ, ···, vk] ϕ show that εkεℓ(ρ∗ϕ ` ρ∗ψ) = εk+ℓρ∗(ψ ` ϕ), since we assume R is commutative. A trivial calculation gives εk+ℓ = (−1)kℓεkεℓ, hence ρ∗ϕ ` ρ∗ψ = (−1)kℓρ∗(ψ ` ϕ). Since ρ is chain homotopic to the identity, the ρ∗ ’s disappear when we pass to cohomology classes, and so we obtain the desired formula α ` β = (−1)kℓβ ` α. The chain map property ∂ρ = ρ∂ can be veriﬁed by calculating, for a singular n simplex σ, ∂ρ(σ ) = εn (−1)i
σ \|\|[vn, ···, vn−i, ···, v0] Xi ρ∂(σ ) = ρ (−1)iσ \|\|[v0, ···, b vi, ···, vn] vn−i, ···, v0] b (−1)n−iσ \|\|[vn, ···, Xi = εn−1 Xi so the result follows from the easily checked identity εn = (−1)nεn−1. b To deﬁne a chain homotopy between ρ and the identity we are motivated by the construction of the prism operator P in the proof that homotopic maps induce the same homomorphism on homology, in Theorem 2.10. The main ingredient in n × I into (n + 1) simplices with vern × {1}, the vertex wi lying directly above vi. Using n be the projection, we now deﬁne the construction of P was a subdivision of n × {0} and wi in tices vi in the same subdivision, and letting π : ∆ P : Cn(X)→Cn+1(X) by ∆ n × I→ ∆ P (σ ) = ∆ (−1)iεn−i(σ π ) \|\| [v0, ···, vi, wn, ···, wi] ∆ Xi Thus the w vertices are written in reverse order, and there is a compensating sign n × I εn−i. One can view this formula as arising from the in which the vertices are ordered v0, ···, vn, wn, ···, w0 rather than the more natural ordering v0, ···, vn, w0, ···, wn. complex structure on ∆ ∆ To show ∂P + P ∂ = ρ − 11 we ﬁrst calculate ∂P, leaving out σ ’s and σ π ’s for notational simplicity: ∂P = (−1)i(−1)j εn−i[v0, ···, vj, ···, vi, wn, ···, wi] Xj≤i + (−1)i(−1)i+1+n−jεn−i[v0, ·
··, vi, wn, ···, b wj, ···, wi] The j = i terms in these two sums give Xj≥i c εn[wn, ···, w0] + εn−i[v0, ···, vi−1, wn, ···, wi] Xi>0 + (−1)n+i+1εn−i[v0, ···, vi, wn, ···, wi+1] − [v0, ···, vn] Xi<n 212 Chapter 3 Cohomology In this expression the two summation terms cancel since replacing i by i − 1 in the second sum produces a new sign (−1)n+iεn−i+1 = −εn−i. The remaining two terms εn[wn, ···, w0] and −[v0, ···, vn] represent ρ(σ ) − σ. So in order to show that ∂P + P ∂ = ρ − 11, it remains to check that in the formula for ∂P above, the terms with j ≠ i give −P ∂. Calculating P ∂ from the deﬁnitions, we have P ∂ = (−1)i(−1)j εn−i−1[v0, ···, vi, wn, ···, wj, ···, wi] Xi<j + (−1)i−1(−1)j εn−i[v0, ···, c vj, ···, vi, wn, ···, wi] Xi>j b Since εn−i = (−1)n−iεn−i−1, this ﬁnishes the veriﬁcation that ∂P + P ∂ = ρ − 11, and so the theorem is proved when A = ∅. The proof also applies when A ≠ ∅ since the maps ρ and P take chains in A to chains in A, so the dual homomorphisms ρ∗ and P ∗ act on relative cochains. ⊔⊓ The Cohomology Ring Since cup product is associative and distributive, it is natural to try to make it the multiplication in a ring structure on the cohomology groups of a space X
. This is easy to do if we simply deﬁne H ∗(X; R) to be the direct sum of the groups H n(X; R). Elements of H ∗(X; R) are ﬁnite sums i αi with αi ∈ H i(X; R), and the product of It is routine to check two such sums is deﬁned to be that this makes H ∗(X; R) into a ring, with identity if R has an identity. Similarly, P H ∗(X, A; R) is a ring via the relative cup product. Taking scalar multiplication by elements of R into account, these rings can also be regarded as R algebras. i,j αiβj. i αi P j βj P P = For example, the calculations in Example 3.8 or 3.9 above show that H ∗(RP2; Z2) consists of the polynomials a0 +a1α+a2α2 with coeﬃcients ai ∈ Z2, so H ∗(RP2; Z2) is the quotient Z2[α]/(α3) of the polynomial ring Z2[α] by the ideal generated by α3. This example illustrates how H ∗(X; R) often has a more compact description than the sequence of individual groups H n(X; R), so there is a certain economy in the change of scale that comes from regarding all the groups H n(X; R) as part of a single object H ∗(X; R). Adding cohomology classes of diﬀerent dimensions to form H ∗(X; R) is a convenient formal device, but it has little topological signiﬁcance. One always regards the cohomology ring as a graded ring: a ring A with a decomposition as a sum k≥0Ak of additive subgroups Ak such that the multiplication takes Ak × Aℓ to Ak+ℓ. To indicate that an element a ∈ A lies in Ak we write \|a\| = k. This applies in particular to elements of H k(X; R). Some authors call \|a\| the ‘degree’ of a, but we will use the term ‘dimension’ which is more geometric and
avoids potential confusion with the L degree of a polynomial. Cup Product Section 3.2 213 A graded ring satisfying the commutativity property of Theorem 3.11, ab = (−1)\|a\|\|b\|ba, is usually called simply commutative in the context of algebraic topology, in spite of the potential for misunderstanding. In the older literature one ﬁnds less ambiguous terms such as graded commutative, anticommutative, or skew com- mutative. Example 3.12: Polynomial Rings. Among the simplest graded rings are polynomial rings R[α] and their truncated versions R[α]/(αn), consisting of polynomials of degree less than n. The example we have seen is H ∗(RP2; Z2) ≈ Z2[α]/(α3). More generally we will show in Theorem 3.19 that H ∗(RPn; Z2) ≈ Z2[α]/(αn+1) and H ∗(RP∞; Z2) ≈ Z2[α]. In these cases \|α\| = 1. We will also show that H ∗(CPn; Z) ≈ Z[α]/(αn+1) and H ∗(CP∞; Z) ≈ Z[α] with \|α\| = 2. The analogous results for quaternionic projective spaces are also valid, with \|α\| = 4. The coeﬃcient ring Z in the complex and quaternionic cases could be replaced by any commutative ring R, but not for RPn and RP∞ since a polynomial ring R[α] is strictly commutative, so for this to be a commutative ring in the graded sense we must have either \|α\| even or 2 = 0 in R. Polynomial rings in several variables also have graded ring structures, and these graded rings can sometimes be realized as cohomology rings of spaces. For example, Z2[α1, ···, αn] is H ∗(X; Z2) for X the product of n copies of RP∞, with \|αi\| = 1 for each i, as we will see in Example 3.20. ··· αik Example 3.13: Exterior Algebras. Another nice example of a commutative
graded R[α1, ···, αn] over a commutative ring R with identity. ring is the exterior algebra, i1 < ··· < ik, with This is the free R module with basis the ﬁnite products αi1 Λ associative, distributive multiplication deﬁned by the rules αiαj = −αjαi for i ≠ j and α2 i = 0. The empty product of αi ’s is allowed, and provides an identity element R[α1, ···, αn]. The exterior algebra becomes a commutative graded ring by 1 in specifying odd dimensions for the generators αi. Λ The example we have seen is the torus T 2 = S 1 × S 1, where H ∗(T 2; Z) ≈ Z[α, β] with \|α\| = \|β\| = 1 by the calculations in Example 3.7. More generally, for the n torus T n, H ∗(T n; R) is the exterior algebra R[α1, ···, αn] as we will see in Example 3.16. The same is true for any product of odd-dimensional spheres, where \|αi\| is the dimension of the ith sphere. Λ Λ Induced homomorphisms are ring homomorphisms by Proposition 3.10. Here is an example illustrating this fact. α H ∗(Xα; R) Example 3.14: Product Rings. The isomorphism H ∗( whose coordinates are induced by the inclusions iα : Xα ֓ α Xα is a ring isomorphism with respect to the usual coordinatewise multiplication in a product ring, because each coordinate function i∗ α is a ring homomorphism. Similarly for a wedge sum H ∗(Xα; R) is a ring isomorphism. Here we take α Xα; R) ≈ the isomorphism α Xα; R) H ∗( ` ` Q ≈-----→ W e α Q e 214 Chapter 3 Cohomology reduced cohomology to be cohomology relative to a basepoint, and we use relative cup products. We should assume the basepoints xα ∈ Xα are deformation retracts of neighborhoods, to be sure that the claimed isomorphism does indeed hold. This product ring structure for wedge sums can
sometimes be used to rule out splittings of a space as a wedge sum up to homotopy equivalence. For example, consider CP2, which is S 2 with a cell e4 attached by a certain map f : S 3→S 2. Using homology or just the additive structure of cohomology it is impossible to conclude that CP2 is not homotopy equivalent to S 2 ∨ S 4, and hence that f is not homotopic to a constant map. However, with cup products we can distinguish these two spaces since the square of each element of H 2(S 2 ∨ S 4; Z) is zero in view of the ring isoH ∗(S 4; Z), but the square of a generator of morphism H 2(CP2; Z) is nonzero since H ∗(CP2; Z) ≈ Z[α]/(α3). H ∗(S 2 ∨ S 4; Z) ≈ H ∗(S 2; Z)⊕ e e e More generally, cup products can be used to distinguish inﬁnitely many diﬀerent homotopy classes of maps S 4n−1→S 2n for all n ≥ 1. This is systematized in the notion of the Hopf invariant, which is studied in §4.B. Here is the evident general question raised by the preceding examples: The Realization Problem. Which graded commutative R algebras occur as cup product algebras H ∗(X; R) of spaces X? This is a diﬃcult problem, with the degree of diﬃculty depending strongly on the coeﬃcient ring R. The most accessible case is R = Q, where essentially every graded commutative Q algebra is realizable, as shown in [Quillen 1969]. Next in order of diﬃculty is R = Zp with p prime. This is much harder than the case of Q, and only partial results, obtained with much labor, are known. Finally there is R = Z, about which very little is known beyond what is implied by the Zp cases. A K¨unneth Formula One might guess that there should be some connection between cup product and product spaces, and indeed this is the case, as we will show in this subsection. To begin, we deﬁne the cross
product, or external cup product as it is sometimes called. This is the map H ∗(X; R) × H ∗(Y ; R) ×-----------------------→ H ∗(X × Y ; R) given by a× b = p∗ 2 (b) where p1 and p2 are the projections of X × Y onto X and Y. Since cup product is distributive, the cross product is bilinear, that is, linear 1 (a) ` p∗ in each variable separately. We might hope that the cross product map would be an isomorphism in many cases, thereby giving a nice description of the cohomology rings of these product spaces. However, a bilinear map is rarely a homomorphism, so it could hardly be an isomorphism. Fortunately there is a nice algebraic solution Cup Product Section 3.2 215 to this problem, and that is to replace the direct product H ∗(X; R)× H ∗(Y ; R) by the tensor product H ∗(X; R) ⊗R H ∗(Y ; R). Let us review the deﬁnition and basic properties of tensor products. For abelian groups A and B the tensor product A ⊗ B is deﬁned to be the abelian group with generators a ⊗ b for a ∈ A, b ∈ B, and relations (a + a′) ⊗ b = a ⊗ b + a′ ⊗ b and a ⊗ (b + b′) = a ⊗ b + a ⊗ b′. The zero element of A ⊗ B is, and −(a ⊗ b) = −a ⊗ b = a ⊗ (−b). More generally n(a ⊗ b) = na ⊗ b = a ⊗ nb for all n ∈ Z. Some readily veriﬁed elementary properties are: (1) A ⊗ B ≈ B ⊗ A. iAi) ⊗ B ≈ L i(Ai ⊗ B). (2) ( (3) (A ⊗ B) ⊗ C ≈ A ⊗ (B ⊗ C). L (4) Z ⊗ A �
� A. (5) Zn ⊗ A ≈ A/nA. (6) A pair of homomorphisms f : A→A′ and g : B→B′ induces a homomorphism f ⊗ g : A ⊗ B→A′ ⊗ B′ via (f ⊗ g)(a ⊗ b) = f (a) ⊗ g(b). (7) A bilinear map ϕ : A× B→C induces a homomorphism A ⊗ B→C sending a ⊗ b to ϕ(a, b). In (1)–(5) the isomorphisms are the obvious ones, for example a ⊗ b ֏ b ⊗ a in (1) and n ⊗ a ֏ na in (4). Properties (1), (2), (4), and (5) allow the calculation of tensor products of ﬁnitely generated abelian groups. The generalization to tensor products of modules over a commutative ring R is easy. One deﬁnes A ⊗R B for R modules A and B to be the quotient group of A ⊗ B obtained by imposing the further relations r a ⊗ b = a ⊗ r b for r ∈ R, a ∈ A, and b ∈ B. This relation guarantees that A ⊗R B is again an R module, with r (a ⊗ b. The statements (1)–(3), (6), and (7) remain valid for tensor products of R modules. The generalization of (4) is the canonical isomorphism R ⊗. However, A ⊗R B is generally diﬀerent from A ⊗ B. For example, if R = Z[ When R is Z or Zm we have A ⊗R B = A ⊗ B, and this is also true when R is Q or q b = 2], 2}, then R ⊗R R = R but any subring of Q since in A ⊗ B we have p a ⊗ p which as a group is free abelian of rank 2 with basis {1, R ⊗ R is free abelian of rank 4 with basis {1 ⊗ 1
, 1 ⊗ q a ⊗ pb = 1 q a ⊗ b = 1 q qb =, √ 2 ⊗ 1, √ Property (7) of tensor products guarantees that the cross product as deﬁned above √ √ √ √ gives rise to a homomorphism of R modules H ∗(X; R) ⊗R H ∗(Y ; R) ×-----------------→ H ∗(X × Y ; R), a ⊗ b ֏ a× b which we shall also call cross product. This map becomes a ring homomorphism if we deﬁne the multiplication in a tensor product of graded rings by (a ⊗ b)(c ⊗ d) = (−1)\|b\|\|c\|ac ⊗ bd where \|x\| denotes the dimension of x. Namely, if we denote the 216 Chapter 3 Cohomology cross product map by µ and we deﬁne (a ⊗ b)(c ⊗ d) = (−1)\|b\|\|c\|ac ⊗ bd, then µ (a ⊗ b)(c ⊗ d) = (−1)\|b\|\|c\|µ(ac ⊗ bd) = (−1)\|b\|\|c\|(a ` c)× (b ` d) 1 (a ` c) ` p∗ = (−1)\|b\|\|c\|p∗ 1 (a) ` p∗ = (−1)\|b\|\|c\|p∗ 2 (b) ` p∗ 1 (a) ` p∗ = p∗ = (a× b)(c × d) = µ(a ⊗ b)µ(c ⊗ d) 1 (c) ` p∗ 1 (c) ` p∗ 2 (b ` d) 2 (d) 2 (b) ` p∗ 2 (d) Theorem 3.15. The cross product H ∗(X; R) ⊗R H ∗(Y ; R)→H ∗(X × Y ; R) is an isomorphism of rings if X and Y are CW complexes and H k(Y ; R) is a ﬁnitely generated free R module for all k. Results of this type, computing homology or cohomology
of a product space, are known as K¨unneth formulas. The hypothesis that X and Y are CW complexes will be shown to be unnecessary in §4.1 when we consider CW approximations to arbitrary spaces. On the other hand, the freeness hypothesis cannot always be dispensed with, as we shall see in §3.B when we obtain a completely general K¨unneth formula for the homology of a product space. When the conclusion of the theorem holds, the ring structure in H ∗(X × Y ; R) is determined by the ring structures in H ∗(X; R) and H ∗(Y ; R). Example 3E.6 shows that some hypotheses are necessary in order for this to be true. Λ Example 3.16. The exterior algebra R[α1, ···, αn] is the graded tensor product over R of the one-variable exterior algebras R[αi] where the αi ’s have odd dimension. The K¨unneth formula then gives an isomorphism H ∗(S k1 × ··· × S kn ; Z) ≈ Z[α1, ···, αn] if the dimensions ki are all odd. With some ki ’s even, one would have the tensor product of an exterior algebra for the odd-dimensional spheres and Λ truncated polynomial rings Z[α]/(α2) for the even-dimensional spheres. Of course, Z[α] and Z[α]/(α2) are isomorphic as rings, but when one takes tensor products in the graded sense it becomes important to distinguish them as graded rings, with α Λ Z[α] and even-dimensional in Z[α]/(α2). These remarks apply odd-dimensional in more generally with any coeﬃcient ring R in place of Z, though when R = Z2 there is no need to distinguish between the odd-dimensional and even-dimensional cases Λ Λ since signs become irrelevant. The idea of the proof of the theorem will be to consider, for a ﬁxed CW complex Y, the functors hn(X, A) = H i(X, A; R)⊗R H n−i(Y ; R) i L kn(X, A) = H n(X × Y, A× Y ; R) The cross
product, or a relative version of it, deﬁnes a map µ : hn(X, A)→kn(X, A) which we would like to show is an isomorphism when X is a CW complex and A = ∅. Cup Product Section 3.2 217 We will show: (1) h∗ and k∗ are cohomology theories on the category of CW pairs. (2) µ is a natural transformation: It commutes with induced homomorphisms and with coboundary homomorphisms in long exact sequences of pairs. It is obvious that µ : hn(X)→kn(X) is an isomorphism when X is a point since it is just the scalar multiplication map R ⊗R H n(Y ; R)→H n(Y ; R). The following general fact will then imply the theorem. Proposition 3.17. If a natural transformation between unreduced cohomology theories on the category of CW pairs is an isomorphism when the CW pair is (point, ∅), then it is an isomorphism for all CW pairs. Proof: Let µ : h∗(X, A)→k∗(X, A) be the natural transformation. By the ﬁve-lemma it will suﬃce to show that µ is an isomorphism when A = ∅. First we do the case of ﬁnite-dimensional X by induction on dimension. The induction starts with the case that X is 0 dimensional, where the result holds by : α (Dn α, ∂Dn hypothesis and by the axiom for disjoint unions. For the induction step, µ gives a map between the two long exact sequences for the pair (X n, X n−1), with commuting squares since µ is a natural transformation. The ﬁve-lemma reduces the inductive step to showing that µ is an isomorphism for (X, A) = (X n, X n−1). Let α)→(X n, X n−1) be a collection of characteristic maps for all the n cells ∗ is an isomorphism for h∗ and k∗, so by naturality it suﬃces of X. By excision, Φ to show that µ is an isomorphism for (X, A) = α)
. The axiom for disjoint unions gives a further reduction to the case of the pair (Dn, ∂Dn). Finally, this case follows by applying the ﬁve-lemma to the long exact sequences of this pair, since Dn is contractible and hence is covered by the 0 dimensional case, and ∂Dn is (n − 1) dimensional. α, ∂Dn α (Dn ` ` Φ The case that X is inﬁnite-dimensional reduces to the ﬁnite-dimensional case by a telescope argument as in the proof of Lemma 2.34. We leave this for the reader since the ﬁnite-dimensional case suﬃces for the h∗ and k∗ we are considering, where the maps hi(X)→hi(X n) and ki(X)→ki(X n) induced by the inclusion X n ֓ X are ⊔⊓ isomorphisms when n is suﬃciently large with respect to i. It remains to check that h∗ and k∗ are cohomology theories, and Proof of 3.15: that µ is a natural transformation. Since we are dealing with unreduced cohomology theories there are four axioms to verify. (1) Homotopy invariance: f ≃ g implies f ∗ = g∗. This is obvious for both h∗ and k∗. (2) Excision: h∗(X, A) ≈ h∗(B, A ∩ B) for A and B subcomplexes of the CW complex X = A ∪ B. This is obvious, and so is the corresponding statement for k∗ since (A× Y ) ∪ (B × Y ) = (A ∪ B)× Y and (A× Y ) ∩ (B × Y ) = (A ∩ B)× Y. 218 Chapter 3 Cohomology (3) The long exact sequence of a pair. This is a triviality for k∗, but a few words of explanation are needed for h∗, where the desired exact sequence is obtained in two steps. For the ﬁrst step, tensor the long exact sequence of ordinary cohomology groups for a pair (X, A) with the free R module H n(Y ; R
), for a ﬁxed n. This yields another exact sequence because H n(Y ; R) is a direct sum of copies of R, so the result of tensoring an exact sequence with this direct sum is simply to produce a direct sum of copies of the exact sequence, which is again an exact sequence. The second step is to let n vary, taking a direct sum of the previously constructed exact sequences for each n, with the nth exact sequence shifted up by n dimensions. (4) Disjoint unions. Again this axiom obviously holds for k∗, but some justiﬁcation is required for h∗. What is needed is the algebraic fact that there is a canonifor R modules Mα and a ﬁnitely cal isomorphism generated free R module N. Since N is a direct product of ﬁnitely many copies Rβ of R, Mα ⊗R N is a direct product of corresponding copies Mαβ = Mα ⊗R Rβ of β Mαβ, which is obviously Mα and the desired relation becomes true. Mα ⊗R N α Mαβ ≈ ⊗R N ≈ α Mα Q Q α Q β Q Q Q α Finally there is naturality of µ to consider. Naturality with respect to maps between spaces is immediate from the naturality of cup products. Naturality with respect to coboundary maps in long exact sequences is commutativity of the following square: To check this, start with an element of the upper left product, represented by cocycles ϕ ∈ C k(A; R) and ψ ∈ C ℓ(Y ; R). Extend ϕ to a cochain ϕ ∈ C k(X; R). Then the pair (ϕ, ψ) maps rightward to (δϕ, ψ) and then downward to p♯ 2 (ψ). Going the other way around the square, (ϕ, ψ) maps downward to p♯ 2(ψ) and then 2 (ψ) over rightward to δ ⊔⊓ X × Y. Finally, δ 2 (ψ) extends p♯ 2 (ψ) since δψ = 0. 1 (ϕ) ` p♯ p♯ 1 (�
�) ` p♯ p♯ 1 (ϕ) ` p♯ 1 (δϕ) ` p♯ 1 (δϕ) ` p♯ 1 (ϕ)` p♯ since p♯ = p♯ 1 (ϕ) ` p♯ 2 (ψ) 2 (ψ) It is sometimes important to have a relative version of the K¨unneth formula in Theorem 3.15. The relative cross product is H ∗(X, A; R) ⊗R H ∗(Y, B; R) ×-----------------→ H ∗(X × Y, A× Y ∪ X × B; R) for CW pairs (X, A) and (Y, B), deﬁned just as in the absolute case by a× b = 2 (b) ∈ H ∗(X × Y, X × B; R). 1 (a) ∈ H ∗(X × Y, A× Y ; R) and p∗ 1 (a) ` p∗ p∗ 2 (b) where p∗ Theorem 3.18. For CW pairs (X, A) and (Y, B) the cross product homomorphism H ∗(X, A; R) ⊗R H ∗(Y, B; R)→H ∗(X × Y, A× Y ∪ X × B; R) is an isomorphism of rings if H k(Y, B; R) is a ﬁnitely generated free R module for each k. Cup Product Section 3.2 219 Proof: The case B = ∅ was covered in the course of the proof of the absolute case, so it suﬃces to deduce the case B ≠ ∅ from the case B = ∅. The following commutative diagram shows that collapsing B to a point reduces the proof to the case that B is a point: The lower map is an isomorphism since the quotient spaces (X × Y )/(A× Y ∪ X × B) / and A× (Y /B) ∪ X × (B/B) X × (Y /B) For the case that B is a point y0 ∈ Y we will use the following commutative diagram: are the same. Since y0 is a
retract of Y, the upper row of this diagram is a split short exact sequence. The lower row is the long exact sequence of a triple, and it too is a split short exact sequence since (X × y0, A× y0) is a retract of (X × Y, A× Y ). The middle and right cross product maps are isomorphisms by the case B = ∅ since H k(Y ; R) is a ﬁnitely generated free R module if H k(Y, y0; R) is. The ﬁve-lemma then implies that the ⊔⊓ left-hand cross product map is an isomorphism as well. The relative cross product for pairs (X, x0) and (Y, y0) gives a reduced cross product H ∗(X; R) ⊗R H ∗(Y ; R) ×-----------------→ H ∗(X ∧ Y ; R) e e e where X ∧ Y is the smash product X × Y /(X × {y0} ∪ {x0}× Y ). The preceding theoH ∗(X; R) or rem implies that this reduced cross product is an isomorphism if either H ∗(Y ; R) is a ﬁnitely generated free R module in each dimension. For example, we H n+k(X ∧ S k; R) via cross product with a generator have isomorphisms e kX of X, of H k(S k; R) ≈ R. The space X ∧ S k is the k fold reduced suspension e kX; R) derivable by so we see that the suspension isomorphisms Σ elementary exact sequence arguments can also be obtained via cross product with a H n(X; R) ≈ H n(X; R) ≈ H n+k( e e e e Σ generator of H ∗(S k; R). e 220 Chapter 3 Cohomology Spaces with Polynomial Cohomology Earlier in this section we mentioned that projective spaces provide examples of spaces whose cohomology rings are polynomial rings. Here is the precise statement: Theorem 3.19. H ∗(RPn; Z2) ≈ Z2[α]/(αn+1) and H ∗(RP∞; Z2) ≈ Z
2[α], where \|α\| = 1. In the complex case, H ∗(CPn; Z) ≈ Z[α]/(αn+1) and H ∗(CP∞; Z) ≈ Z[α] where \|α\| = 2. This turns out to be a quite important result, and it can be proved in a number of diﬀerent ways. The proof we give here uses the geometry of projective spaces to reduce the result to a very special case of the K¨unneth formula. Another proof using Poincar´e duality will be given in Example 3.40. A third proof is contained in Example 4D.5 as an application of the Gysin sequence. Proof: Let us do the case of RPn ﬁrst. To simplify notation we abbreviate RPn to P n and we let the coeﬃcient group Z2 be implicit. Since the inclusion P n−1֓P n induces an isomorphism on H i for i ≤ n − 1, it suﬃces by induction on n to show that the cup product of a generator of H n−1(P n) with a generator of H 1(P n) is a generator of H n(P n). It will be no more work to show more generally that the cup product of a generator of H i(P n) with a generator of H n−i(P n) is a generator of H n(P n). As a further notational aid, we let j = n − i, so i + j = n. The proof uses some of the geometric structure of P n. Recall that P n consists of nonzero vectors (x0, ···, xn) ∈ Rn+1 modulo multiplication by nonzero scalars. Inside P n is a copy of P i represented by vectors whose last j coordinates xi+1, ···, xn are zero. We also have a copy of P j represented by points whose ﬁrst i coordinates x0, ···, xi−1 are zero. The intersection P i ∩ P j is a single point p, represented by vectors whose only nonzero coordinate is xi. Let U be the subspace of P n represented by vectors with nonzero coordinate xi. Each point in U may be represented by a unique vector with xi =
1 and the other n coordinates arbitrary, so U is homeomorphic to Rn, with p corresponding to 0 under this homeomorphism. We can write this Rn as Ri × Rj, with Ri as the coordinates x0, ···, xi−1 and Rj as the coordinates xi+1, ···, xn. In the ﬁgure P n is represented as a disk with antipodal points of its boundary sphere identiﬁed to form a P n−1 ⊂ P n with U = P n − P n−1 the interior of the disk. Consider the diagram (i) Cup Product Section 3.2 221 which commutes by naturality of cup product. We will show that the four vertical maps are isomorphisms and that the lower cup product map takes generator cross generator to generator. Commutativity of the diagram will then imply that the upper cup product map also takes generator cross generator to generator. The lower map in the right column is an isomorphism by excision. For the upper map in this column, the fact that P n − {p} deformation retracts to a P n−1 gives an isomorphism H n(P n, P n −{p}) ≈ H n(P n, P n−1) via the ﬁve-lemma applied to the long exact sequences for these pairs. And H n(P n, P n−1) ≈ H n(P n) by cellular cohomology. To see that the vertical maps in the left column of (i) are isomorphisms we will use the following commutative diagram: (ii) If we can show all these maps are isomorphisms, then the same argument will apply with i and j interchanged, and the vertical maps in the left column of (i) will be isomorphisms. The left-hand square in (ii) consists of isomorphisms by cellular cohomology. The right-hand vertical map is obviously an isomorphism. The lower right horizontal map is an isomorphism by excision, and the map to the left of this is an isomorphism since P i − {p} deformation retracts onto P i−1. The remaining maps will be isomorphisms if the middle map in the upper row is an isomorphism. And this map is in fact an isomorphism because P n − P j deformation
retracts onto P i−1 by the following argument. The subspace P n − P j ⊂ P n consists of points represented by vectors v = (x0, ···, xn) with at least one of the coordinates x0, ···, xi−1 nonzero. The formula ft(v) = (x0, ···, xi−1, txi, ···, txn) for t decreasing from 1 to 0 gives a well-deﬁned deformation retraction of P n − P j onto P i−1 since ft(λv) = λft(v) for scalars λ ∈ R. The cup product map in the bottom row of (i) is equivalent to the cross product H i(Ii, ∂Ii)× H j(Ij, ∂Ij)→H n(In, ∂In), where the cross product of generators is a generator by the relative form of the K¨unneth formula in Theorem 3.18. Alternatively, if one wishes to use only the absolute K¨unneth formula, the cross product for cubes is equivalent to the cross product H i(S i)× H j(S j )→H n(S i × S j) by means of the quotient maps Ii→S i and Ij→S j collapsing the boundaries of the cubes to points. This ﬁnishes the proof for RPn. The case of RP∞ follows from this since the inclusion RPn ֓ RP∞ induces isomorphisms on H i(−; Z2) for i ≤ n by cellular cohomology. Complex projective spaces are handled in precisely the same way, using Z coef⊔⊓ ﬁcients and replacing each H k by H 2k and R by C. 222 Chapter 3 Cohomology There are also quaternionic projective spaces HPn and HP∞, deﬁned exactly as in the complex case, with CW structures of the form e0 ∪ e4 ∪ e8 ∪ ···. Associativity of quaternion multiplication is needed for the identiﬁcation v ∼ λv to be an equivalence relation, so the deﬁnition does not extend to octonionic projective spaces, though there is an octonionic projective plane OP2 de
ﬁned in Example 4.47. The cup product structure in quaternionic projective spaces is just like that in complex pro- jective spaces, except that the generator is 4 dimensional: H ∗(HP∞; Z) ≈ Z[α] and H ∗(HPn; Z) ≈ Z[α]/(αn+1), with \|α\| = 4 The same proof as in the real and complex cases works here as well. The cup product structure for RP∞ with Z coeﬃcients can easily be deduced from the cup product structure with Z2 coeﬃcients, as follows. In general, a ring homomorphism R→S induces a ring homomorphism H ∗(X, A; R)→H ∗(X, A; S). In the case of the projection Z→Z2 we get for RP∞ an induced chain map of cellular cochain complexes with Z and Z2 coeﬃcients: From this we see that the ring homomorphism H ∗(RP∞; Z)→H ∗(RP∞; Z2) is injective in positive dimensions, with image the even-dimensional part of H ∗(RP∞; Z2). Alternatively, this could be deduced from the universal coeﬃcient theorem. Hence we have H ∗(RP∞; Z) ≈ Z[α]/(2α) with \|α\| = 2. The cup product structure in H ∗(RPn; Z) can be computed in a similar fashion, though the description is a little cumbersome: H ∗(RP2k; Z) ≈ Z[α]/(2α, αk+1), \|α\| = 2 H ∗(RP2k+1; Z) ≈ Z[α, β]/(2α, αk+1, β2, αβ), \|α\| = 2, \|β\| = 2k + 1 Here β is a generator of H 2k+1(RP2k+1; Z) ≈ Z. From this calculation we see that the rings H ∗(RP2k+1; Z) and H ∗(RP2k ∨ S 2k+1; Z) are isomorphic, though with Z2 coeﬃcients this is no longer
true, as the generator α ∈ H 1(RP2k+1; Z2) has α2k+1 ≠ 0, while α2k+1 = 0 for the generator α ∈ H 1(RP2k ∨ S 2k+1; Z2). Example 3.20. Combining the calculation H ∗(RP∞; Z2) ≈ Z2[α] with the K¨unneth formula, we see that H ∗(RP∞ × RP∞; Z2) is isomorphic to Z2[α1] ⊗ Z2[α2], which is just the polynomial ring Z2[α1, α2]. More generally it follows by induction that for a product of n copies of RP∞, the Z2 cohomology is a polynomial ring in n variables. Similar remarks apply to CP∞ and HP∞ with coeﬃcients in Z or any commutative ring. Cup Product Section 3.2 223 The following theorem of Hopf is a nice algebraic application of the cup product structure in H ∗(RPn × RPn; Z2). Theorem 3.21. If Rn has the structure of a division algebra over the scalar ﬁeld R, then n must be a power of 2. Proof: For a division algebra structure on Rn the multiplication maps x ֏ ax and x ֏ xa are linear isomorphisms for each nonzero a, so the multiplication map Rn × Rn→Rn induces a map h : RPn−1× RPn−1→RPn−1 which is a homeomorphism when restricted to each subspace RPn−1 × {y} and {x}× RPn−1. The map h is continuous since it is a quotient of the multiplication map which is bilinear and hence continuous. The induced homomorphism h∗ on Z2 cohomology is a ring homomorphism Z2[α]/(αn)→Z2[α1, α2]/(αn 2 ) determined by the element h∗(α) = k1α1 + k2α2. The inclusion RPn−1 ֓ RPn−1 × RPn−1 onto the ﬁrst factor sends α1 to α and α2 to 0
, as one sees by composing with the projections of RPn−1 × RPn−1 onto its two factors. The fact that h restricts to a homeomorphism on the ﬁrst factor then implies that k1 is nonzero. Similarly k2 is nonzero, so since these coeﬃcients lie in Z2 we have h∗(α) = α1 + α2. 1, αn n k αk n k k P 2 ), so the coeﬃcient Since αn = 0 we must have h∗(αn) = 0, so (α1+α2)n = 1αn−k 2 = 0. This is an equation in the ring Z2[α1, α2]/(αn 1, αn must be zero in Z2 for all k in the range 0 < k < n. It is a rather easy number theory fact that this happens only when n is a power of 2. Namely, an obviously equivalent statement is that in the polynomial ring Z2[x], the equality (1+x)n = 1+xn holds only when n is a power of 2. To prove the latter statement, write n as a sum of powers of 2, n = n1 +···+nk with n1 < ··· < nk. Then (1 + x)n = (1 + x)n1 ··· (1 + x)nk = (1 + xn1 ) ··· (1 + xnk ) since squaring is an additive homomorphism with Z2 coeﬃcients. If one multiplies the product (1 + xn1 ) ··· (1 + xnk ) out, no terms combine or cancel since ni ≥ 2ni−1 for each i, and so the resulting polynomial has 2k terms. Thus if this polynomial equals 1 + xn we must have k = 1, which means that n is a power of 2. ⊔⊓ The same argument can be applied with C in place of R, to show that if Cn is a division algebra over C then = 0 for all k in the range 0 < k < n, but now we can use Z rather than Z2 coeﬃcients, so we deduce that n = 1. Thus there are no higher-dimensional division algebras
over C. This is assuming we are talking about n k ﬁnite-dimensional division algebras. For inﬁnite dimensions there is for example the ﬁeld of rational functions C(x). We saw in Theorem 3.19 that RP∞, CP∞, and HP∞ have cohomology rings that are polynomial algebras. We will describe now a construction for enlarging S 2n to a space J(S 2n) whose cohomology ring H ∗(J(S 2n); Z) is almost the polynomial ring Z[x] on a generator x of dimension 2n. And if we change from Z to Q coeﬃcients, then H ∗(J(S 2n); Q) is exactly the polynomial ring Q[x]. This construction, known 224 Chapter 3 Cohomology as the James reduced product, is also of interest because of its connections with loopspaces described in §4.J. b For a space X, let X k be the product of k copies of X. From the disjoint union k≥1 X k, let us form a quotient space J(X) by identifying (x1, ···, xi, ···, xk) with (x1, ···, xi, ···, xk) if xi = e, a chosen basepoint of X. Points of J(X) can thus ` be thought of as k tuples (x1, ···, xk), k ≥ 0, with no xi = e. Inside J(X) is the subspace Jm(X) consisting of the points (x1, ···, xk) with k ≤ m. This can be viewed as a quotient space of Xm under the identiﬁcations (x1, ···, xi, e, ···, xm) ∼ (x1, ···, e, xi, ···, xm). For example, J1(X) = X and J2(X) = X × X/(x, e) ∼ (e, x). If X is a CW complex with e a 0 cell, the quotient map Xm→Jm(X) glues together the m subcomplexes of the product complex Xm
where one coordinate is e. These glueings are by homeomorphisms taking cells onto cells, so Jm(X) inherits a CW structure from Xm. There are natural inclusions Jm(X) ⊂ Jm+1(X) as subcomplexes, and J(X) is the union of these subcomplexes, hence is also a CW complex. Proposition 3.22. For n > 0, H ∗ multiple of n. If n is even, the ith power of a generator of H n a generator of H in isomorphic as a graded ring to H ∗(S n; Z) ⊗ H ∗ J(S n); Z, for each i ≥ 1. When n is odd, H ∗ J(S 2n); Z J(S n); Z J(S n); Z. is i! times J(S n); Z is consists of a Z in each dimension a It follows that for n even, H ∗ can be identiﬁed with the subring of the polynomial ring Q[x] additively generated by the monomials xi/i!. This subring is is called a divided polynomial algebra and is denoted J(S n); Z isomorphic to Z[x] when n is even and to Z[x] ⊗ Z[x]. Thus H ∗(J(S n); Z Z[y] when n is odd. Γ Γ Γ Λ J(S n); Z Proof: Giving S n its usual CW structure, the resulting CW structure on J(S n) consists of exactly one cell in each dimension a multiple of n. If n > 1 we deduce immediately from cellular cohomology that H ∗ consists exactly of Z ’s in dimensions a multiple of n. For an alternative argument that works also when n = 1, consider the quotient map q : (S n)m→Jm(S n). This carries each cell of (S n)m homeomorphically onto a cell of Jm(S n). In particular q is a cellular map, taking k skeleton to k skeleton for each k, so q induces a chain map of cellular chain complexes. This chain map is surjective since each cell of Jm(S n) is the homeomorphic image of a cell of (S n)m. Hence the cellular boundary
maps for Jm(S n) will be trivial if they are trivial for (S n)m, as indeed they are since H ∗ is free with basis in one-to-one correspondence with the cells, by Theorem 3.15. (S n)m; Z We can compute cup products in H ∗ q∗. Let xk denote the generator of H kn by the cellular cocycle assigning the value 1 to the kn cell. Since q identiﬁes all the n cells of (S n)m to form the n cell of Jm(S n), we see from cellular cohomology that q∗(x1) is the sum α1 +···+αm of the generators of H n dual to the n cells of (S n)m. By the same reasoning we have q∗(xk) = ··· αik Jm(S n); Z Jm(S n); Z dual to the kn cell, represented (S n)m; Z αi1 i1<···<ik. by computing their images under P Cup Product Section 3.2 225 If n is even, the cup product structure in H ∗ is strictly commutative and H ∗ (S n)m; Z ≈ Z[α1, ···, αm]/(α2 1, ···, α2 (S n)m; Z m). Then we have q∗(xm 1 ) = (α1 + ··· + αm)m = m!α1 ··· αm = m!q∗(xm) Since q∗ is an isomorphism on Hmn this implies xm Jm(S n); Z. The inclusion Jm(S n) ֓ J(S n) induces isomorphisms on H i for i ≤ mn so we have xm as well, where x1 and xm are interpreted now as elements of H ∗ 1 = m!xm in H ∗ J(S n); Z 1 = m!xm in Hmn J(S n); Z. When n is odd we have x2 1 = 0 by commutativity, and it will suﬃce to prove the following two formulas: (a) x1x2m = x2m+1 in H ∗ J2m+1(S n); Z
(b) x2x2m−2 = mx2m in H ∗ J2m(S n); Z For (a) we apply q∗ and compute in the exterior algebra.. Z[α1, ···, α2m+1] : q∗(x1x2m) = αi α1 ··· αi ··· α2m+1 Λ Xi Xi αiα1 ··· = Xi b b αi ··· α2m+1 = (−1)i−1α1 ··· α2m+1 Xi The coeﬃcients in this last summation are +1, −1, ···, +1, so their sum is +1 and (a) follows. For (b) we have q∗(x2x2m−2) = αi1 αi2 α1 ··· αi1 ··· αi2 ··· α2m Xi1<i2 αi1 αi2 α1 ··· Xi1<i2 αi1 ··· αi2 b ··· α2m = b (−1)i1−1(−1)i2−2α1 ··· α2m = Xi1<i2 b b Xi1<i2 i1<i2 (−1)i1−1(−1)i2−2 for a ﬁxed i1 have i2 varying from The terms in the coeﬃcient i1 + 1 to 2m. These terms are +1, −1, ··· and there are 2m − i1 of them, so their sum is 0 if i1 is even and 1 if i1 is odd. Now letting i1 vary, it takes on the odd values 1, 3, ···, 2m − 1, so the whole summation reduces to m 1 ’s and we have the desired relation x2x2m−2 = mx2m. P ⊔⊓ Z[x] ⊂ Q[x], if we let xi = xi/i! then the multiplicative structure is given by In i+j xixj = xi+j. More generally, for a commutative ring R we could deﬁne R[x] i to be the free R module with basis x0 = 1, x1,
x2, ··· and multiplication deﬁned by Γ xixj = R[x]. Q[x] is just Q[x]. However, for R = Zp with p prime When R = Q it is clear that xi+j. The preceding proposition implies that H ∗ J(S 2n); R i+j i Γ ≈ something quite diﬀerent happens: There is an isomorphism [x] ≈ Zp[x1, xp, xp2, ···]/(x Zp p2, ···) = Zp[xpi ]/(x p pi ) Oi≥0 Γ as we show in §3.C, where we will also see that divided polynomial algebras are in a Γ certain sense dual to polynomial algebras. 226 Chapter 3 Cohomology The examples of projective spaces lead naturally to the following question: Given a coeﬃcient ring R and an integer d > 0, is there a space X having H ∗(X; R) ≈ R[α] with \|α\| = d? Historically, it took major advances in the theory to answer this simple- looking question. Here is a table giving all the possible values of d for some of R d Z Q Z2 Zp 2, 4 any even number 1, 2, 4 any even divisor of 2(p − 1) the most obvious and important choices of R, namely Z, Q, Z2, and Zp with p an odd prime. As we have seen, projective spaces give the examples for Z and Z2. Examples for Q are the spaces J(S d), and examples for Zp are constructed in §3.G. Showing that no other d ’s are possible takes considerably more work. The fact that d must be even when R ≠ Z2 is a consequence of the commutativity property of cup product. In Theorem 4L.9 and Corollary 4L.10 we will settle the case R = Z and show that d must be a power of 2 for R = Z2 and a power of p times an even divisor of 2(p − 1) for R = Zp, p odd. Ruling out the remaining cases is best done using K–theory, as in [VBKT] or the classical reference [Adams
& Atiyah 1966]. However there is one slightly anomalous case, R = Z2, d = 8, which must be treated by special arguments; see [Toda 1963]. It is an interesting fact that for each even d there exists a CW complex Xd which is simultaneously an example for all the admissible choices of coeﬃcients R in the table. Moreover, Xd can be chosen to have the simplest CW structure consistent with its cohomology, namely a single cell in each dimension a multiple of d. For example, we may take X2 = CP∞ and X4 = HP∞. The next space X6 would have H ∗(X6; Zp) ≈ Zp[α] for p = 7, 13, 19, 31, ···, primes of the form 3s + 1, the condition 6\|2(p − 1) being equivalent to p = 3s + 1. (By a famous theorem of Dirichlet there are inﬁnitely many primes in any such arithmetic progression.) Note that, in terms of Z coeﬃcients, Xd must have the property that for a generator α of H d(Xd; Z), each power αi is an integer ai times a generator of H di(Xd; Z), with ai ≠ 0 if H ∗(Xd; Q) ≈ Q[α] and ai relatively prime to p if H ∗(Xd; Zp) ≈ Zp[α]. A construction of Xd is given in [SSAT], or in the original source [Hoﬀman & Porter 1973]. One might also ask about realizing the truncated polynomial ring R[α]/(αn+1), in view of the examples provided by RPn, CPn, and HPn, leaving aside the trivial case n = 1 where spheres provide examples. The analysis for polynomial rings also settles which truncated polynomial rings are realizable; there are just a few more than for the full polynomial rings. There is also the question of realizing polynomial rings R[α1, ···, αn] with generators αi in speciﬁed dimensions di. Since R[α1, ···, αm] ⊗R R[β1, ···, βn] is equal to R
[α1, ···, αm, β1, ···, βn], the product of two spaces with polynomial cohomology is again a space with polynomial cohomology, assuming the number of polynomial generators is ﬁnite in each dimension. For example, the n fold product (CP∞)n has H ∗ ≈ Z[α1, ···, αn] with each αi 2 dimensional. Similarly, products of (CP∞)n; Z Cup Product Section 3.2 227 the spaces J(S di) realize all choices of even di ’s with Q coeﬃcients. However, with Z and Zp coeﬃcients, products of one-variable examples do not exhaust all the possibilities. As we show in §4.D, there are three other basic examples with Z coeﬃcients: 1. Generalizing the space CP∞ of complex lines through the origin in C∞, there is the Grassmann manifold Gn(C∞) of n dimensional vector subspaces of C∞, and this has H ∗(Gn(C∞); Z) ≈ Z[α1, ···, αn] with \|αi\| = 2i. This space is also known as BU(n), the ‘classifying space’ of the unitary group U(n). It is central to the study of vector bundles and K–theory. 2. Replacing C by H, there is the quaternionic Grassmann manifold Gn(H∞), also known as BSp(n), the classifying space for the symplectic group Sp(n), with H ∗(Gn(H∞); Z) ≈ Z[α1, ···, αn] with \|αi\| = 4i. 3. There is a classifying space BSU(n) for the special unitary group SU(n), whose cohomology is the same as for BU(n) but with the ﬁrst generator α1 omitted, so H ∗(BSU(n); Z) ≈ Z[α2, ···, αn] with \|αi\| = 2i. These examples and their products account for all the realizable polynomial cup product rings with Z coeﬃ
cients, according to a theorem in [Andersen & Grodal 2008]. The situation for Zp coeﬃcients is more complicated and will be discussed in §3.G. Polynomial algebras are examples of free graded commutative algebras, where ‘free’ means loosely ‘having no unnecessary relations’. In general, a free graded com- mutative algebra is a tensor product of single-generator free graded commutative algebras. The latter are either polynomial algebras R[α] on even-dimension generators α or quotients R[α]/(2α2) with α odd-dimensional. Note that if R is a ﬁeld then R[α]/(2α2) is either the exterior algebra R[α] if the characteristic of R is not 2, or the polynomial algebra R[α] otherwise. Every graded commutative algebra is a quotient of a free one, clearly. Λ Example 3.23: Subcomplexes of the n Torus. To give just a small hint of the endless variety of nonfree cup product algebras that can be realized, consider subcomplexes of the n torus T n, the product of n copies of S 1. Here we give S 1 its standard minimal cell structure and T n the resulting product cell structure. We know that H ∗(T n; Z) Z[α1, ···, αn], with the monomial αi1 corresponding is the exterior algebra via cellular cohomology to the k cell e1. So if we pass to a subcomplex i1 X ⊂ T n by omitting certain cells, then H ∗(X; Z) is the quotient of Z[α1, ···, αn] obtained by setting the monomials corresponding to the omitted cells equal to zero. Λ Z[α1, ···, αn], the ideal generated by the monomials corresponding to the ‘minimal’ omitted cells, Since we are dealing with rings, we are factoring out by an ideal in × ··· × e1 ik ··· αik Λ those whose boundary is entirely contained in X. For example, if we take X to be the subcomplex of T 3 obtained by deleting the cells e1 3, then H ∗(X; Z)
≈ 3 and e1 Z[α1, α2, α3]/(α2α3). 1 × e1 2 × e1 2 × e1 Λ Λ 228 Chapter 3 Cohomology we get a simplex n−1 ∩ q−1(X). This is a subcomplex of How many diﬀerent subcomplexes of T n are there? To each subcomplex X ⊂ T n we can associate a ﬁnite simplicial complex CX by the following procedure. View T n as the quotient of the n cube In = [0, 1]n ⊂ Rn obtained by identifying opposite If we intersect In with the hyperplane x1 + ··· + xn = ε for small ε > 0, faces. n−1. Then for q : In→T n the quotient map, we take CX to be n−1 whose k simplices correspond exactly to the (k + 1) cells of X. In this way we get a one-to-one correspondence between ∆ ∆ n−1. Every simplicial complex with subcomplexes X ⊂ T n and subcomplexes CX ⊂ n−1, so we see that T n has quite a large number n vertices is a subcomplex of of subcomplexes if n is not too small. The cohomology rings H ∗(X; Z) are of a type that was completely classiﬁed in [Gubeladze 1998], Theorem 3.1, and from this classiﬁcation it follows that the ring H ∗(X; Z) (or even H ∗(X; Z2) ) determines the subcomplex X uniquely, up to permutation of the n circle factors of T n. ∆ ∆ ∆ More elaborate examples could be produced by looking at subcomplexes of the product of n copies of CP∞. polynomial rings modulo ideals generated by monomials, and it is again true that In this case the cohomology rings are isomorphic to the cohomology ring determines the subcomplex up to permutation of factors. How- ever, these cohomology rings are still a whole lot less complicated than the general case, where one takes free algebras modulo ideals generated by arbitrary polynomials having all their terms of the
same dimension. Let us conclude this section with an example of a cohomology ring that is not too far removed from a polynomial ring. Example 3.24: Cohen–Macaulay Rings. Let X be the quotient space CP∞/CPn−1. The quotient map CP∞→X induces an injection H ∗(X; Z)→H ∗(CP∞; Z) embedding H ∗(X; Z) in Z[α] as the subring generated by 1, αn, αn+1, ···. If we view this subring as a module over Z[αn], it is free with basis {1, αn+1, αn+2, ···, α2n−1}. Thus H ∗(X; Z) is an example of a Cohen–Macaulay ring: a ring containing a polynomial subring over which it is a ﬁnitely generated free module. While polynomial cup product rings are rather rare, Cohen–Macauley cup product rings occur much more frequently. Exercises 1. Assuming as known the cup product structure on the torus S 1 × S 1, compute the cup product structure in H ∗(Mg) for Mg the closed orientable surface of genus g by using the quotient map from Mg to a wedge sum of g tori, shown below. Cup Product Section 3.2 229 2. Using the cup product H k(X, A; R)× H ℓ(X, B; R)→H k+ℓ(X, A ∪ B; R), show that if X is the union of contractible open subsets A and B, then all cup products of positive-dimensional classes in H ∗(X; R) are zero. This applies in particular if X is a suspension. Generalize to the situation that X is the union of n contractible open f (x) − f (−x) subsets, to show that all n fold cup products of positive-dimensional classes are zero. (a) Using the cup product structure, show there is no map RPn→RPm inducing 3. a nontrivial map H 1(RPm; Z2)→H 1(RPn; Z2) if n > m. What is the corresponding result for maps CPn→CPm? (b
) Prove the Borsuk–Ulam theorem by the following argument. Suppose on the contrary that f : S n→Rn satisﬁes f (x) ≠ f (−x) for all x. Then deﬁne g : S n→S n−1 by g(x) = /\|f (x) − f (−x)\|, so g(−x) = −g(x) and g induces a map RPn→RPn−1. Show that part (a) applies to this map. 4. Apply the Lefschetz ﬁxed point theorem to show that every map f : CPn→CPn has a ﬁxed point if n is even, using the fact that f ∗ : H ∗(CPn; Z)→H ∗(CPn; Z) is a ring homomorphism. When n is odd show there is a ﬁxed point unless f ∗(α) = −α, for α a generator of H 2(CPn; Z). [See Exercise 3 in §2.C for an example of a map without ﬁxed points in this exceptional case.] 5. Show the ring H ∗(RP∞; Z2k) is isomorphic to Z2k[α, β]/(2α, 2β, α2 − kβ) where \|α\| = 1 and \|β\| = 2. [Use the coeﬃcient map Z2k→Z2 and the proof of Theorem 3.19.] 6. Use cup products to compute the map H ∗(CPn; Z)→H ∗(CPn; Z) induced by the map CPn→CPn that is a quotient of the map Cn+1→Cn+1 raising each coordinate to the d th power, (z0, ···, zn) ֏ (zd n), for a ﬁxed integer d > 0. [First do the case n = 1.] 7. Use cup products to show that RP3 is not homotopy equivalent to RP2 ∨ S 3. 8. Let X be CP2 with a cell e3 attached by a map S 2→CP1 ⊂ CP2 of degree p, and let Y = M(Zp, 2) ∨ S
4. Thus X and Y have the same 3 skeleton but diﬀer in the way their 4 cells are attached. Show that X and Y have isomorphic cohomology rings with Z coeﬃcients but not with Zp coeﬃcients. 9. Show that if Hn(X; Z) is free for each n, then H ∗(X; Zp) and H ∗(X; Z) ⊗ Zp are isomorphic as rings, so in particular the ring structure with Z coeﬃcients determines the ring structure with Zp coeﬃcients. 10. Show that the cross product map H ∗(X; Z) ⊗ H ∗(Y ; Z)→H ∗(X × Y ; Z) is not an isomorphism if X and Y are inﬁnite discrete sets. [This shows the necessity of the 0, ···, zd hypothesis of ﬁnite generation in Theorem 3.15.] 11. Using cup products, show that every map S k+ℓ→S k × S ℓ induces the trivial homomorphism Hk+ℓ(S k+ℓ)→Hk+ℓ(S k × S ℓ), assuming k > 0 and ℓ > 0. 12. Show that the spaces (S 1 × CP∞)/(S 1 × {x0}) and S 3 × CP∞ have isomorphic cohomology rings with Z or any other coeﬃcients. [An exercise for §4.L is to show these two spaces are not homotopy equivalent.] 230 Chapter 3 Cohomology 13. Describe H ∗(CP∞/CP1; Z) as a ring with ﬁnitely many multiplicative generators. How does this ring compare with H ∗(S 6 × HP∞; Z)? 14. Let q : RP∞→CP∞ be the natural quotient map obtained by regarding both spaces as quotients of S ∞, modulo multiplication by real scalars in one case and complex scalars in the other. Show that the induced map q∗ : H ∗(CP∞; Z)→H ∗(RP∞; Z) is surjective in even dimensions by showing ﬁr
st by a geometric argument that the restriction q : RP2→CP1 induces a surjection on H 2 and then appealing to cup product structures. Next, form a quotient space X of RP∞∐CPn by identifying each point x ∈ RP2n with q(x) ∈ CPn. Show there are ring isomorphisms H ∗(X; Z) ≈ Z[α]/(2αn+1) and H ∗(X; Z2) ≈ Z2[α, β]/(β2 − α2n+1), where \|α\| = 2 and \|β\| = 2n + 1. Make a similar construction and analysis for the quotient map q : CP∞→HP∞. 15. For a ﬁxed coeﬃcient ﬁeld F, deﬁne the Poincar´e series of a space X to be i aiti where ai is the dimension of H i(X; F ) as a the formal power series p(t) = vector space over F, assuming this dimension is ﬁnite for all i. Show that p(X × Y ) = p(X)p(Y ). Compute the Poincar´e series for S n, RPn, RP∞, CPn, CP∞, and the spaces in the preceding three exercises. 16. Show that if X and Y are ﬁnite CW complexes such that H ∗(X; Z) and H ∗(Y ; Z) contain no elements of order a power of a given prime p, then the same is true for X × Y. [Apply Theorem 3.15 with coeﬃcients in various ﬁelds.] P 17. [This has now been incorporated into Proposition 3.22.] 18. For the closed orientable surface M of genus g ≥ 1, show that for each nonzero α ∈ H 1(M; Z) there exists β ∈ H 1(M; Z) with αβ ≠ 0. Deduce that M is not homotopy equivalent to a wedge sum X ∨ Y of CW complexes with nontrivial reduced homology. Do the same for closed nonorientable surfaces using cohomology with Z2 coeﬃcients. Algebraic topology is most often concerned with properties of
spaces that depend only on homotopy type, so local topological properties do not play much of a role. Digressing somewhat from this viewpoint, we study in this section a class of spaces whose most prominent feature is their local topology, namely manifolds, which are locally homeomorphic to Rn. It is somewhat miraculous that just this local homogeneity property, together with global compactness, is enough to impose a strong symmetry on the homology and cohomology groups of such spaces, as well as strong nontriviality of cup products. This is the Poincar´e duality theorem, one of the earliest theorems in the subject. In fact, Poincar´e’s original work on the duality property came before homology and cohomology had even been properly deﬁned, and it took many Poincar´e Duality Section 3.3 231 years for the concepts of homology and cohomology to be reﬁned suﬃciently to put Poincar´e duality on a ﬁrm footing. Let us begin with some deﬁnitions. A manifold of dimension n, or more concisely an n manifold, is a Hausdorﬀ space M in which each point has an open neighborhood homeomorphic to Rn. The dimension of M is intrinsically characterized by the fact that for x ∈ M, the local homology group Hi(M, M −{x}; Z) is nonzero only for i = n : Hi(M, M − {x}; Z) ≈ Hi(Rn, Rn − {0}; Z) ≈ by excision since Rn is contractible Hi−1(Rn − {0}; Z) Hi−1(S n−1; Z) e ≈ since Rn − {0} ≃ S n−1 A compact manifold is called closed, to distinguish it from the more general notion e of a compact manifold with boundary, considered later in this section. For example S n is a closed manifold, as are RPn and lens spaces since they have S n as a covering space. Another closed manifold is CPn. This is compact since it is a quotient space of S 2n+1, and the manifold property is satisﬁed since there is an open cover by subsets homeomorphic to R2n, the sets Ui = { [z0
, ···, zn] ∈ CPn \| zi = 1 }. The same reasoning applies also for quaternionic projective spaces. Further examples of closed manifolds can be generated from these using the obvious fact that the product of closed manifolds of dimensions m and n is a closed manifold of dimension m + n. Poincar´e duality in its most primitive form asserts that for a closed orientable manifold M of dimension n, there are isomorphisms Hk(M; Z) ≈ H n−k(M; Z) for all k. Implicit here is the convention that homology and cohomology groups of neg- ative dimension are zero, so the duality statement includes the fact that all the non- trivial homology and cohomology of M lies in the dimension range from 0 to n. The deﬁnition of ‘orientable’ will be given below. Without the orientability hypothesis there is a weaker statement that Hk(M; Z2) ≈ H n−k(M; Z2) for all k. As we show in Corollaries A.8 and A.9 in the Appendix, the homology groups of a closed manifold are all ﬁnitely generated. So via the universal coeﬃcient theorem, Poincar´e duality for a closed orientable n manifold M can be stated just in terms of homology: Modulo their torsion subgroups, Hk(M; Z) and Hn−k(M; Z) are isomorphic, and the torsion subgroups of Hk(M; Z) and Hn−k−1(M; Z) are isomorphic. However, the statement in terms of cohomology is really more natural. Poincar´e duality thus expresses a certain symmetry in the homology of closed orientable manifolds. For example, consider the n dimensional torus T n, the product of n circles. By induction on n it follows from the K¨unneth formula, or from the easy special case Hi(X × S 1; Z) ≈ Hi(X; Z)⊕ Hi−1(X; Z) which was an exercise in §2.2, that Hk(T n; Z) is isomorphic to the direct sum of copies of Z. So Poincar´e duality is reﬂected
in the relation. The reader might also check that Poincar´e n k = n k n n−k duality is consistent with our calculations of the homology of projective spaces and lens spaces, which are all orientable except for RPn with n even. 232 Chapter 3 Cohomology For many manifolds there is a very nice geometric proof of Poincar´e duality using the notion of dual cell structures. The germ of this idea can be traced back to the ﬁve regular Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Each of these polyhedra has a dual polyhedron whose vertices are the center points of the faces of the given polyhedron. Thus the dual of the cube is the octahedron, and vice versa. Similarly the dodecahedron and icosahedron are dual to each other, and the tetrahedron is its own dual. One can regard each of these polyhedra as deﬁning a cell structure C on S 2 with a dual cell structure C ∗ determined by the dual polyhedron. Each vertex of C lies in a dual 2 cell of C ∗, each edge of C crosses a dual edge of C ∗, and each 2 cell of C contains a dual vertex of C ∗. The ﬁrst ﬁgure at the right shows the case of the cube and octahe- dron. There is no need to restrict to regular polyhedra here, and we can generalize further by replacing S 2 by any surface. A portion of a more-or-less random pair of dual cell structures is shown in the second ﬁgure. On the torus, if we lift a dual pair of cell structures to the universal cover R2, we get a dual pair of periodic tilings of the plane, as in the next three ﬁgures. The last two ﬁgures show that the standard CW structure on the sur- face of genus g, obtained from a 4g gon by identifying edges via the product of commutators [a1, b1] ··· [ag, bg], is homeomorphic to its own dual. Given a pair of dual cell structures C and C ∗ on a closed surface M, the pairing of cells with dual cells gives identiﬁc
ations of cellular chain groups C ∗ 0 = C2, 1 = C1, and C ∗ C ∗ 2 = C0. If we use Z coeﬃcients these identiﬁcations are not quite canonical since there is an ambiguity of sign for each cell, the choice of a generator for the corresponding Z summand of the cellular chain complex. We can avoid this ambiguity by considering the simpler situation of Z2 coeﬃcients, where the identiﬁcations Ci = C ∗ 2−i are completely canonical. The key observation now is that under these identiﬁcations, the cellular boundary map ∂ : Ci→Ci−1 becomes the cellular coboundary map δ : C ∗ 2−i+1 since ∂ assigns to a cell the sum of the cells which are faces of it, while δ assigns to a cell the sum of the cells of which it is a face. Thus Hi(C; Z2) ≈ H 2−i(C ∗; Z2), and hence Hi(M; Z2) ≈ H 2−i(M; Z2) since C and C ∗ are cell structures on the same surface M. 2−i→C ∗ Poincar´e Duality Section 3.3 233 To reﬁne this argument to Z coeﬃcients the problem of signs must be addressed. After analyzing the situation more closely, one sees that if M is orientable, it is possible to make consistent choices of orientations of all the cells of C and C ∗ so that the boundary maps in C agree with the coboundary maps in C ∗, and therefore one gets Hi(C; Z) ≈ H 2−i(C ∗; Z), hence Hi(M; Z) ≈ H 2−i(M; Z). For manifolds of higher dimension the situation is entirely analogous. One would consider dual cell structures C and C ∗ on a closed n manifold M, each i cell of C being dual to a unique (n−i) cell of C ∗ which it intersects in one point ‘transversely’. For example on the 3 dimensional torus S 1 × S 1 × S 1 one could take the standard cell structure lifting to the decomposition of the universal cover R3 into cubes with vertices at the integer
lattice points Z3, and then the dual cell structure is obtained by translating this by the vector (1/2, 1/2, 1/2). Each edge in either cell structure then has a dual 2 cell which it pierces orthogonally, and each vertex lies in a dual 3 cell. All the manifolds one commonly meets, for example all diﬀerentiable manifolds, have dually paired cell structures with the properties needed to carry out the proof of Poincar´e duality we have just sketched. However, to construct these cell structures requires a certain amount of manifold theory. To avoid this, and to get a theorem that applies to all manifolds, we will take a completely diﬀerent approach, using algebraic topology to replace the geometry of dual cell structures. Orientations and Homology Let us consider the question of how one might deﬁne orientability for manifolds. First there is the local question: What is an orientation of Rn? Whatever an orientation of Rn is, it should have the property that it is preserved under rotations and reversed by reﬂections. For example, in R2 the notions of ‘clockwise’ and ‘counterclockwise’ certainly have this property, as do ‘right-handed’ and ‘left-handed’ in R3. We shall take the viewpoint that this property is what characterizes orientations, so anything satisfying the property can be regarded as an orientation. With this in mind, we propose the following as an algebraic-topological deﬁnition: An orientation of Rn at a point x is a choice of generator of the inﬁnite cyclic group Hn(Rn, Rn − {x}), where the absence of a coeﬃcient group from the notation means that we take coeﬃcients in Z. To verify that the characteristic property of orientations is satisﬁed we use the isomorphisms Hn(Rn, Rn − {x}) ≈ Hn−1(Rn − {x}) ≈ Hn−1(S n−1) where S n−1 is a sphere centered at x. Since these isomorphisms are natural, and rotations of S n−1 have degree 1, being homotopic to the identity, while reﬂections
have degree −1, we see that a rotation ρ of Rn ﬁxing x takes a generator α of Hn(Rn, Rn − {x}) to itself, ρ∗(α) = α, while a reﬂection takes α to −α. Note that with this deﬁnition, an orientation of Rn at a point x determines an orientation at every other point y via the canonical isomorphisms Hn(Rn, Rn−{x}) ≈ Hn(Rn, Rn − B) ≈ Hn(Rn, Rn − {y}) where B is any ball containing both x and y. 234 Chapter 3 Cohomology An advantage of this deﬁnition of local orientation is that it can be applied to any n dimensional manifold M : A local orientation of M at a point x is a choice of generator µx of the inﬁnite cyclic group Hn(M, M − {x}). Notational Convention. In what follows we will very often be looking at homology groups of the form Hn(X, X − A). To simplify notation we will write Hn(X, X − A) as Hn(X \|\| A), or more generally Hn(X \|\| A; G) if a coeﬃcient group G needs to be speciﬁed. By excision, Hn(X \|\| A) depends only on a neighborhood of the closure of A in X, so it makes sense to view Hn(X \|\| A) as local homology of X at A. Having settled what local orientations at points of a manifold are, a global orien- tation ought to be ‘a consistent choice of local orientations at all points’. We make this precise by the following deﬁnition. An orientation of an n dimensional manifold M is a function x ֏ µx assigning to each x ∈ M a local orientation µx ∈ Hn(M \|\| x), satisfying the ‘local consistency’ condition that each x ∈ M has a neighborhood Rn ⊂ M containing an open ball B of ﬁnite radius about x such that all the local orientations µy at points y ∈ B are the images of one generator µB of Hn(M \|\| B) �
� Hn(Rn \|\| B) under the natural maps Hn(M \|\| B)→Hn(M \|\| y). If an orientation exists for M, then M is called orientable. Every manifold M has an orientable two-sheeted covering space M. For example, RP2 is covered by S 2, and the Klein bottle has the torus as a two-sheeted covering space. The general construction goes as follows. As a set, let f M = µx \|\|\|\| x ∈ M and µx is a local orientation of M at x f M→M, and we wish to topologize The map µx ֏ x deﬁnes a two-to-one surjection M to make this a covering space projection. Given an open ball B ⊂ Rn ⊂ M of ﬁnite radius and a generator µB ∈ Hn(M \|\| B), let U(µB ) be the set of all µx ∈ M such that f x ∈ B and µx is the image of µB under the natural map Hn(M \|\| B)→Hn(M \|\| x). It is easy to check that these sets U(µB) form a basis for a topology on M, and that the M is orientable since each point projection M \|\| µx) corµx ∈ Hn( µx ∈ M \|\| µx) ≈ Hn(U(µB ) \|\| µx) ≈ Hn(B \|\| x), responding to µx under the isomorphisms Hn( and by construction these local orientations satisfy the local consistency condition M→M is a covering space. The manifold M has a canonical local orientation given by the element f f f f f f f e necessary to deﬁne a global orientation. f Proposition 3.25. If M is connected, then M is orientable iﬀ M has two components. In particular, M is orientable if it is simply-connected, or more generally if π1(M) has no subgroup of index two. f The ﬁrst statement is a formulation of the intuitive notion of nonorientability as being able to go around some closed loop and come back with the opposite orientation, M→M this corresponds to a loop in M that lifts since in terms of the covering space f f
Poincar´e Duality Section 3.3 235 to a path in M connecting two distinct points with the same image in M. The existence of such paths is equivalent to M being connected. f Proof: If M is connected, covering space of M. If it has two components, they are each mapped homeomorphi- M has either one or two components since it is a two-sheeted f cally to M by the covering projection, so M is orientable, being homeomorphic to a component of the orientable manifold M. Conversely, if M is orientable, it has exactly two orientations since it is connected, and each of these orientations deﬁnes a component of M. The last statement of the proposition follows since connected two-sheeted covering spaces of M correspond to index-two subgroups of π1(M), by f ⊔⊓ the classiﬁcation of covering spaces. f f The covering space M→M can be embedded in a larger covering space MZ→M where MZ consists of all elements αx ∈ Hn(M \|\| x) as x ranges over M. As before, we topologize MZ via the basis of sets U(αB ) consisting of αx ’s with x ∈ B and αx the image of an element αB ∈ Hn(M \|\| B) under the map Hn(M \|\| B)→Hn(M \|\| x). The covering space MZ→M is inﬁnite-sheeted since for ﬁxed x ∈ M, the αx ’s range over the inﬁnite cyclic group Hn(M \|\| x). Restricting αx to be zero, we get a copy M0 of M M, k = 1, 2, ···, in MZ. The rest of MZ consists of an inﬁnite sequence of copies Mk of where Mk consists of the αx ’s that are k times either generator of Hn(M \|\| x). A continuous map M→MZ of the form x ֏ αx ∈ Hn(M \|\| x) is called a section of the covering space. An orientation of M is the same thing as a section x ֏ µx such that µx is a generator of Hn(M \|\| x) for each
x. f One can generalize the deﬁnition of orientation by replacing the coeﬃcient group Z by any commutative ring R with identity. Then an R orientation of M assigns to each x ∈ M a generator of Hn(M \|\| x; R) ≈ R, subject to the corresponding local consistency condition, where a ‘generator’ of R is an element u such that Ru = R. Since we assume R has an identity element, this is equivalent to saying that u is a unit, an invertible element of R. The deﬁnition of the covering space MZ generalizes immediately to a covering space MR→M, and an R orientation is a section of this covering space whose value at each x ∈ M is a generator of Hn(M \|\| x; R). The structure of MR is easy to describe. In view of the canonical isomorphism Hn(M \|\| x; R) ≈ Hn(M \|\| x) ⊗ R, each r ∈ R determines a subcovering space Mr of MR consisting of the points ±µx ⊗ r ∈ Hn(M \|\| x; R) for µx a generator of Hn(M \|\| x). If r has order 2 in R then r = −r so Mr is just a copy of M, and otherwise Mr is isomorphic to the two-sheeted cover M. The covering space MR is the union of these Mr ’s, which are disjoint except for the equality Mr = M−r. In particular we see that an orientable manifold is R orientable for all R, while f a nonorientable manifold is R orientable iﬀ R contains a unit of order 2, which is equivalent to having 2 = 0 in R. Thus every manifold is Z2 orientable. In practice this means that the two most important cases are R = Z and R = Z2. In what follows 236 Chapter 3 Cohomology the reader should keep these two cases foremost in mind, but we will usually state results for a general R. The orientability of a closed manifold is reﬂected in the structure of its homology, according to the following result. Theorem 3.26. Let M be a closed connected n manifold. Then : (a) If M is R orientable, the map Hn(M; R)→H
n(M \|\| x; R) ≈ R is an isomorphism for all x ∈ M. (b) If M is not R orientable, the map Hn(M; R)→Hn(M \|\| x; R) ≈ R is injective with image { r ∈ R \| 2r = 0 } for all x ∈ M. (c) Hi(M; R) = 0 for i > n. In particular, Hn(M; Z) is Z or 0 depending on whether M is orientable or not, and in either case Hn(M; Z2) = Z2. An element of Hn(M; R) whose image in Hn(M \|\| x; R) is a generator for all x is called a fundamental class for M with coeﬃcients in R. By the theorem, a fundamen- tal class exists if M is closed and R orientable. To show that the converse is also true, let µ ∈ Hn(M; R) be a fundamental class and let µx denote its image in Hn(M \|\| x; R). The function x ֏ µx is then an R orientation since the map Hn(M; R)→Hn(M \|\| x; R) factors through Hn(M \|\| B; R) for B any open ball in M containing x. Furthermore, M must be compact since µx can only be nonzero for x in the image of a cycle representing µ, and this image is compact. In view of these remarks a fundamental class could also be called an orientation class for M. The theorem will follow fairly easily from a more technical statement: Lemma 3.27. Let M be a manifold of dimension n and let A ⊂ M be a compact subset. Then : (a) If x ֏ αx is a section of the covering space MR→M, then there is a unique class αA ∈ Hn(M \|\| A; R) whose image in Hn(M \|\| x; R) is αx for all x ∈ A. (b) Hi(M \|\| A; R) = 0 for i > n. To deduce the theorem from this, choose A = M, a compact set by assumption. theorem, let Part (c) of the theorem is immediate from (b) of the lemma. To obtain (a)
and (b) of the R(M) be the set of sections of MR→M. The sum of two sections is a R(M) is an R module. There section, and a scalar multiple of a section is a section, so is a homomorphism Hn(M; R)→ R(M) sending a class α to the section x ֏ αx, where αx is the image of α under the map Hn(M; R)→Hn(M \|\| x; R). Part (a) of the lemma asserts that this homomorphism is an isomorphism. If M is connected, each Γ Γ Γ section is uniquely determined by its value at one point, so statements (a) and (b) of the theorem are apparent from the earlier discussion of the structure of MR. ⊔⊓ Proof of 3.27: The coeﬃcient ring R will play no special role in the argument so we shall omit it from the notation. We break the proof up into four steps. Poincar´e Duality Section 3.3 237 (1) First we observe that if the lemma is true for compact sets A, B, and A ∩ B, then it is true for A ∪ B. To see this, consider the Mayer–Vietoris sequence 0 -→ Hn(M \|\| A ∪ B) -----→ Hn(M \|\| A) ⊕ Hn(M \|\| B) -----→ Hn(M \|\| A ∩ B) Φ Ψ Ψ Ψ Φ Φ is is (α) = (α, −α) and Here the zero on the left comes from the assumption that Hn+1(M \|\| A ∩ B) = 0. The (α, β) = α + β, where we omit notation for map maps on homology induced by inclusion. The terms Hi(M \|\| A ∪ B) farther to the left in this sequence are sandwiched between groups that are zero by assumption, so Hi(M \|\| A ∪ B) = 0 for i > n. This gives (b). For the existence half of (a), if x ֏ αx is a section, the hypothesis gives unique classes αA ∈ Hn(M \|\| A), αB ∈ Hn(M \|\| B), and αA∩B
∈ Hn(M \|\| A ∩ B) having image αx for all x in A, B, or A ∩ B respectively. The images of αA and αB in Hn(M \|\| A ∩ B) satisfy the deﬁning property of αA∩B, hence must equal αA∩B. Exactness of the sequence then implies that (αA, −αB) = (αA∪B ) for some αA∪B ∈ Hn(M \|\| A ∪ B). This means that αA∪B maps to αA and αB, so αA∪B has image αx for all x ∈ A ∪ B since αA and αB have this property. To see that αA∪B is unique, observe that if a class α ∈ Hn(M \|\| A ∪ B) has image zero in Hn(M \|\| x) for all x ∈ A ∪ B, then its images in Hn(M \|\| A) and Hn(M \|\| B) have the same property, hence are zero by hypothesis, so α itself must be zero since is injective. Uniqueness of αA∪B follows by applying this observation to the diﬀerence between two choices for αA∪B. Φ Φ (2) A compact set A ⊂ M can be written as the union of ﬁnitely many compact sets A1, ···, Am each contained in an open Rn ⊂ M. We now apply (1) to A1 ∪ ··· ∪ Am−1 and Am. The intersection of these two sets is (A1 ∩ Am) ∪ ··· ∪ (Am−1 ∩ Am), a union of m − 1 compact sets each contained in an open Rn ⊂ M. By induction on m this gives a reduction to the case m = 1, so A ⊂ Rn ⊂ M. (3) When A ⊂ Rn ⊂ M and A is a union of convex compact sets A1, ···, Am in Rn, an inductive argument as in (2) reduces to the case that A itself is convex. When A is convex the result is evident since the map Hi(M \|\| A)→Hi(M \|\| x)
is equivalent to Hi(Rn \|\| A)→Hi(Rn \|\| x) by excision, and the latter map is an isomorphism for any x ∈ A, as both Rn − A and Rn − {x} deformation retract onto a sphere centered at x. (4) For an arbitrary compact set A ⊂ Rn ⊂ M let α ∈ Hi(M \|\| A) be represented by a relative cycle z in Rn with ∂z in Rn − A, and let C be the union of the images of the singular simplices in ∂z. Since C is compact, it has a positive distance δ from A in Rn. We can cover A by ﬁnitely many closed balls in Rn of radius less than δ centered at points of A. Let K be the union of these balls, so K is disjoint from C. The relative cycle z deﬁnes an element αK ∈ Hi(M \|\| K) mapping to the given α ∈ Hi(M \|\| A). If i > n then by (3) we have Hi(M \|\| K) = 0, so αK = 0, which implies α = 0 and hence Hi(M \|\| A) = 0. For the uniqueness half of (a) when i = n it suﬃces to show that α = 0 if αx is zero in Hn(M \|\| x) for all x ∈ A. Since K is a union of balls B meeting A and Hn(M \|\| B)→Hn(M \|\| x) is an isomorphism for all x ∈ B, it follows that αK maps to 238 Chapter 3 Cohomology zero in Hn(M \|\| x) for each x ∈ K since this is true when x ∈ A. Applying step (3) to the zero section of MR→M we conclude that αK is zero, hence also α. This ﬁnishes the uniqueness statement in (a). The existence statement is easy since we can let αA be the image of the element αB associated to any ball B with A ⊂ B ⊂ Rn. ⊔⊓ ∆ For a closed n manifold having the structure of a complex there is a more explicit construction for a fundamental class. Consider the case of Z coeﬃcients. In P simplicial hom
ology a fundamental class must be represented by some linear comi kiσi of the n simplices σi of M. The condition that the fundamental bination class maps to a generator of Hn(M \|\| x; Z) for points x in the interiors of the σi ’s means that each coeﬃcient ki must be ±1. The ki ’s must also be such that i kiσi is a cycle. This implies that if σi and σj share a common (n − 1) dimensional face, then ki determines kj and vice versa. Analyzing the situation more closely, one can i kiσi a cycle is possible iﬀ M is show that a choice of signs for the ki ’s making i kiσi deﬁnes a fundaorientable, and if such a choice is possible, then the cycle mental class. With Z2 coeﬃcients there is no issue of signs, and i σi always deﬁnes a fundamental class. P P P P Some information about Hn−1(M) can also be squeezed out of the preceding theorem: Corollary 3.28. If M is a closed connected n manifold, the torsion subgroup of Hn−1(M; Z) is trivial if M is orientable and Z2 if M is nonorientable. Proof: This is an application of the universal coeﬃcient theorem for homology, using the fact that the homology groups of M are ﬁnitely generated, from Corollaries A.8 and A.9 in the Appendix. In the orientable case, if Hn−1(M; Z) contained torsion, then for some prime p, Hn(M; Zp) would be larger than the Zp coming from Hn(M; Z). In the nonorientable case, Hn(M; Zm) is either Z2 or 0 depending on whether m is even or odd. This forces the torsion subgroup of Hn−1(M; Z) to be Z2. ⊔⊓ The reader who is familiar with Bockstein homomorphisms, which are discussed in §3.E, will recognize that the Z2 in Hn−1(M; Z) in the nonorientable case is the image of the Bock
stein homomorphism Hn(M; Z2)→Hn−1(M; Z) coming from the short exact sequence of coeﬃcient groups 0→Z→Z→Z2→0. The structure of Hn(M; G) and Hn−1(M; G) for a closed connected n manifold M can be explained very nicely in terms of cellular homology when M has a CW structure with a single n cell, which is the case for a large number of manifolds. Note that there can be no cells of higher dimension since a cell of maximal dimension produces nontrivial local homology in that dimension. Consider the cellular boundary map d : Cn(M)→Cn−1(M) with Z coeﬃcients. Since M has a single n cell we have Cn(M) = Z. If M is orientable, d must be zero since Hn(M; Z) = Z. Then since d Poincar´e Duality Section 3.3 239 is zero, Hn−1(M; Z) must be free. On the other hand, if M is nonorientable then d must take a generator of Cn(M) to twice a generator α of a Z summand of Cn−1(M), in order for Hn(M; Zp) to be zero for odd primes p and Z2 for p = 2. The cellular chain α must be a cycle since 2α is a boundary and hence a cycle. It follows that the torsion subgroup of Hn−1(M; Z) must be a Z2 generated by α. Concerning the homology of noncompact manifolds there is the following general statement. Proposition 3.29. If M is a connected noncompact n manifold, then Hi(M; R) = 0 for i ≥ n. Proof: Represent an element of Hi(M; R) by a cycle z. This has compact image in M, so there is an open set U ⊂ M containing the image of z and having compact closure U ⊂ M. Let V = M − U. Part of the long exact sequence of the triple (M, U ∪ V, V ) ﬁts into a commutative diagram When i > n, the two groups on either side of Hi(U ∪ V, V ; R) are zero
by Lemma 3.27 since U ∪ V and V are the complements of compact sets in M. Hence Hi(U; R) = 0, so z is a boundary in U and therefore in M, and we conclude that Hi(M; R) = 0. When i = n, the class [z] ∈ Hn(M; R) deﬁnes a section x֏[z]x of MR. Since M is connected, this section is determined by its value at a single point, so [z]x will be zero for all x if it is zero for some x, which it must be since z has compact image and M is noncompact. By Lemma 3.27, z then represents zero in Hn(M, V ; R), hence also in Hn(U; R) since the ﬁrst term in the upper row of the diagram above is zero when i = n, by Lemma 3.27 again. So [z] = 0 in Hn(M; R), and therefore Hn(M; R) = 0 ⊔⊓ since [z] was an arbitrary element of this group. The Duality Theorem The form of Poincar´e duality we will prove asserts that for an R orientable closed n manifold, a certain naturally deﬁned map H k(M; R)→Hn−k(M; R) is an isomorphism. The deﬁnition of this map will be in terms of a more general construction called cap product, which has close connections with cup product. For an arbitrary space X and coeﬃcient ring R, deﬁne an R bilinear cap product a : Ck(X; R)× C ℓ(X; R)→Ck−ℓ(X; R) for k ≥ ℓ by setting for σ : k→X and ϕ ∈ C ℓ(X; R). To see that this induces a cap product in homology σ a ϕ = ϕ σ \|\| [v0, ···, vℓ] σ \|\| [vℓ, ···, vk] ∆ 240 Chapter 3 Cohomology and cohomology we use the formula ∂(σ a ϕ) = (−1)�
��(∂σ a ϕ − σ a δϕ) which is checked by a calculation: ∂σ a ϕ = ℓ Xi = 0 ℓ+1 (−1)iϕ σ \|\|[v0, ···, k + (−1)iϕ Xi = ℓ+1 vi, ···, vℓ+1] b σ \|\|[v0, ···, vℓ] σ \|\|[vℓ+1, ···, vk] σ \|\|[vℓ, ···, vi, ···, vk] b σ a δϕ = (−1)iϕ Xi = 0 k σ \|\|[v0, ···, σ \|\|[vℓ+1, ···, vk] vi, ···, vℓ+1] b ∂(σ a ϕ) = (−1)i−ℓϕ Xi = ℓ σ \|\|[v0, ···, vℓ] σ \|\|[vℓ, ···, vi, ···, vk] From the relation ∂(σ a ϕ) = ±(∂σ a ϕ − σ a δϕ) it follows that the cap product of a cycle σ and a cocycle ϕ is a cycle. Further, if ∂σ = 0 then ∂(σ a ϕ) = ±(σ a δϕ), so the cap product of a cycle and a coboundary is a boundary. And if δϕ = 0 then ∂(σ a ϕ) = ±(∂σ a ϕ), so the cap product of a boundary and a cocycle is a boundary. These facts imply that there is an induced cap product b Hk(X; R)× H ℓ(X; R) a-----------------------→ Hk−ℓ(X; R) which is R linear in each variable. Using the same formulas, one checks that cap product has the relative forms Hk(X, A; R)× H ℓ(X; R) Hk(X, A; R)× H ℓ(X, A; R) a-----------------------→ Hk−ℓ(X
, A; R) a-----------------------→ Hk−ℓ(X; R) For example, in the second case the cap product Ck(X; R)× C ℓ(X; R)→Ck−ℓ(X; R) restricts to zero on the submodule Ck(A; R)× C ℓ(X, A; R), so there is an induced cap product Ck(X, A; R)× C ℓ(X, A; R)→Ck−ℓ(X; R). The formula for ∂(σ a ϕ) still holds, so we can pass to homology and cohomology groups. There is also a more general relative cap product Hk(X, A ∪ B; R)× H ℓ(X, A; R) a-----------------------→ Hk−ℓ(X, B; R), deﬁned when A and B are open sets in X, using the fact that Hk(X, A ∪ B; R) can be computed using the chain groups Cn(X, A + B; R) = Cn(X; R)/Cn(A + B; R), as in the derivation of relative Mayer–Vietoris sequences in §2.2. Cap product satisﬁes a naturality property that is a little more awkward to state than the corresponding result for cup product since both covariant and contravariant functors are involved. Given a map f : X→Y, the relevant induced maps on homology It does not quite make sense and cohomology ﬁt into the diagram shown below. Poincar´e Duality Section 3.3 241 to say this diagram commutes, but the spirit of commutativity is contained in the formula f∗(α) a ϕ = f∗ α a f ∗(ϕ) which is obtained by substituting f σ for σ in the deﬁnition of cap product: f σ aϕ = ϕ f σ \|\| [vℓ, ···, vk]. There are evident relative versions as well. f σ \|\| [v0, ···, vℓ] Now we can state Poincar´e duality for closed manifolds:
Theorem 3.30 (Poincar´e Duality). If M is a closed R orientable n manifold with fundamental class [M] ∈ Hn(M; R), then the map D : H k(M; R) -→ Hn−k(M; R) deﬁned by D(α) = [M] a α is an isomorphism for all k. Recall that a fundamental class for M is an element of Hn(M; R) whose image in Hn(M \|\| x; R) is a generator for each x ∈ M. The existence of such a class was shown in Theorem 3.26. Example 3.31: Surfaces. Let M be the closed orientable surface of genus g, obtained as usual from a 4g gon by identifying pairs of edges according to the word a1b1a−1 g b−1 the 4g gon to its center, as indicated in the ﬁgure complex structure on M is obtained by coning oﬀ 1 ··· agbga−1 1 b−1 g. A for the case g = 2. We can compute cap products ∆ using simplicial homology and cohomology since cap products are deﬁned for simplicial homology and co- homology by exactly the same formula as for singular homology and cohomology, so the isomorphism be- tween the simplicial and singular theories respects cap products. A fundamental class [M] generating H2(M) is represented by the 2 cycle formed by the sum of all 4g 2 simplices with the signs indicated. The edges ai and bi form a basis for H1(M). Under the isomorphism H 1(M) ≈ Hom(H1(M), Z), the cohomology class αi corresponding to ai assigns the value 1 to ai and 0 to the other basis elements. This class αi is represented by the cocycle ϕi assigning the value 1 to the 1 simplices meeting the arc labeled αi in the ﬁgure and 0 to the other 1 simplices. Similarly we have a class βi corresponding to bi, represented by the cocycle ψi assigning the value 1 to the 1 simplices meeting the arc βi and 0 to the other 1 simplices. Applying the deﬁnition of cap product, we have
[M] a ϕi = bi and [M] a ψi = −ai since in both cases there is just one 2 simplex [v0, v1, v2] where ϕi or ψi is nonzero on the edge [v0, v1]. Thus bi is the Poincar´e dual of αi and −ai is the Poincar´e dual of βi. If we interpret Poincar´e duality entirely in terms of homology, identifying αi with its Hom-dual ai and βi with bi, then the classes ai and bi are Poincar´e duals of each other, up to sign at least. Geometrically, Poincar´e duality is reﬂected in the fact that the loops αi and bi are homotopic, as are the loops βi and ai. 242 Chapter 3 Cohomology The closed nonorientable surface N of genus g can be treated in the same way if we use Z2 coefﬁcients. We view N as obtained from a 2g gon by 1 ··· a2 identifying consecutive pairs of edges according to the g. We have classes αi ∈ H 1(N; Z2) repword a2 resented by cocycles ϕi assigning the value 1 to the edges meeting the arc αi. Then [N] a ϕi = ai, so ai is the Poincar´e dual of αi. In terms of homology, ai is the Hom-dual of αi, so ai is its own Poincar´e dual. Geometrically, the loops ai on N are homotopic to their Poincar´e dual loops αi. Our proof of Poincar´e duality, like the construction of fundamental classes, will be by an inductive argument using Mayer–Vietoris sequences. The induction step requires a version of Poincar´e duality for open subsets of M, which are noncompact and can satisfy Poincar´e duality only when a diﬀerent kind of cohomology called cohomology with compact supports is used. Cohomology with Compact Supports Before giving the general deﬁnition, let us look at the conceptually simpler notion of simplicial cohomology with compact supports. Here one starts
with a complex ∆ borhood that meets only ﬁnitely many simplices. Consider the subgroup X which is locally compact. This is equivalent to saying that every point has a neighi c(X; G) i(X; G) consisting of cochains that are compactly supported in the sense that they take nonzero values on only ﬁnitely many sim- of the simplicial cochain group ∆ plices. The coboundary of such a cochain ϕ can have a nonzero value only on those (i+1) simplices having a face on which ϕ is nonzero, and there are only ﬁnitely many such simplices by the local compactness assumption, so δϕ lies in i+1 c (X; G). Thus we have a subcomplex of the simplicial cochain complex. The cohomology groups for this subcomplex will be denoted temporarily by H i ∆ c(X; G). Example 3.32. Let us compute these cohomology groups when X = R with the complex structure having vertices at the integer points. For a simplicial 0 cochain 0 nonzero. Namely, consider the map c (X) it must be identically zero. Thus H 0 to be a cocycle it must take the same value on all vertices, but then if the cochain ∆ c (R; G) = 0. However, H 1 c (R; G) is lies in c (R; G)→G sending each cochain to the sum 1(X), just c (R; G)→G. ∆ 1 c (X) is a cocycle. It is an easy exercise to is not deﬁned on all of vanishes on coboundaries, so it induces a map H 1 Σ of its values on all the 1 simplices. Note that 1 c (X). The map on ∆ ∆ Σ 1 : This is surjective since every element of verify that it is also injective, so H 1 ∆ Σ c (R; G) ≈ G. ∆ Compactly supported cellular cohomology for a locally compact CW complex could be deﬁned in a similar fashion, using cellular cochains that are nonzero on ∆ Poincar´e Duality Section 3.3 243 only ﬁnitely many cells.
However, what we really need is singular cohomology with compact supports for spaces without any simplicial or cellular structure. The quickest deﬁnition of this is the following. Let C i c(X; G) be the subgroup of C i(X; G) consisting of cochains ϕ : Ci(X)→G for which there exists a compact set K = Kϕ ⊂ X such that ϕ is zero on all chains in X − K. Note that δϕ is then also zero on chains in X − K, so δϕ lies in C i+1 c(X; G) ’s for varying i form a subcomplex of the singular cochain complex of X. The cohomology groups H i c(X; G) of this subcomplex are the cohomology groups with compact supports. (X; G) and the C i c Cochains in C i c(X; G) have compact support in only a rather weak sense. A stronger and perhaps more natural condition would have been to require cochains to be nonzero only on singular simplices contained in some compact set, depending on the cochain. However, cochains satisfying this condition do not in general form a subcomplex of the singular cochain complex. For example, if X = R and ϕ is a 0 cochain assigning a nonzero value to one point of R and zero to all other points, then δϕ assigns a nonzero value to arbitrarily large 1 simplices. It will be quite useful to have an alternative deﬁnition of H i c(X; G) in terms of algebraic limits, which enter the picture in the following way. The cochain group C i c(X; G) is the union of its subgroups C i(X, X − K; G) as K ranges over compact subsets of X. Each inclusion K ֓ L induces inclusions C i(X, X − K; G) ֓ C i(X, X − L; G) for all i, so there are induced maps H i(X, X − K; G)→H i(X, X − L; G). These need not be injective, but one might still hope that H i c(X; G) is somehow describable in terms of the system of groups H i(X, X − K; G) for
varying K. This is indeed the case, and it is algebraic limits that provide the description. Suppose one has abelian groups Gα indexed by some partially ordered index set I having the property that for each pair α, β ∈ I there exists γ ∈ I with α ≤ γ and β ≤ γ. Such an I is called a directed set. Suppose also that for each pair α ≤ β one has a homomorphism fαβ : Gα→Gβ, such that fαα = 11 for each α, and if α ≤ β ≤ γ then fαγ is the composition of fαβ and fβγ. Given this data, which is called a directed system of groups, there are two equivalent ways of deﬁning the direct limit group lim--→Gα. The shorter deﬁnition is that lim--→Gα is the quotient of the direct sum α Gα by the subgroup generated by all elements of the form a − fαβ(a) for a ∈ Gα, where α Gα. The other deﬁnition, which is often we are viewing each Gα as a subgroup of more convenient to work with, runs as follows. Deﬁne an equivalence relation on the L α Gα by a ∼ b if fαγ(a) = fβγ(b) for some γ, where a ∈ Gα and b ∈ Gβ. This is clearly reﬂexive and symmetric, and transitivity follows from the directed set set L ` property. It could also be described as the equivalence relation generated by setting a ∼ fαβ(a). Any two equivalence classes [a] and [b] have representatives a′ and b′ lying in the same Gγ, so deﬁne [a] + [b] = [a′ + b′]. One checks this is welldeﬁned and gives an abelian group structure to the set of equivalence classes. It is easy to check further that the map sending an equivalence class [a] to the coset of a 244 Chapter 3 Cohomology in lim--→Gα is a homomorphism, with an inverse induced by the map for ai ∈ Gαi i[ai]. Thus we can identify lim--→Gα with the group of equivalence classes
[a]. A useful consequence of this is that if we have a subset J ⊂ I with the property that for each α ∈ I there exists a β ∈ J with α ≤ β, then lim--→Gα is the same whether we compute it with α varying over I or just over J. In particular, if I has a maximal element γ, we can take J = {γ} and then lim--→Gα = Gγ. P P i ai ֏ Suppose now that we have a space X expressed as the union of a collection of subspaces Xα forming a directed set with respect to the inclusion relation. Then the groups Hi(Xα; G) for ﬁxed i and G form a directed system, using the homomorphisms induced by inclusions. The natural maps Hi(Xα; G)→Hi(X; G) induce a homomorphism lim--→Hi(Xα; G)→Hi(X; G). Proposition 3.33. If a space X is the union of a directed set of subspaces Xα with the property that each compact set in X is contained in some Xα, then the natural map lim--→Hi(Xα; G)→Hi(X; G) is an isomorphism for all i and G. Proof: For surjectivity, represent a cycle in X by a ﬁnite sum of singular simplices. The union of the images of these singular simplices is compact in X, hence lies in some Xα, so the map lim--→Hi(Xα; G)→Hi(X; G) is surjective. Injectivity is similar: If a cycle in some Xα is a boundary in X, compactness implies it is a boundary in some Xβ ⊃ Xα, hence represents zero in lim--→Hi(Xα; G). ⊔⊓ Now we can give the alternative deﬁnition of cohomology with compact supports in terms of direct limits. For a space X, the compact subsets K ⊂ X form a directed set under inclusion since the union of two compact sets is compact. To each compact K ⊂ X we associate the group H i(X, X − K; G), with a ﬁxed i and coeﬃcient group G, and to each inclusion
K ⊂ L of compact sets we associate the natural homomorphism H i(X, X −K; G)→H i(X, X −L; G). The resulting limit group lim--→H i(X, X −K; G) is then equal to H i c(X; G) since each element of this limit group is represented by a cocycle in C i(X, X − K; G) for some compact K, and such a cocycle is zero in lim--→H i(X, X − K; G) iﬀ it is the coboundary of a cochain in C i−1(X, X − L; G) for some compact L ⊃ K. Note that if X is compact, then H i c(X; G) = H i(X; G) since there is a unique maximal compact set K ⊂ X, namely X itself. This is also immediate from the original deﬁnition since C i c(X; G) = C i(X; G) if X is compact. Example 3.34: H ∗ c (Rn; G). To compute lim--→H i(Rn, Rn − K; G) it suﬃces to let K range over balls Bk of integer radius k centered at the origin since every compact set is contained in such a ball. Since H i(Rn, Rn − Bk; G) is nonzero only for i = n, when it is G, and the maps H n(Rn, Rn − Bk; G)→H n(Rn, Rn − Bk+1; G) are isomorphisms, we deduce that H i c(Rn; G) = 0 for i ≠ n and H n c (Rn; G) ≈ G. This example shows that cohomology with compact supports is not an invariant of homotopy type. This can be traced to diﬃculties with induced maps. For example, Poincar´e Duality Section 3.3 245 the constant map from Rn to a point does not induce a map on cohomology with compact supports. The maps which do induce maps on H ∗ c are the proper maps, In the proof of those for which the inverse image of each compact set is compact. Po
incar´e duality, however, we will need induced maps of a diﬀerent sort going in the opposite direction from what is usual for cohomology, maps H i c(V ; G) associated to inclusions U ֓ V of open sets in the ﬁxed manifold M. c(U; G)→H i The group H i(X, X−K; G) for K compact depends only on a neighborhood of K in X by excision, assuming X is Hausdorﬀ so that K is closed. As convenient shorthand notation we will write this group as H i(X \|\| K; G), in analogy with the similar notation used earlier for local homology. One can think of cohomology with compact supports as the limit of these ‘local cohomology groups at compact subsets’. Duality for Noncompact Manifolds For M an R orientable n manifold, possibly noncompact, we can deﬁne a dualc (M; R)→Hn−k(M; R) by a limiting process in the following way. For ity map DM : H k compact sets K ⊂ L ⊂ M we have a diagram where Hn(M \|\| A; R) = Hn(M, M − A; R) and H k(M \|\| A; R) = H k(M, M − A; R). By Lemma 3.27 there are unique elements µK ∈ Hn(M \|\| K; R) and µL ∈ Hn(M \|\| L; R) restricting to a given orientation of M at each point of K and L, respectively. From the uniqueness we have i∗(µL) = µK. The naturality of cap product implies that ai∗(x). Therefore, leti∗(µL)ax = µL ting K vary over compact sets in M, the homomorphisms H k(M \|\| K; R)→Hn−k(M; R), c (M; R)→Hn−k(M; R). x֏µK c (M; R) = H ∗(M; R) if M is compact, the following theorem generalizes ax, induce in the limit a duality homomorphism DM : H k ai�
�(x) for all x ∈ H k(M \|\| K; R), so µK ax = µL Since H ∗ Poincar´e duality for closed manifolds: Theorem 3.35. The duality map DM : H k for all k whenever M is an R oriented n manifold. c (M; R)→Hn−k(M; R) is an isomorphism The proof will not be diﬃcult once we establish a technical result stated in the next lemma, concerning the commutativity of a certain diagram. Commutativity state- ments of this sort are usually routine to prove, but this one seems to be an exception. The reader who consults other books for alternative expositions will ﬁnd somewhat uneven treatments of this technical point, and the proof we give is also not as simple as one would like. The coeﬃcient ring R will be ﬁxed throughout the proof, and for simplicity we will omit it from the notation for homology and cohomology. 246 Chapter 3 Cohomology Lemma 3.36. If M is the union of two open sets U and V, then there is a diagram of Mayer–Vietoris sequences, commutative up to sign : Proof: Compact sets K ⊂ U and L ⊂ V give rise to the Mayer–Vietoris sequence in the upper row of the following diagram, whose lower row is also a Mayer–Vietoris sequence: The two maps labeled isomorphisms come from excision. Assuming this diagram commutes, consider passing to the limit over compact sets K ⊂ U and L ⊂ V. Since each compact set in U ∩V is contained in an intersection K ∩L of compact sets K ⊂ U and L ⊂ V, and similarly for U ∪ V, the diagram induces a limit diagram having the form stated in the lemma. The ﬁrst row of this limit diagram is exact since a direct limit of exact sequences is exact; this is an exercise at the end of the section, and follows easily from the deﬁnition of direct limits. It remains to consider the commutativity of the preceding diagram involving K and L. In the two squares shown, not involving boundary or coboundary maps, it is a triviality to check commutativity at the level of cycles and coc
ycles. Less trivial is the third square, which we rewrite in the following way: (∗) Letting A = M − K and B = M − L, the map δ is the coboundary map in the Mayer– Vietoris sequence obtained from the short exact sequence of cochain complexes 0 -→ C ∗(M, A + B) -→ C ∗(M, A) ⊕ C ∗(M, B) -→ C ∗(M, A ∩ B) -→ 0 where C ∗(M, A + B) consists of cochains on M vanishing on chains in A and chains in B. To evaluate the Mayer–Vietoris coboundary map δ on a cohomology class represented by a cocycle ϕ ∈ C ∗(M, A ∩ B), the ﬁrst step is to write ϕ = ϕA − ϕB Poincar´e Duality Section 3.3 247 for ϕA ∈ C ∗(M, A) and ϕB ∈ C ∗(M, B). Then δ[ϕ] is represented by the cocycle δϕA = δϕB ∈ C ∗(M, A + B), where the equality δϕA = δϕB comes from the fact that ϕ is a cocycle, so δϕ = δϕA − δϕB = 0. Similarly, the boundary map ∂ in the homology Mayer–Vietoris sequence is obtained by representing an element of Hi(M) by a cycle z that is a sum of chains zU ∈ Ci(U) and zV ∈ Ci(V ), and then ∂[z] = [∂zU ]. Via barycentric subdivision, the class µK∪L can be represented by a chain α that is a sum αU −L + αU ∩V + αV −K of chains in U − L, U ∩ V, and V − K, respectively, since these three open sets cover M. The chain αU ∩V represents µK∩L since the other two chains αU −L and αV −K lie in the complement of K ∩ L, hence vanish in Hn(M \|\| K ∩ L
) ≈ Hn(U ∩ V \|\| K ∩ L). Similarly, αU −L + αU ∩V represents µK. In the square (∗) let ϕ be a cocycle representing an element of H k(M \|\| K ∪ L). Under δ this maps to the cohomology class of δϕA. Continuing on to Hn−k−1(U ∩ V ) a δϕA, which is in the same homology class as ∂αU ∩V we obtain αU ∩V a ϕA since ∂(αU ∩V a ϕA) = (−1)k(∂αU ∩V a ϕA − αU ∩V a δϕA) and αU ∩V a ϕA is a chain in U ∩ V. Going around the square (∗) the other way, ϕ maps ﬁrst to α a ϕ. To apply the Mayer–Vietoris boundary map ∂ to this, we ﬁrst write α a ϕ as a sum of a chain in U and a chain in V : α a ϕ = (αU −L a ϕ) + (αU ∩V a ϕ + αV −K a ϕ) Then we take the boundary of the ﬁrst of these two chains, obtaining the homology a ϕA], we have a ϕ)] ∈ Hn−k−1(U ∩ V ). To compare this with [∂αU ∩V class [∂(αU −L ∂(αU −L a ϕ) = (−1)k∂αU −L = (−1)k∂αU −L a ϕ a ϕA since δϕ = 0 since ∂αU −L zero on chains in B = M − L a ϕB = 0, ϕB being = (−1)k+1∂αU ∩V a ϕA where this last equality comes from the fact that ∂(αU −L + αU ∩V ) a ϕA = 0 since ∂(αU −L + αU ∩V ) is a chain in U − K by the earlier observation that αU −
L + αU ∩V represents µK, and ϕA vanishes on chains in A = M − K. Thus the square (∗) commutes up to a sign depending only on k. ⊔⊓ Proof of Poincar´e Duality: There are two inductive steps, ﬁnite and inﬁnite: (A) If M is the union of open sets U and V and if DU, DV, and DU ∩V are isomorphisms, then so is DM. Via the ﬁve-lemma, this is immediate from the preceding lemma. 248 Chapter 3 Cohomology : H k (B) If M is the union of a sequence of open sets U1 ⊂ U2 ⊂ ··· and each duality map c (Ui)→Hn−k(Ui) is an isomorphism, then so is DM. To show this we notice DUi c (Ui) can be regarded as the limit of the groups H k(M \|\| K) as K ﬁrst that by excision, H k ranges over compact subsets of Ui. Then there are natural maps H k c (Ui+1) since the second of these groups is a limit over a larger collection of K ’s. Thus we can form lim--→H k c (M) since the compact sets in M are just the compact sets in all the Ui ’s. By Proposition 3.33, Hn−k(M) ≈ lim--→Hn−k(Ui). The map DM is thus the limit of the isomorphisms DUi c (Ui) which is obviously isomorphic to H k, hence is an isomorphism. c (Ui)→H k Now after all these preliminaries we can prove the theorem in three easy steps: ∆ ∆ ∆ ∆ ∆ n, ∂ n, ∂ ∆ n). n aϕ is the last vertex of n, ∂ n] ∈ Hn( (1) The case M = Rn can be proved by regarding Rn as the interior of n)→Hn−k( then the map DM can be identiﬁed with the map H k( n, ∂ by cap product with a unit times the generator [ ∆ n, and
a maximal open set U for which the theorem holds. If U ≠ M, choose a point x ∈ M − U and an open neighborhood V of x homeomorphic to Rn. The theorem holds for V and U ∩ V by (1) and (2), and it holds for U by assumption, so by (A) it holds for U ∪V, contradicting ⊔⊓ the maximality of U. Poincar´e Duality Section 3.3 249 Corollary 3.37. A closed manifold of odd dimension has Euler characteristic zero. Proof: Let M be a closed n manifold. If M is orientable, we have rank Hi(M; Z) = rank H n−i(M; Z), which equals rank Hn−i(M; Z) by the universal coeﬃcient theorem. Thus if n is odd, all the terms of i(−1)i rank Hi(M; Z) cancel in pairs. P If M is not orientable we apply the same argument using Z2 coeﬃcients, with rank Hi(M; Z) replaced by dim Hi(M; Z2), the dimension as a vector space over Z2, i(−1)i dim Hi(M; Z2) = 0. It remains to check that this alternating to conclude that i(−1)i rank Hi(M; Z). We can do this by using sum equals the Euler characteristic P the isomorphisms Hi(M; Z2) ≈ H i(M; Z2) and applying the universal coeﬃcient theorem for cohomology. Each Z summand of Hi(M; Z) gives a Z2 summand of H i(M; Z2). Each Zm summand of Hi(M; Z) with m even gives Z2 summands of H i(M; Z2) and H i+1(M, Z2), whose contributions to i(−1)i dim Hi(M; Z2) cancel. And Zm summands of Hi(M; Z) with m odd contribute nothing to H ∗(M; Z2). ⊔⊓ P P Connection with Cup Product Cup and cap product are related by the formula (∗) ψ(α a ϕ) = (ϕ ` ψ)(α) for
α ∈ Ck+ℓ(X; R), ϕ ∈ C k(X; R), and ψ ∈ C ℓ(X; R). This holds since for a singular (k + ℓ) simplex σ : k+ℓ→X we have ψ(σ a ϕ) = ψ ∆ = ϕ ϕ σ \|\|[v0, ···, vk] σ \|\|[v0, ···, vk] ψ σ \|\|[vk, ···, vk+ℓ] σ \|\|[vk, ···, vk+ℓ] = (ϕ ` ψ)(σ ) The formula (∗) says that the map ϕ` : C ℓ(X; R)→C k+ℓ(X; R) is equal to the map HomR(Cℓ(X; R), R)→HomR(Ck+ℓ(X; R), R) dual to aϕ. Passing to homology and cohomology, we obtain the commutative di- agram at the right. When the maps h are isomorphisms, for example when R is a ﬁeld or when R = Z and the homology groups of X are free, then the map ϕ ` is the dual of a ϕ. Thus in these cases cup and cap product determine each other, at least if one assumes ﬁnite generation so that cohomology determines homology as well as vice versa. However, there are examples where cap and cup products are not equivalent when R = Z and there is torsion in homology. By means of the formula (∗), Poincar´e duality has nontrivial implications for the cup product structure of manifolds. For a closed R orientable n manifold M, consider the cup product pairing H k(M; R) × H n−k(M; R) ---------→ R, (ϕ, ψ) ֏ (ϕ ` ψ)[M] 250 Chapter 3 Cohomology Such a bilinear pairing A× B→R is said to be nonsingular if the maps A→HomR(B, R) and B→HomR(A, R), obtained by viewing the pairing as
a function of each variable separately, are both isomorphisms. Proposition 3.38. The cup product pairing is nonsingular for closed R orientable manifolds when R is a ﬁeld, or when R = Z and torsion in H ∗(M; Z) is factored out. Proof: Consider the composition H n−k(M; R) h-----→ HomR(Hn−k(M; R), R) D∗-----→ HomR(H k(M; R), R) where h is the map appearing in the universal coeﬃcient theorem, induced by evaluation of cochains on chains, and D∗ is the Hom dual of the Poincar´e duality map D : H k→Hn−k. The composition D∗h sends ψ ∈ H n−k(M; R) to the homomorphism ϕ ֏ ψ([M] a ϕ) = (ϕ ` ψ)[M]. For ﬁeld coeﬃcients or for integer coeﬃcients with torsion factored out, h is an isomorphism. Nonsingularity of the pairing in one of its variables is then equivalent to D being an isomorphism. Nonsingularity in the other variable follows by commutativity of cup product. ⊔⊓ Corollary 3.39. If M is a closed connected orientable n manifold, then an element α ∈ H k(M; Z) generates an inﬁnite cyclic summand of H k(M; Z) iﬀ there exists an element β ∈ H n−k(M; Z) such that α ` β is a generator of H n(M; Z) ≈ Z. With coeﬃcients in a ﬁeld this holds for any α ≠ 0. Proof: For α to generate a Z summand of H k(M; Z) is equivalent to the existence of a homomorphism ϕ : H k(M; Z)→Z with ϕ(α) = ±1. By the nonsingularity of the cup product pairing, ϕ is realized by taking cup product with an element β ∈ H n−k(M; Z) and evaluating on [M], so having a β with
α ` β generating H n(M; Z) is equivalent to having ϕ with ϕ(α) = ±1. The case of ﬁeld coeﬃcients is similar but easier. ⊔⊓ Example 3.40: Projective Spaces. The cup product structure of H ∗(CPn; Z) as a truncated polynomial ring Z[α]/(αn+1) with \|α\| = 2 can easily be deduced from this as follows. The inclusion CPn−1֓CPn induces an isomorphism on H i for i ≤ 2n−2, so by induction on n, H 2i(CPn; Z) is generated by αi for i < n. By the corollary, there is an integer m such that the product α ` mαn−1 = mαn generates H 2n(CPn; Z). This can only happen if m = ±1, and therefore H ∗(CPn; Z) ≈ Z[α]/(αn+1). The same argument shows H ∗(HPn; Z) ≈ Z[α]/(αn+1) with \|α\| = 4. For RPn one can use the same argument with Z2 coeﬃcients to deduce that H ∗(RPn; Z2) ≈ Z2[α]/(αn+1) with \|α\| = 1. The cup product structure in inﬁnite-dimensional projective spaces follows from the ﬁnite-dimensional case, as we saw in the proof of Theorem 3.19. Could there be a closed manifold whose cohomology is additively isomorphic to that of CPn but with a diﬀerent cup product structure? For n = 2 the answer is no since duality implies that the square of a generator of H 2 must be a generator of Poincar´e Duality Section 3.3 251 H 4. For n = 3, duality says that the product of generators of H 2 and H 4 must be a generator of H 6, but nothing is said about the square of a generator of H 2. Indeed, for S 2 × S 4, whose cohomology has the same additive structure as CP3, the square of the generator of H 2(S 2 × S 4; Z) is zero
since it is the pullback of a generator of H 2(S 2; Z) under the projection S 2 × S 4→S 2, and in H ∗(S 2; Z) the square of the generator of H 2 is zero. More generally, an exercise for §4.D describes closed 6 manifolds having the same cohomology groups as CP3 but where the square of the generator of H 2 is an arbitrary multiple of a generator of H 4. Example 3.41: Lens Spaces. Cup products in lens spaces can be computed in the same way as in projective spaces. For a lens space L2n+1 of dimension 2n + 1 with fundamental group Zm, we computed Hi(L2n+1; Z) in Example 2.43 to be Z for i = 0 and 2n + 1, Zm for odd i < 2n + 1, and 0 otherwise. In particular, this implies that L2n+1 is orientable, which can also be deduced from the fact that L2n+1 is the orbit space of an action of Zm on S 2n+1 by orientation-preserving homeomorphisms, using an exercise at the end of this section. By the universal coeﬃcient theorem, H i(L2n+1; Zm) is Zm for each i ≤ 2n+1. Let α ∈ H 1(L2n+1; Zm) and β ∈ H 2(L2n+1; Zm) be generators. The statement we wish to prove is: H j(L2n+1; Zm) is generated by βi αβi ( for j = 2i for j = 2i + 1 By induction on n we may assume this holds for j ≤ 2n−1 since we have a lens space L2n−1 ⊂ L2n+1 with this inclusion inducing an isomorphism on H j for j ≤ 2n − 1, as one sees by comparing the cellular chain complexes for L2n−1 and L2n+1. The preceding corollary does not apply directly for Zm coeﬃcients with arbitrary m, but its proof does since the maps h : H i(L2n+1; Zm)→Hom(Hi(L2n+1; Zm), Zm) are isomorphisms. We conclude that β
` kαβn−1 generates H 2n+1(L2n+1; Zm) for some integer k. We must have k relatively prime to m, otherwise the product β ` kαβn−1 = kαβn would have order less than m and so could not generate H 2n+1(L2n+1; Zm). Then since k is relatively prime to m, αβn is also a generator of H 2n+1(L2n+1; Zm). From this it follows that βn must generate H 2n(L2n+1; Zm), otherwise it would have order less than m and so therefore would αβn. The rest of the cup product structure on H ∗(L2n+1; Zm) is determined once α2 is expressed as a multiple of β. When m is odd, the commutativity formula for cup product implies α2 = 0. When m is even, commutativity implies only that α2 is either zero or the unique element of H 2(L2n+1; Zm) ≈ Zm of order two. In fact it is the latter possibility which holds, since the 2 skeleton L2 is the circle L1 with a 2 cell attached by a map of degree m, and we computed the cup product structure in this 2 complex in Example 3.9. It does not seem to be possible to deduce the nontriviality of α2 from Poincar´e duality alone, except when m = 2. The cup product structure for an inﬁnite-dimensional lens space L∞ follows from the ﬁnite-dimensional case since the restriction map H j(L∞; Zm)→H j(L2n+1; Zm) is 252 Chapter 3 Cohomology an isomorphism for j ≤ 2n + 1. As with RPn, the ring structure in H ∗(L2n+1; Z) is determined by the ring structure in H ∗(L2n+1; Zm), and likewise for L∞, where one has the slightly simpler structure H ∗(L∞; Z) ≈ Z[α]/(mα) with \|α\| = 2. The case of L2n+1 is obtained from this by setting αn+1 = 0 and adjoining
the extra Z ≈ H 2n+1(L2n+1; Z). A diﬀerent derivation of the cup product structure in lens spaces is given in Example 3E.2. Using the ad hoc notation H k f r ee(M) for H k(M) modulo its torsion subgroup, the preceding proposition implies that for a closed orientable manifold M of dimenf r ee(M)→Z is a sion 2n, the middle-dimensional cup product pairing H n nonsingular bilinear form on H n f r ee(M). This form is symmetric or skew-symmetric according to whether n is even or odd. The algebra in the skew-symmetric case is f r ee(M)× H n rather simple: With a suitable choice of basis, the matrix of a skew-symmetric nonsingular bilinear form over Z can be put into the standard form consisting of 2× 2 blocks −1 0 0 along the diagonal and zeros elsewhere, according to an algebra exercise at the 1 end of the section. In particular, the rank of H n(M 2n) must be even when n is odd. We are already familiar with these facts in the case n = 1 by the explicit computations of cup products for surfaces in §3.2. The symmetric case is much more interesting algebraically. There are only ﬁnitely many isomorphism classes of symmetric nonsingular bilinear forms over Z of a ﬁxed rank, but this ‘ﬁnitely many’ grows rather rapidly, for example it is more than 80 million for rank 32; see [Serre 1973] for an exposition of this beautiful chapter of number theory. One can ask whether all these forms actually occur as cup product pairings in closed manifolds M 4k for a given k. The answer is yes for 4k = 4, 8, 16 but seems to be unknown in other dimensions. In dimensions 4, 8, and 16 one can even take M 4k to be simply-connected and have the bare minimum of homology: Z ’s in dimensions 0 and 4k and a free abelian group in dimension 2k. In dimension 4 there are at most two nonhomeomorphic simply-connected closed 4 manifolds with the same bilinear form. Namely, there are two manifolds
with the same form if the square α ` α of some α ∈ H 2(M 4) is an odd multiple of a generator of H 4(M 4), for example for CP2, and otherwise the M 4 is unique, for example for S 4 or S 2 × S 2 ; see [Freedman & Quinn 1990]. In §4.C we take the ﬁrst step in this direction by proving a classical result of J. H. C. Whitehead that the homotopy type of a simply-connected closed 4 manifold is uniquely determined by its cup product structure. Other Forms of Duality Generalizing the deﬁnition of a manifold, an n manifold with boundary is a Hausdorﬀ space M in which each point has an open neighborhood homeomorphic either to Rn or to the half-space Rn x ∈ M corresponds under such a homeomorphism to a point (x1, ···, xn) ∈ Rn + = { (x1, ···, xn) ∈ Rn \| xn ≥ 0 }. If a point + with Poincar´e Duality Section 3.3 253 xn = 0, then by excision we have Hn(M, M − {x}; Z) ≈ Hn(Rn + − {0}; Z) = 0, whereas if x corresponds to a point (x1, ···, xn) ∈ Rn + with xn > 0 or to a point of Rn, then Hn(M, M − {x}; Z) ≈ Hn(Rn, Rn − {0}; Z) ≈ Z. Thus the points x with Hn(M, M − {x}; Z) = 0 form a well-deﬁned subspace, called the boundary of M and + = Rn−1 and ∂Dn = S n−1. It is evident that ∂M is an denoted ∂M. For example, ∂Rn (n − 1) dimensional manifold with empty boundary. +, Rn If M is a manifold with boundary, then a collar neighborhood of ∂M in M is an open neighborhood homeomorphic to ∂M × [0, 1) by a homeomorphism taking ∂M to ∂M × {0}. Proposition 3.42. If M is a compact manifold
with boundary, then ∂M has a collar neighborhood. Proof: Let M ′ be M with an external collar attached, the quotient of the disjoint union of M and ∂M × [0, 1] in which x ∈ ∂M is identiﬁed with (x, 0) ∈ ∂M × [0, 1]. It will suﬃce to construct a homeomorphism h : M→M ′ since ∂M ′ clearly has a collar neighborhood. Since M is compact, so is the closed subspace ∂M. This implies that we can choose a ﬁnite number of continuous functions ϕi : ∂M→[0, 1] such that the sets Vi = ϕ−1 i (0, 1] form an open cover of ∂M and each Vi has closure contained in an open set Ui ⊂ M homeomorphic to the half-space Rn +. After dividing each ϕi by j ϕj we may assume i ϕi = 1. P Let ψk = ϕ1 + ··· + ϕk and let Mk ⊂ M ′ be the union of M with the points P (x, t) ∈ ∂M × [0, 1] with t ≤ ψk(x). By deﬁnition ψ0 = 0 and M0 = M. We construct a homeomorphism hk : Mk−1→Mk as follows. The homeomorphism Uk ≈ Rn + gives a collar neighborhood ∂Uk × [−1, 0] of ∂Uk in Uk, with x ∈ ∂Uk corresponding to (x, 0) ∈ ∂Uk × [−1, 0]. Via the external collar ∂M × [0, 1] we then have an embedding ∂Uk × [−1, 1] ⊂ M ′. We deﬁne hk to be the identity outside this ∂Uk × [−1, 1], and for x ∈ ∂Uk we let hk stretch the segment {x}× [−1, ψk−1(x)] linearly onto {x}× [−1, ψk(x)]. The composition of all the hk ’s then gives a homeomorphism M ≈
M ′, ﬁnishing the proof. ⊔⊓ More generally, collars can be constructed for the boundaries of paracompact manifolds in the same way. A compact manifold M with boundary is deﬁned to be R orientable if M − ∂M is R orientable as a manifold without boundary. If ∂M × [0, 1) is a collar neighborhood of ∂M in M then Hi(M, ∂M; R) is naturally isomorphic to Hi(M − ∂M, ∂M × (0, ε); R), so when M is R orientable, Lemma 3.27 gives a relative fundamental class [M] in Hn(M, ∂M; R) restricting to a given orientation at each point of M − ∂M. It will not be diﬃcult to deduce the following generalization of Poincar´e duality to manifolds with boundary from the version we have already proved for noncompact manifolds: 254 Chapter 3 Cohomology Theorem 3.43. Suppose M is a compact R orientable n manifold whose boundary ∂M is decomposed as the union of two compact (n−1) dimensional manifolds A and B with a common boundary ∂A = ∂B = A ∩ B. Then cap product with a fundamental class [M] ∈ Hn(M, ∂M; R) gives isomorphisms DM : H k(M, A; R)→Hn−k(M, B; R) for all k. The possibility that A, B, or A ∩ B is empty is not excluded. The cases A = ∅ and B = ∅ are sometimes called Lefschetz duality. Proof: The cap product map DM : H k(M, A; R)→Hn−k(M, B; R) is deﬁned since the existence of collar neighborhoods of A ∩ B in A and B and ∂M in M implies that A and B are deformation retracts of open neighborhoods U and V in M such that U ∪ V deformation retracts onto A ∪ B = ∂M and U ∩ V deformation retracts onto A ∩ B. The case B = ∅ is proved by applying Theorem 3.35 to M −∂M
. Via a collar neighc (M − ∂M; R), and there are obvious borhood of ∂M we see that H k(M, ∂M; R) ≈ H k isomorphisms Hn−k(M; R) ≈ Hn−k(M − ∂M; R). The general case reduces to the case B = ∅ by applying the ﬁve-lemma to the following diagram, where coeﬃcients in R are implicit: For commutativity of the middle square one needs to check that the boundary map Hn(M, ∂M)→Hn−1(∂M) sends a fundamental class for M to a fundamental class for ⊔⊓ ∂M. We leave this as an exercise at the end of the section. Here is another kind of duality which generalizes the calculation of the local ho- mology groups Hi(M, M − {x}; Z) : Theorem 3.44. If K is a compact, locally contractible subspace of a closed orientable n manifold M, then Hi(M, M − K; Z) ≈ H n−i(K; Z) for all i. Proof: Let U be an open neighborhood of K in M. Consider the following diagram whose rows are long exact sequences of pairs: Poincar´e Duality Section 3.3 255 The second vertical map is the Poincar´e duality isomorphism given by cap products with a fundamental class [M]. This class can be represented by a cycle which is the sum of a chain in M − K and a chain in U representing elements of Hn(M − K, U − K) and Hn(U, U − K) respectively, and the ﬁrst and third vertical maps are given by It is not hard to check that the diagram relative cap products with these classes. commutes up to sign, where for the square involving boundary and coboundary maps one uses the formula for the boundary of a cap product. Passing to the direct limit over decreasing U ⊃ K, the ﬁrst vertical arrow become the Poincar´e duality isomorphism Hi(M − K) ≈ H n−i (M − K). The ﬁve-lemma then gives an isomorphism Hi(M, M − K
) ≈ lim--→H n−i(U). We will show that the natural map from this limit to H n−i(K) is an isomorphism. This is easy when K has a neighborhood that is a mapping cylinder of some map X→K, as in the ‘letter examples’ at the beginning of Chapter 0, since in this case we can compute the direct limit us- c ing neighborhoods U which are segments of the mapping cylinder that deformation retract to K. For the general case we use Theorem A.7 and Corollary A.9 in the Appendix. The latter says that M can be embedded in some Rk as a retract of a neighborhood N in Rk, and then Theorem A.7 says that K is a retract of a neighborhood in Rk and hence, by restriction, of a neighborhood W in M. We can compute lim--→H n−i(U) using just neighborhoods U in W, so these also retract to K and hence the map lim--→H n−i(U)→H n−i(K) is surjective. To show that it is injective, note ﬁrst that the retraction U→K is homotopic to the identity U→U through maps U→Rk, via the standard linear homotopy. Choosing a smaller U if necessary, we may assume this homotopy is through maps U→N since K is stationary during the homotopy. Applying the retraction N→M gives a homotopy through maps U→M ﬁxed on K. Restricting to suﬃciently small V ⊂ U, we then obtain a homotopy in U from the inclusion map V→U to the retraction V→K. Thus the map H n−i(U)→H n−i(V ) factors as H n−i(U)→H n−i(K)→H n−i(V ) where the ﬁrst map is induced by inclusion and the second by the retraction. This implies that the kernel of lim--→H n−i(U)→H n−i(K) is ⊔⊓ trivial. From this theorem we can easily deduce Alexander duality: Corollary 3.45. If K is a compact, locally contractible, nonempty, proper subspace of S n
, then H n−i−1(K; Z) for all i. Hi(S n − K; Z) ≈ Proof: The long exact sequence of reduced homology for the pair (S n, S n − K) gives e Hi(S n −K; Z) ≈ Hi+1(S n, S n −K; Z) for most values of i. The exception isomorphisms e is when i = n − 1 and we have only a short exact sequence e 0 -→ Hn(S n; Z) -→ Hn(S n, S n − K; Z) -→ Hn−1(S n − K; Z) -→ 0 e e 256 Chapter 3 Cohomology where the initial 0 is Hn(S n − K; Z) which is zero since the components of S n − K are noncompact n manifolds. This short exact sequence splits since we can map it to Hn−1(S n − K; Z) H 0(K; Z) is H 0(K; Z) with a e ⊔⊓ the corresponding sequence with K replaced by a point in K. Thus is Hn(S n, S n − K; Z) with a Z summand canceled, just as Z summand canceled. e e The special case of Alexander duality when K is a sphere or disk was treated by more elementary means in Proposition 2B.1. As remarked there, it is interesting that the homology of S n − K does not depend on the way that K is embedded in S n. There can be local pathologies as in the case of the Alexander horned sphere, or global complications as with knotted circles in S 3, but these have no eﬀect on the homology of the complement. The only requirement is that K is not too bad a space itself. An example where the theorem fails without the local contractibility assumption is the ‘quasi-circle’, deﬁned in an exercise for §1.3. This compact subspace K ⊂ R2 can be regarded as a subspace of S 2 by adding a point at inﬁnity. Then we have H 1(K; Z) = 0 since K H0(S 2 − K; Z) ≈ Z since S 2 − K has two path-components, but is simply-connected.
e Corollary 3.46. If X ⊂ Rn is compact and locally contractible then Hi(X; Z) is 0 for i ≥ n and torsionfree for i = n − 1 and n − 2. e For example, a closed nonorientable n manifold M cannot be embedded as a subspace of Rn+1 since Hn−1(M; Z) contains a Z2 subgroup, by Corollary 3.28. Thus the Klein bottle cannot be embedded in R3. More generally, the 2 dimensional complex Xm,n studied in Example 1.24, the quotient spaces of S 1 × I under the identiﬁcations (z, 0) ∼ (e2π i/mz, 0) and (z, 1) ∼ (e2π i/nz, 1), cannot be embedded in R3 if m and n are not relatively prime, since H1(Xm,n; Z) is Z× Zd where d is the greatest common divisor of m and n. The Klein bottle is the case m = n = 2. Proof: Viewing X as a subspace of the one-point compactiﬁcation S n, Alexander Hn−i−1(S n − X; Z). The latter group is zero duality gives isomorphisms for i ≥ n and torsionfree for i = n − 1, so the result follows from the universal e ⊔⊓ coeﬃcient theorem since X has ﬁnitely generated homology groups. H i(X; Z) ≈ e There is a way of extending Alexander duality and the duality in Theorem 3.44 to compact sets K that are not locally contractible, by replacing the singular cohomology of K with another kind of cohomology called ˇCech cohomology. This is deﬁned in the following way. To each open cover U = {Uα} of a given space X we can associate a simplicial complex N(U) called the nerve of U. This has a vertex vα for each Uα, and a set of k + 1 vertices spans a k simplex whenever the k + 1 corresponding Uα ’s have nonempty intersection. When another cover V = {Vβ} is a reﬁnement of U, so each
Vβ is contained in some Uα, then these inclusions induce a simplicial map Poincar´e Duality Section 3.3 257 N(V)→N(U) that is well-deﬁned up to homotopy. We can then form the direct limit lim--→H i(N(U); G) with respect to ﬁner and ﬁner open covers U. This limit group is by deﬁnition the ˇCech cohomology group ˇH i(X; G). For a full exposition of this cohomology theory see [Eilenberg & Steenrod 1952]. With an analogous deﬁnition of relative groups, ˇCech cohomology turns out to satisfy the same axioms as singular cohomology. For spaces homotopy equivalent to CW complexes, ˇCech cohomology coincides with singular cohomology, but for spaces with local complexities it often behaves more reasonably. For example, if X is the subspace of R3 consisting of the spheres of radius 1/n and center (1/n, 0, 0) for n = 1, 2, ···, then contrary to what one might expect, H 3(X; Z) is nonzero, as shown in [Barratt & Milnor 1962]. But ˇH 3(X; Z) = 0 and ˇH 2(X; Z) = Z∞, the direct sum of countably many copies of Z. Oddly enough, the corresponding ˇCech homology groups deﬁned using inverse limits are not so well-behaved. This is because the exactness axiom fails due to the algebraic fact that an inverse limit of exact sequences need not be exact, as a direct limit would be; see §3.F. However, there is a way around this problem using a more reﬁned deﬁnition. This is Steenrod homology theory, which the reader can learn about in [Milnor 1995]. Exercises 1. Show that there exist nonorientable 1 dimensional manifolds if the Hausdorﬀ condition is dropped from the deﬁnition of a manifold. 2. Show that deleting a point from a manifold of dimension greater than 1 does not aﬀect orientability of the manifold. 3
. Show that every covering space of an orientable manifold is an orientable manifold. 4. Given a covering space action of a group G on an orientable manifold M by orientation-preserving homeomorphisms, show that M/G is also orientable. 5. Show that M × N is orientable iﬀ M and N are both orientable. 6. Given two disjoint connected n manifolds M1 and M2, a connected n manifold M1♯M2, their connected sum, can be constructed by deleting the interiors of closed n balls B1 ⊂ M1 and B2 ⊂ M2 and identifying the resulting boundary spheres ∂B1 and ∂B2 via some homeomorphism between them. (Assume that each Bi embeds nicely in a larger ball in Mi.) (a) Show that if M1 and M2 are closed then there are isomorphisms Hi(M1♯M2; Z) ≈ Hi(M1; Z)⊕ Hi(M2; Z) for 0 < i < n, with one exception: If both M1 and M2 are nonorientable, then Hn−1(M1♯M2; Z) is obtained from Hn−1(M1; Z)⊕ Hn−1(M2; Z) by replacing one of the two Z2 summands by a Z summand. [Euler characteristics may help in the exceptional case.] (b) Show that χ (M1♯M2) = χ (M1) + χ (M2) − χ (S n) if M1 and M2 are closed. 258 Chapter 3 Cohomology 7. For a map f : M→N between connected closed orientable n manifolds with fundamental classes [M] and [N], the degree of f is deﬁned to be the integer d such that f∗([M]) = d[N], so the sign of the degree depends on the choice of fundamental classes. Show that for any connected closed orientable n manifold M there is a degree 1 map M→S n. 8. For a map f : M→N between connected closed orientable n manifolds, suppose there is a ball B ⊂ N such that f −1(B) is the disjoint union of balls Bi each mapped i εi
where εi is +1 or −1 homeomorphically by f onto B. Show the degree of f is according to whether f : Bi→B preserves or reverses local orientations induced from given fundamental classes [M] and [N]. P 9. Show that a p sheeted covering space projection M→N has degree ±p, when M and N are connected closed orientable manifolds. 10. Show that for a degree 1 map f : M→N of connected closed orientable manifolds, the induced map f∗ : π1M→π1N is surjective, hence also f∗ : H1(M)→H1(N). [Lift N→N corresponding to the subgroup Im f∗ ⊂ π1N, then f to the covering space consider the two cases that this covering is ﬁnite-sheeted or inﬁnite-sheeted.] e 11. If Mg denotes the closed orientable surface of genus g, show that degree 1 maps Mg→Mh exist iﬀ g ≥ h. 12. As an algebraic application of the preceding problem, show that in a free group F with basis x1, ···, x2k, the product of commutators [x1, x2] ··· [x2k−1, x2k] is not equal to a product of fewer than k commutators [vi, wi] of elements vi, wi ∈ F. [Recall that the 2 cell of Mk is attached by the product [x1, x2] ··· [x2k−1, x2k]. From a relation [x1, x2] ··· [x2k−1, x2k] = [v1, w1] ··· [vj, wj] in F, construct a degree 1 map Mj→Mk.] h ⊂ Mg be a compact subsurface of genus h with one boundary circle, so h is homeomorphic to Mh with an open disk removed. Show there is no retraction h if h > g/2. [Apply the previous problem, using the fact that Mg − M ′ h has 13. Let M ′ M ′ Mg→M ′ genus g − h.] 14. Let X be the shrinking wedge of circles in Example 1.25
, the subspace of R2 consisting of the circles of radius 1/n and center (1/n, 0) for n = 1, 2, ···. (a) If fn : I→X is the loop based at the origin winding once around the nth circle, show that the inﬁnite product of commutators [f1, f2][f3, f4] ··· deﬁnes a loop in X that is nontrivial in H1(X). [Use Exercise 12.] (b) If we view X as the wedge sum of the subspaces A and B consisting of the odd- numbered and even-numbered circles, respectively, use the same loop to show that the map H1(X)→H1(A)⊕ H1(B) induced by the retractions of X onto A and B is not an isomorphism. Poincar´e Duality Section 3.3 259 15. For an n manifold M and a compact subspace A ⊂ M, show that Hn(M, M −A; R) R(A) of sections of the covering space MR→M over A, is isomorphic to the group that is, maps A→MR whose composition with MR→M is the identity. 16. Show that (α a ϕ) a ψ = α a (ϕ ` ψ) for all α ∈ Ck(X; R), ϕ ∈ C ℓ(X; R), and ψ ∈ C m(X; R). Deduce that cap product makes H∗(X; R) a right H ∗(X; R) module. Γ 17. Show that a direct limit of exact sequences is exact. More generally, show that If {Cα, fαβ} is a directed system of chain homology commutes with direct limits: complexes, with the maps fαβ : Cα→Cβ chain maps, then Hn(lim--→Cα) = lim--→Hn(Cα). 18. Show that a direct limit lim--→Gα of torsionfree abelian groups Gα is torsionfree. More generally, show that any ﬁnitely generated subgroup of lim--→Gα is realized as a subgroup of some Gα. 19. Show that a direct
limit of countable abelian groups over a countable indexing set is countable. Apply this to show that if X is an open set in Rn then Hi(X; Z) is countable for all i. c (X; G) = 0 if X is path-connected and noncompact. 20. Show that H 0 21. For a space X, let X + be the one-point compactiﬁcation. If the added point, denoted ∞, has a neighborhood in X + that is a cone with ∞ the cone point, show that the evident map H n [Question: Does this result hold when X = Z× R?] c (X; G)→H n(X +, ∞; G) is an isomorphism for all n. c (X; G) for all n. 22. Show that H n c(X; G) are isomorphic. This can be done by showing that complex X the simplicial and singular cohoi c(X; G) i(X, A; G) as A ranges over subcomplexes of X that c(X; G) is the union of its sub- c (X × R; G) ≈ H n−1 23. Show that for a locally compact mology groups H i is the union of its subgroups contain all but ﬁnitely many simplices, and likewise C i ∆ groups C i(X, A; G) for the same family of subcomplexes A. 24. Let M be a closed connected 3 manifold, and write H1(M; Z) as Zr ⊕ F, the direct sum of a free abelian group of rank r and a ﬁnite group F. Show that H2(M; Z) is Zr if M is orientable and Zr −1 ⊕ Z2 if M is nonorientable. In particular, r ≥ 1 when M is nonorientable. Using Exercise 6, construct examples showing there are no other ∆ ∆ restrictions on the homology groups of closed 3 manifolds. [In the nonorientable case consider the manifold N obtained from S 2 × I by identifying S 2 × {0} with S 2 × {1} via a reﬂection of S 2.] 25. Show that if a closed orientable manifold M of dimension 2k has Hk−1
(M; Z) torsionfree, then Hk(M; Z) is also torsionfree. 26. Compute the cup product structure in H ∗(S 2 × S 8♯S 4 × S 6; Z), and in particular show that the only nontrivial cup products are those dictated by Poincar´e duality. [See Exercise 6. The result has an evident generalization to connected sums of S i × S n−i ’s for ﬁxed n and varying i.] 260 Chapter 3 Cohomology 27. Show that after a suitable change of basis, a skew-symmetric nonsingular bilinear form over Z can be represented by a matrix consisting of 2× 2 blocks along 0 1 −1 0 the diagonal and zeros elsewhere. [For the matrix of a bilinear form, the following operation can be realized by a change of basis: Add an integer multiple of the ith row to the j th row and add the same integer multiple of the ith column to the j th column. Use this to ﬁx up each column in turn. Note that a skew-symmetric matrix must have zeros on the diagonal.] 28. Show that a nonsingular symmetric or skew-symmetric bilinear pairing over a ﬁeld F, of the form F n × F n→F, cannot be identically zero when restricted to all pairs of vectors v, w in a k dimensional subspace V ⊂ F n if k > n/2. 29. Use the preceding problem to show that if the closed orientable surface Mg of genus g retracts onto a graph X ⊂ Mg, then H1(X) has rank at most g. Deduce an alternative proof of Exercise 13 from this, and construct a retraction of Mg onto a wedge sum of k circles for each k ≤ g. 30. Show that the boundary of an R orientable manifold is also R orientable. 31. Show that if M is a compact R orientable n manifold, then the boundary map Hn(M, ∂M; R)→Hn−1(∂M; R) sends a fundamental class for (M, ∂M) to a fundamental class for ∂M. 32. Show that a compact manifold does not retract onto its boundary. 33. Show that if M is a compact
contractible n manifold then ∂M is a homology (n − 1) sphere, that is, Hi(∂M; Z) ≈ Hi(S n−1; Z) for all i. 34. For a compact manifold M verify that the following diagram relating Poincar´e duality for M and ∂M is commutative, up to sign at least: 35. If M is a noncompact R orientable n manifold with boundary ∂M having a collar neighborhood in M, show that there are Poincar´e duality isomorphisms H k c (M; R) ≈ Hn−k(M, ∂M; R) for all k, using the ﬁve-lemma and the following diagram: Universal Coeﬃcients for Homology Section 3.A 261 The main goal in this section is an algebraic formula for computing homology with arbitrary coeﬃcients in terms of homology with Z coeﬃcients. The theory parallels rather closely the universal coeﬃcient theorem for cohomology in §3.1. The ﬁrst step is to formulate the deﬁnition of homology with coeﬃcients in terms of tensor products. The chain group Cn(X; G) as deﬁned in §2.2 consists of the ﬁnite n→X. This means that Cn(X; G) is a formal sums direct sum of copies of G, with one copy for each singular n simplex in X. More generally, the relative chain group Cn(X, A; G) = Cn(X; G)/Cn(A; G) is also a direct sum of copies of G, one for each singular n simplex in X not contained in A. From the i giσi with gi ∈ G and σi : P ∆ P P i giσi ֏ basic properties of tensor products listed in the discussion of the K¨unneth formula in §3.2 it follows that Cn(X, A; G) is naturally isomorphic to Cn(X, A) ⊗ G, via the i σi ⊗ gi. Under this isomorphism the boundary map correspondence Cn(X, A; G)→Cn−1
(X, A; G) becomes the map ∂ ⊗ 11 : Cn(X, A) ⊗ G→Cn−1(X, A) ⊗ G where ∂ : Cn(X, A)→Cn−1(X, A) is the usual boundary map for Z coeﬃcients. Thus we have the following algebraic problem: ∂n-----→ Cn−1 -→ ··· of free abelian groups Cn, Given a chain complex ··· -→ Cn is it possible to compute the homology groups Hn(C; G) of the associated ∂n ⊗11 ----------------------------→ Cn−1 ⊗ G -----→ ··· just in terms of G and chain complex ··· -----→ Cn ⊗ G the homology groups Hn(C) of the original complex? To approach this problem, the idea will be to compare the chain complex C with two simpler subcomplexes, the subcomplexes consisting of the cycles and the boundaries in C, and see what happens upon tensoring all three complexes with G. Let Zn = Ker ∂n ⊂ Cn and Bn = Im ∂n+1 ⊂ Cn. The restrictions of ∂n to these two subgroups are zero, so they can be regarded as subcomplexes Z and B of C with trivial boundary maps. Thus we have a short exact sequence of chain complexes consisting of the commutative diagrams (i) The rows in this diagram split since each Bn is free, being a subgroup of the free group Cn. Thus Cn ≈ Zn ⊕ Bn−1, but the chain complex C is not the direct sum of the chain complexes Z and B since the latter have trivial boundary maps but the boundary maps in C may be nontrivial. Now tensor with G to get a commutative diagram 262 Chapter 3 Cohomology (ii) The rows are exact since the rows in (i) split and tensor products satisfy (A⊕ B, so the rows in (ii) are split exact sequences too. Thus we have a short exact sequence of chain complexes 0→Z ⊗ G→C ⊗ G→B ⊗ G→0. Since the boundary maps are trivial in Z ⊗ G and B ⊗ G, the associated long exact
sequence of homology groups has the form (iii) ··· -→ Bn ⊗ G -→ Zn ⊗ G -→ Hn(C; G) -→ Bn−1 ⊗ G -→ Zn−1 ⊗ G -→ ··· The ‘boundary’ maps Bn ⊗ G→Zn ⊗ G in this sequence are simply the maps in ⊗ 11 where in : Bn→Zn is the inclusion. This is evident from the deﬁnition of the boundary map in a long exact sequence of homology groups: In diagram (ii) one takes an element of Bn−1 ⊗ G, pulls it back via (∂n ⊗ 11)−1 to Cn ⊗ G, then applies ∂n ⊗ 11 to get into Cn−1 ⊗ G, then pulls back to Zn−1 ⊗ G. The long exact sequence (iii) can be broken up into short exact sequences (iv) 0 -→ Coker(in ⊗ 11) -→ Hn(C; G) -→ Ker(in−1 ⊗ 11) -→ 0 where Coker(in ⊗ 11) = (Zn ⊗ G)/ Im(in ⊗ 11). The next lemma shows this cokernel is just Hn(C) ⊗ G. Lemma 3A.1. If the sequence of abelian groups A j⊗11 ------------------→ C ⊗ G -----→ 0. so is A ⊗ G i⊗11 ------------------→ B ⊗ G i-----→ B j-----→ C -----→ 0 is exact, then Proof: Certainly the compositions of two successive maps in the latter sequence are zero. Also, j ⊗ 11 is clearly surjective since j is. To check exactness at B ⊗ G it suﬃces to show that the map B ⊗ G/ Im(i ⊗ 11)→C ⊗ G induced by j ⊗ 11 is an isomorphism, which we do by constructing its inverse. Deﬁne a map ϕ : C × G→B ⊗ G/ Im(i ⊗ 11) by ϕ(c, g) = b
⊗ g where j(b) = c. This ϕ is well-deﬁned since if j(b) = j(b′) = c then b − b′ = i(a) for some a ∈ A by exactness, so b ⊗ g − b′ ⊗ g = (b − b′) ⊗ g = i(a) ⊗ g ∈ Im(i ⊗ 11). Since ϕ is a homomorphism in each variable separately, it induces a homomorphism C ⊗ G→B ⊗ G/ Im(i ⊗ 11). This is clearly an inverse to the map B ⊗ G/ Im(i ⊗ 11)→C ⊗ G. ⊔⊓ It remains to understand Ker(in−1 ⊗ 11), or equivalently Ker(in ⊗ 11). The situation is that tensoring the short exact sequence (v) 0 --------→ Bn in------------→ Zn --------→ Hn(C) --------→ 0 with G produces a sequence which becomes exact only by insertion of the extra term Ker(in ⊗ 11) : (vi) 0 -→ Ker(in ⊗ 11) -----→ Bn ⊗ G in ⊗11 -------------------------→ Zn ⊗ G -----→ Hn(C) ⊗ G -→ 0 Universal Coeﬃcients for Homology Section 3.A 263 What we will show is that Ker(in ⊗ 11) does not really depend on Bn and Zn but only on their quotient Hn(C), and of course G. The sequence (v) is a free resolution of Hn(C), where as in §3.1 a free resolution of an abelian group H is an exact sequence ··· -----→ F2 f2------------→ F1 f1------------→ F0 f0------------→ H -----→ 0 with each Fn free. Tensoring a free resolution of this form with a ﬁxed group G produces a chain complex ··· -----→ F1 ⊗ G f1 ⊗11 -------------------------→ F0 ⊗ G f0 ⊗11 -------------------------→ H ⊗ G -----→ 0 By the preceding lemma this is exact at F0
⊗ G and H ⊗ G, but to the left of these two terms it may not be exact. For the moment let us write Hn(F ⊗ G) for the homology group Ker(fn ⊗ 11)/ Im(fn+1 ⊗ 11). Lemma 3A.2. For any two free resolutions F and F ′ of H there are canonical isomorphisms Hn(F ⊗ G) ≈ Hn(F ′ ⊗ G) for all n. Proof: We will use Lemma 3.1(a). In the situation described there we have two free resolutions F and F ′ with a chain map between them. If we tensor the two free resolutions with G we obtain chain complexes F ⊗ G and F ′ ⊗ G with the maps αn ⊗ 11 forming a chain map between them. Passing to homology, this chain map induces homomorphisms α∗ : Hn(F ⊗ G)→Hn(F ′ ⊗ G) which are independent of the choice of αn ’s since if αn and α′ n are chain homotopic via a chain homotopy λn then αn ⊗ 11 and α′ n ⊗ 11 are chain homotopic via λn ⊗ 11. For a composition H α-----→ H ′ β-----→ H ′′ with free resolutions F, F ′, and F ′′ of these three groups also given, the induced homomorphisms satisfy (βα)∗ = β∗α∗ since we can choose for the chain map F→F ′′ the composition of chain maps F→F ′→F ′′. In particular, if we take α to be an isomorphism, with β its inverse and F ′′ = F, then β∗α∗ = (βα)∗ = 11∗ = 11, and similarly with β and α reversed. So α∗ is an isomorphism if α is an isomorphism. Specializing further, taking α to be the identity but with two diﬀerent free resolutions F and F ′, we get a canonical isomorphism 11∗ : Hn(F ⊗ G)→Hn(F ′ ⊗ G).
⊔⊓ The group Hn(F ⊗ G), which depends only on H and G, is denoted Torn(H, G). Since a free resolution 0→F1→F0→H→0 always exists, as noted in §3.1, it follows that Torn(H, G) = 0 for n > 1. Usually Tor1(H, G) is written simply as Tor(H, G). As we shall see later, Tor(H, G) provides a measure of the common torsion of H and G, hence the name ‘ Tor’. Is there a group Tor0(H, G)? With the deﬁnition given above it would be zero since Lemma 3A.1 implies that F1 ⊗ G→F0 ⊗ G→H ⊗ G→0 is exact. It is probably better to modify the deﬁnition of Hn(F ⊗ G) to be the homology groups of the sequence 264 Chapter 3 Cohomology ··· →F1 ⊗ G→F0 ⊗ G→0, omitting the term H ⊗ G which can be regarded as a kind of augmentation. With this revised deﬁnition, Lemma 3A.1 then gives an isomorphism Tor0(H, G) ≈ H ⊗ G. We should remark that Tor(H, G) is a functor of both G and H : Homomorphisms α : H→H ′ and β : G→G′ induce homomorphisms α∗ : Tor(H, G)→Tor(H ′, G) and β∗ : Tor(H, G)→Tor(H, G′), satisfying (αα′)∗ = α∗α′ ∗, and 11∗ = 11. The induced map α∗ was constructed in the proof of Lemma 3A.2, while for β the construction of β∗ is obvious. ∗, (ββ′)∗ = β∗β′ Before going into calculations of Tor(H, G) let us ﬁnish analyzing the earlier exact sequence (iv). Recall that we have a chain complex C of free abelian groups, with homology groups denoted Hn(C), and tens
oring C with G gives another complex C ⊗ G whose homology groups are denoted Hn(C; G). The following result is known as the universal coeﬃcient theorem for homology since it describes homology with arbitrary coeﬃcients in terms of homology with the ‘universal’ coeﬃcient group Z. Theorem 3A.3. If C is a chain complex of free abelian groups, then there are natural short exact sequences 0 -→ Hn(C) ⊗ G -→ Hn(C; G) -→ Tor(Hn−1(C), G) -→ 0 for all n and all G, and these sequences split, though not naturally. Naturality means that a chain map C→C ′ induces a map between the correspond- ing short exact sequences, with commuting squares. Proof: This exact sequence is (iv) since we can identify Coker(in ⊗ 11) with Hn(C) ⊗ G and Ker(in−1 ⊗ 11) with Tor(Hn−1(C), G). Verifying naturality is a mental exercise in deﬁnition-checking, left to the reader. The splitting is obtained as follows. We observed earlier that the short exact sequence 0→Zn→Cn→Bn−1→0 splits, so there is a projection p : Cn→Zn restricting to the identity on Zn. The map p gives an extension of the quotient map Zn→Hn(C) to a homomorphism Cn→Hn(C). Letting n vary, we then have a chain map C→H(C) where the groups Hn(C) are regarded as a chain complex with trivial boundary maps, so the chain map condition is automatic. Now tensor with G to get a chain map C ⊗ G→H(C) ⊗ G. Taking homology groups, we then have induced homomorphisms Hn(C; G)→Hn(C) ⊗ G since the boundary maps in the chain complex H(C) ⊗ G are trivial. The homomorphisms Hn(C; G)→Hn(C) ⊗ G give the desired splitting since at ⊔⊓ the level of chains they are the identity on

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