Split (1)

train · 21.9k rows

text stringlengths 270 6.81k
cycles in C, by the deﬁnition of p. Corollary 3A.4. For each pair of spaces (X, A) there are split exact sequences 0 -→ Hn(X, A) ⊗ G -→ Hn(X, A; G) -→ Tor(Hn−1(X, A), G) -→ 0 for all n, and these sequences are natural with respect to maps (X, A)→(Y, B). ⊔⊓ The splitting is not natural, for if it were, a map X→Y that induced trivial maps Hn(X)→Hn(Y ) and Hn−1(X)→Hn−1(Y ) would have to induce the trivial map Universal Coeﬃcients for Homology Section 3.A 265 Hn(X; G)→Hn(Y ; G) for all G, but in Example 2.51 we saw an instance where this fails, namely the quotient map M(Zm, n)→S n+1 with G = Zm. The basic tools for computing Tor are given by: Proposition 3A.5. (1) Tor(A, B) ≈ Tor(B, A). (2) Tor( iAi, B) ≈ iTor(Ai, B). L L (3) Tor(A, B) = 0 if A or B is free, or more generally torsionfree. (4) Tor(A, B) ≈ Tor(T (A), B) where T (A) is the torsion subgroup of A. (5) Tor(Zn, A) ≈ Ker(A (6) For each short exact sequence 0→B→C→D→0 there is a naturally associated n-----→ A). exact sequence 0→Tor(A, B)→Tor(A, C)→Tor(A, D)→A ⊗ B→A ⊗ C→A ⊗ D→0 Proof: Statement (2) is easy since one can choose as a free resolution of i Ai the direct sum of free resolutions of the Ai ’s. Also easy is (5), which comes from tensoring the free resolution 0→Z n-----→ Z→Zn→0 with A. L For (3), if A is
free, it has a free resolution with Fn = 0 for n ≥ 1, so Tor(A, B) = 0 for all B. On the other hand, if B is free, then tensoring a free resolution of A with B preserves exactness, since tensoring a sequence with a direct sum of Z ’s produces just a direct sum of copies of the given sequence. So Tor(A, B) = 0 in this case too. The generalization to torsionfree A or B will be given below. For (6), choose a free resolution 0→F1→F0→A→0 and tensor with the given short exact sequence to get a commutative diagram The rows are exact since tensoring with a free group preserves exactness. Extending the three columns by zeros above and below, we then have a short exact sequence of chain complexes whose associated long exact sequence of homology groups is the desired six-term exact sequence. To prove (1) we apply (6) to a free resolution 0→F1→F0→B→0. Since Tor(A, F1) and Tor(A, F0) vanish by the part of (3) which we have proved, the six-term sequence in (6) reduces to the ﬁrst row of the following diagram: The second row comes from the deﬁnition of Tor(B, A). The vertical isomorphisms come from the natural commutativity of tensor product. Since the squares commute, there is induced a map Tor(A, B)→Tor(B, A), which is an isomorphism by the ﬁvelemma. 266 Chapter 3 Cohomology 0 -→ F1 if B is torsionfree. Suppose Now we can prove the statement (3) in the torsionfree case. For a free resolution ϕ-----→ F0 -→ A -→ 0 we wish to show that ϕ ⊗ 11 : F1 ⊗ B→F0 ⊗ B is injective i xi ⊗ bi lies in the kernel of ϕ ⊗ 11. This means that i ϕ(xi) ⊗ bi can be reduced to 0 by a ﬁnite number of applications of the deﬁning relations for tensor products. Only a ﬁnite number of elements of B are involved in P this
process. These lie in a ﬁnitely generated subgroup B0 ⊂ B, so i xi ⊗ bi lies in the kernel of ϕ ⊗ 11 : F1 ⊗ B0→F0 ⊗ B0. This kernel is zero since Tor(A, B0) = 0, as B0 is ﬁnitely generated and torsionfree, hence free. P P Finally, we can obtain statement (4) by applying (6) to the short exact sequence 0→T (A)→A→A/T (A)→0 since A/T (A) is torsionfree. ⊔⊓ In particular, (5) gives Tor(Zm, Zn) ≈ Zq where q is the greatest common divisor of m and n. Thus Tor(Zm, Zn) is isomorphic to Zm ⊗ Zn, though somewhat by accident. Combining this isomorphism with (2) and (3) we see that for ﬁnitely generated A and B, Tor(A, B) is isomorphic to the tensor product of the torsion subgroups of A and B, or roughly speaking, the common torsion of A and B. This is one reason for the ‘ Tor’ designation, further justiﬁcation being (3) and (4). Homology calculations are often simpliﬁed by taking coeﬃcients in a ﬁeld, usually Q or Zp for p prime. In general this gives less information than taking Z coeﬃcients, but still some of the essential features are retained, as the following result indicates: Corollary 3A.6. (a) Hn(X; Q) ≈ Hn(X; Z) ⊗ Q, so when Hn(X; Z) is ﬁnitely generated, the dimension of Hn(X; Q) as a vector space over Q equals the rank of Hn(X; Z). (b) If Hn(X; Z) and Hn−1(X; Z) are ﬁnitely generated, then for p prime, Hn(X; Zp) consists of (i) a Zp summand for each Z summand of Hn(X;
Z), (ii) a Zp summand for each Zpk summand in Hn(X; Z), k ≥ 1, (iii) a Zp summand for each Zpk summand in Hn−1(X; Z), k ≥ 1. ⊔⊓ Even in the case of nonﬁnitely generated homology groups, ﬁeld coeﬃcients still give good qualitative information: Hn(X; Z) = 0 for all n iﬀ Corollary 3A.7. (a) all n and all primes p. e (b) A map f : X→Y induces isomorphisms on homology with Z coeﬃcients iﬀ it induces isomorphisms on homology with Q and Zp coeﬃcients for all primes p. Hn(X; Q) = 0 and Hn(X; Zp) = 0 for e e Proof: Statement (b) follows from (a) by passing to the mapping cone of f. The universal coeﬃcient theorem gives the ‘only if’ half of (a). For the ‘if’ implication it suﬃces to show that if an abelian group A is such that A ⊗ Q = 0 and Tor(A, Zp) = 0 Universal Coeﬃcients for Homology Section 3.A 267 p-----→ Z→Zp→0 and for all primes p, then A = 0. For the short exact sequences 0→Z 0→Z→Q→Q/Z→0, the six-term exact sequences in (6) of the proposition become 0 -→ Tor(A, Zp) -→ A p-----→ A -→ A⊗ Zp -→ 0 0 -→ Tor(A, Q/Z) -→ A -→ A⊗ Q -→ A⊗ Q/Z -→ 0 p-----→ A If Tor(A, Zp) = 0 for all p, then exactness of the ﬁrst sequence implies that A is injective for all p, so A is torsionfree. Then Tor(A, Q/Z) = 0 by (3) or (4) of the proposition, so the second sequence implies that A→A �
� Q is injective, hence A = 0 if A ⊗ Q = 0. ⊔⊓ The algebra by means of which the Tor functor is derived from tensor products has a very natural generalization in which abelian groups are replaced by modules over a ﬁxed ring R with identity, using the deﬁnition of tensor product of R modules given in §3.2. Free resolutions of R modules are deﬁned in the same way as for abelian groups, using free R modules, which are direct sums of copies of R. Lemmas 3A.1 and 3A.2 carry over to this context without change, and so one has functors TorR n(A, B). However, it need not be true that TorR n(A, B) = 0 for n > 1. The reason this was true when R = Z was that subgroups of free groups are free, but submodules of free R modules need not be free in general. If R is a principal ideal domain, submodules of free R modules are free, so in this case the rest of the algebra, in particular the universal coeﬃcient theorem, goes through without change. When R is a ﬁeld F, every n(A, B) = 0 for n > 0 via the free resolution 0→A→A→0. module is free and TorF Thus Hn(C ⊗F G) ≈ Hn(C) ⊗F G if F is a ﬁeld. Exercises 1. Use the universal coeﬃcient theorem to show that if H∗(X; Z) is ﬁnitely generated, n(−1)n rank Hn(X; Z) is deﬁned, then for any so the Euler characteristic χ (X) = n(−1)n dim Hn(X; F ). coeﬃcient ﬁeld F we have χ (X) = P 2. Show that Tor(A, Q/Z) is isomorphic to the torsion subgroup of A. Deduce that A is torsionfree iﬀ Tor(A, B) = 0 for all B. P H n(X; Q) and H n(X; Zp) are zero
for all n and all primes p, then 3. Show that if Hn(X; Z) = 0 for all n, and hence e e 4. Show that ⊗ and Tor commute with direct limits: (lim--→Aα) ⊗ B = lim--→(Aα ⊗ B) and e Tor(lim--→Aα, B) = lim--→ Tor(Aα, B). 5. From the fact that Tor(A, B) = 0 if A is free, deduce that Tor(A, B) = 0 if A H n(X; G) = 0 for all G and n. e is torsionfree by applying the previous problem to the directed system of ﬁnitely generated subgroups Aα of A. 6. Show that Tor(A, B) is always a torsion group, and that Tor(A, B) contains an element of order n iﬀ both A and B contain elements of order n. 268 Chapter 3 Cohomology K¨unneth formulas describe the homology or cohomology of a product space in terms of the homology or cohomology of the factors. In nice cases these formulas take the form H∗(X × Y ; R) ≈ H∗(X; R) ⊗ H∗(Y ; R) or H ∗(X × Y ; R) ≈ H ∗(X; R) ⊗ H ∗(Y ; R) for a coeﬃcient ring R. For the case of cohomology, such a formula was given in Theorem 3.15, with hypotheses of ﬁnite generation and freeness on the cohomology of one factor. To obtain a completely general formula without these hypotheses it turns out that homology is more natural than cohomology, and the main aim in this section is to derive the general K¨unneth formula for homology. The new feature of the general case is that an extra Tor term is needed to describe the full homology of a product. The Cross Product in Homology A major component of the K¨unneth formula is a cross product map Hi(X; R)× Hj (Y ; R) ×--------------------→ Hi+j (X × Y ; R) There are two ways to deﬁne this. One is a direct de�
��nition for singular homology, involving explicit simplicial formulas. More enlightening, however, is the deﬁnition in terms of cellular homology. This necessitates assuming X and Y are CW complexes, but this hypothesis can later be removed by the technique of CW approximation in §4.1. We shall focus therefore on the cellular deﬁnition, leaving the simplicial deﬁni- tion to later in this section for those who are curious to see how it goes. The key ingredient in the deﬁnition of the cellular cross product will be the fact that the cellular boundary map satisﬁes d(ei × ej) = dei × ej + (−1)iei × dej. Implicit in the right side of this formula is the convention of treating the symbol × as a bilinear operation on cellular chains. With this convention we can then say more generally that d(a× b) = da× b + (−1)ia× db whenever a is a cellular i chain and b is a cellular j chain. From this formula it is obvious that the cross product of two cycles is a cycle. Also, the product of a boundary and a cycle is a boundary since da× b = d(a× b) if db = 0, and similarly a× db = (−1)id(a× b) if da = 0. Hence there is an induced bilinear map Hi(X; R)× Hj(Y ; R)→Hi+j (X × Y ; R), which is by deﬁnition the cross product in cellular homology. Since it is bilinear, it could also be viewed as a homomorphism Hi(X; R) ⊗RHj(Y ; R)→Hi+j(X × Y ; R). In either form, this cross product turns out to be independent of the cell structures on X and Y. Our task then is to express the boundary maps in the cellular chain complex C∗(X × Y ) for X × Y in terms of the boundary maps in the cellular chain complexes C∗(X) and C∗(Y ). For simplicity we consider homology with Z coeﬃcients here, but the same formula for arbitrary coeﬃcients follows immediately from this special case. With Z coeﬃcients, the cellular
chain group Ci(X) is free with basis the i cells of X, but there is a sign ambiguity for the basis element corresponding to each cell ei, The General K¨unneth Formula Section 3.B 269 namely the choice of a generator for the Z summand of Hi(X i, X i−1) corresponding to ei. Only when i = 0 is this choice canonical. We refer to these choices as ‘choosing orientations for the cells’. A choice of such orientations allows cellular i chains to be written unambiguously as linear combinations of i cells. The formula d(ei × ej) = dei × ej +(−1)iei × dej is not completely canonical since it contains the sign (−1)i but not (−1)j. Evidently there is some distinction being made between the two factors of ei × ej. Since the signs arise from orientations, we need to make explicit how an orientation of cells ei and ej determines an orientation of ei × ej. Via characteristic maps, orientations can be obtained from orientations of It will be convenient to choose these the domain disks of the characteristic maps. domains to be cubes since the product of two cubes is again a cube. Thus for a cell ei α α : Ii→X where Ii is the product of i intervals [0, 1]. we take a characteristic map An orientation of Ii is a generator of Hi(Ii, ∂Ii), and the image of this generator under Φ α. We can identify Hi(Ii, ∂Ii) with Hi(Ii, Ii − {x}) for α∗ gives an orientation of ei any point x in the interior of Ii, and then an orientation is determined by a linear Φ i→Ii with x chosen in the interior of the image of this embedding. embedding The embedding is determined by its sequence of vertices v0, ···, vi. The vectors v1 −v0, ···, vi −v0 are linearly independent in Ii, thought of as the unit cube in Ri, so an orientation in our sense is equivalent to an orientation in the sense of linear algebra, ∆ that is, an equivalence class of ordered bases, two ordered bases being equivalent if they diﬀer by a linear transformation of positive determinant. (An ordered basis
can be continuously deformed to an orthonormal basis, by the Gram–Schmidt process, and two orthonormal bases are related either by a rotation or a rotation followed by a reﬂection, according to the sign of the determinant of the transformation taking one to the other.) With this in mind, we adopt the convention that an orientation of Ii × Ij = Ii+j is obtained by choosing an ordered basis consisting of an ordered basis for Ii followed by an ordered basis for Ij. Notice that reversing the orientation for either Ii or Ij then reverses the orientation for Ii+j, so all that really matters is the order of the two factors of Ii × Ij. Proposition 3B.1. The boundary map in the cellular chain complex C∗(X × Y ) is determined by the boundary maps in the cellular chain complexes C∗(X) and C∗(Y ) via the formula d(ei × ej) = dei × ej + (−1)iei × dej. Proof: Let us ﬁrst consider the special case of the cube In. We give I the CW structure with two vertices and one edge, so the ith copy of I has a 1 cell ei and 0 cells 0i and 1i, with dei = 1i − 0i. The n cell in the product In is e1 × ··· × en, and we claim that the boundary of this cell is given by the formula (∗) d(e1 × ··· × en) = (−1)i+1e1 × ··· × dei × ··· × en Xi 270 Chapter 3 Cohomology This formula is correct modulo the signs of the individual terms e1 × ··· × 0i × ··· × en and e1 × ··· × 1i × ··· × en since these are exactly the (n − 1) cells in the boundary sphere ∂In of In. To obtain the signs in (∗), note that switching the two ends of an I factor of In produces a reﬂection of ∂In, as does a transposition of two adjacent I factors. Since reﬂections have degree −1, this implies that (∗) is correct up to an overall sign. This ﬁnal sign can be determined by looking at any term, say the term 01 × e
2 × ··· × en, which has a minus sign in (∗). To check that this is right, consider the n simplex [v0, ···, vn] with v0 at the origin and vk the unit vector along the k th coordinate axis for k > 0. This simplex deﬁnes the ‘positive’ orientation of In as described earlier, and in the usual formula for its boundary the face [v0, v2, ···, vn], which deﬁnes the positive orientation for the face 01 × e2× ··· × en of In, has a minus sign. If we write In = Ii × Ij with i + j = n and we set ei = e1 × ··· × ei and ej = ei+1 × ··· × en, then the formula (∗) becomes d(ei × ej) = dei × ej + (−1)iei × dej. We will use naturality to reduce the general case of the boundary formula to this special case. When dealing with cellular homology, the maps f : X→Y that induce chain maps f∗ : C∗(X)→C∗(Y ) of the cellular chain complexes are the cellular maps, taking X n to Y n for all n, hence (X n, X n−1) to (Y n, Y n−1). The naturality statement we want is then: Lemma 3B.2. For cellular maps f : X→Z and g : Y →W, the cellular chain maps f∗ : C∗(X)→C∗(Z), g∗ : C∗(Y )→C∗(W ), and (f × g)∗ : C∗(X × Y )→C∗(Z × W ) are related by the formula (f × g)∗ = f∗ × g) = δ nβδe γ mαγei δ, then (f × g)∗(ei Proof: The relation (f × g)∗ = f∗ × g∗ means that if f∗(ei α) = γ and if j j γδ mαγnβδ(ei γ × e β) = δ)
. The coeﬃcient g∗(e mαγ is the degree of the composition fαγ : S i→X i/X i−1→Z i/Z i−1→S i where the ﬁrst and third maps are induced by characteristic maps for the cells ei γ, and the middle map is induced by the cellular map f. With the natural choices of basepoints in these quotient spaces, fαγ is basepoint-preserving. The nβδ ’s are obtained similarly from maps gβδ : S j→S j. For f × g, the map (f × g)αβ,γδ : S i+j→S i+j whose degree is the coeﬃcient of ei β) is obtained from the product map fαγ × gβδ : S i × S j→S i × S j by collapsing the (i + j − 1) skeleton of S i × S j to a point. In other words, (f × g)αβ,γδ is the smash product map fαγ ∧ gβδ. What we need to show is the formula deg(f ∧ g) = deg(f ) deg(g) for basepoint-preserving maps f : S i→S i and g : S j→S j. δ in (f × g)∗(ei α and ei α × ej γ × ej Since f ∧ g is the composition of f ∧ 11 and 11 ∧ g, it suﬃces to show that deg(f ∧11) = deg(f ) and deg(11∧g) = deg(g). We do this by relating smash products to suspension. The smash product X ∧S 1 can be viewed as X × I/(X × ∂I ∪{x0}× I), so it is the reduced suspension X, the quotient of the ordinary suspension SX obtained by collapsing the segment {x0}× I to a point. If X is a CW complex with x0 a 0 cell, Σ The General K¨unneth Formula Section 3.B 271 the quotient map SX→X ∧S 1 induces an isomorphism on homology since it collapses a contractible subcomplex to a point. Taking X
= S i, we have the commutative diagram at the right, and from the induced commutative diagram of homology groups Hi+1 we deduce that Sf and f ∧ 11 have the same degree. Since suspension preserves degree by Proposition 2.33, we conclude that deg(f ∧ 11) = deg(f ). The 11 in this formula is the identity map on S 1, and by iteration we obtain the same result for 11 the identity map on S j since S j is the smash product of j copies of S 1. This implies also that deg(11 ∧ g) = deg(g) since a permutation of coordinates in S i+j does not aﬀect the degree of maps S i+j→S i+j. ⊔⊓ α ⊂ X and ej Now to ﬁnish the proof of the proposition, let : Ij→Y j be char: Ii→X i and to ∂Ii is the atacteristic maps of cells ei β ⊂ Y. The restriction of Φ taching map of ei α. We may perform a preliminary homotopy of this attaching map ∂Ii→X i−1 to make it cellular. There is no need to appeal to the cellular approximation theorem to do this since a direct argument is easy: First deform the attaching map so that it sends all but one face of Ii to a point, which is possible since the union of these faces is contractible, then do a further deformation so that the image point of this union of faces is a 0 cell. A homotopy of the attaching map ∂Ii→X i−1 does not aﬀect the cellular boundary dei α, since dei α is determined by the induced map Hi−1(∂Ii)→Hi−1(X i−1)→Hi−1(X i−1, X i−2). So we may assume is cellular, and likewise. The map of cellular chain complexes induced by a cellular, hence also × Φ Ψ map between CW complexes is a chain map, commuting with the cellular boundary Φ Ψ maps. Φ Ψ If ei is the i cell of Ii and ej the j cell of Ij, then ∗(ei) = ei α, ∗(ej ) = ej β,
and ( × )∗(ei × ej) = ei α × e j β, hence Φ Ψ d(ei Ψ Φ α × e j β) = d = ( ( = ( since ( )∗(ei × ej) × )∗d(ei × ej) × Ψ )∗(dei × ej + (−1)iei × dej) Ψ Φ ∗(ei)× ∗(ei)× d Φ Ψ ∗(ej) + (−1)i ∗(ej) + (−1)i Ψ × Φ × Ψ Φ ∗(dei)× Ψ Φ ∗(ei)× = d Φ = ∗(dej) ∗(ej ) α× ej = dei Φ Ψ β + (−1)iei α × dej β Φ Ψ )∗ is a chain map by the special case by the lemma ∗ and since chain maps Φ ∗ are Ψ which completes the proof of the proposition. ⊔⊓ Example 3B.3. Consider X × S k where we give S k its usual CW structure with two cells. The boundary formula in C∗(X × S k) takes the form d(a× b) = da× b since d = 0 in C∗(S k). So the chain complex C∗(X × S k) is just the direct sum of two copies of the chain complex C∗(X), one of the copies having its dimension shifted 272 Chapter 3 Cohomology upward by k. Hence Hn(X × S k; Z) ≈ Hn(X; Z)⊕ Hn−k(X; Z) for all n. In particular, we see that all the homology classes in X × S k are cross products of homology classes in X and S k. Example 3B.4. More subtle things can happen when X and Y both have torsion in their homology. To take the simplest case, let X be S 1 with a cell e2 attached by a map S 1→S 1 of degree m, so H1(X; Z) ≈ Zm and Hi(X; Z) = 0 for i > 1. Similarly, let Y be obtained from
S 1 by attaching a 2 cell by a map of degree n. Thus X and Y each have CW structures with three cells and so X × Y has nine cells. These are indicated by the dots in the diagram at the right, with X in the horizontal direction and Y in the vertical direction. The arrows denote the nonzero cellular boundary maps. For example the two arrows leaving the dot in the upper right corner indicate that ∂(e2 × e2) = m(e1 × e2) + n(e2 × e1). Obviously H1(X × Y ; Z) is Zm ⊕ Zn. In dimension 2, Ker ∂ is generated by e1 × e1, and the image of the boundary map from dimension 3 consists of the multiples (ℓm − kn)(e1 × e1). These form a cyclic group generated by q(e1 × e1) where q is the greatest common divisor of m and n, so H2(X × Y ; Z) ≈ Zq. In dimension 3 the cycles are the multiples of (m/q)(e1 × e2) + (n/q)(e2 × e1), and the smallest such multiple that is a boundary is q[(m/q)(e1 × e2) + (n/q)(e2 × e1)] = m(e1 × e2) + n(e2 × e1), so H3(X × Y ; Z) ≈ Zq. Since X and Y have no homology above dimension 1, this 3 dimensional homology of X × Y cannot be realized by cross products. As the general theory will show, H2(X × Y ; Z) is H1(X; Z) ⊗ H1(Y ; Z) and H3(X × Y ; Z) is Tor(H1. This example generalizes easily to higher dimensions, with X = S i ∪ ei+1 and Y = S j ∪ ej+1, the attaching maps having degrees m and n, respectively. Essentially the same calculation shows that X × Y has both Hi+j and Hi+j+1 isomorphic to Zq. X; Z), H1(Y ; Z) We should say a few words about why the cross product is independent of CW structures. For this we will need a fact proved in the
next chapter in Theorem 4.8, that every map between CW complexes is homotopic to a cellular map. As we mentioned earlier, a cellular map induces a chain map between cellular chain complexes. It is easy to see from the equivalence between cellular and singular homology that the map on cellular homology induced by a cellular map is the same as the map induced on singular homology. Now suppose we have cellular maps f : X→Z and g : Y →W. Then Lemma 3B.2 implies that we have a commutative diagram Now take Z and W to be the same spaces as X and Y but with diﬀerent CW structures, and let f and g be cellular maps homotopic to the identity. The vertical maps in the The General K¨unneth Formula Section 3.B 273 diagram are then the identity, and commutativity of the diagram says that the cross products deﬁned using the diﬀerent CW structures coincide. Cross product is obviously bilinear, or in other words, distributive. It is not hard to check that it is also associative. What about commutativity? If T : X × Y →Y × X is transposition of the factors, then we can ask whether T∗(a× b) equals b× a. The only eﬀect transposing the factors has on the deﬁnition of cross product is in the convention for orienting a product Ii × Ij by taking an ordered basis in the ﬁrst factor followed by an ordered basis in the second factor. Switching the two factors can be achieved by moving each of the i coordinates of Ii past each of the coordinates of Ij. This is a total of ij transpositions of adjacent coordinates, each realizable by a reﬂection, so a sign of (−1)ij is introduced. Thus the correct formula is T∗(a× b) = (−1)ij b× a for a ∈ Hi(X) and b ∈ Hj (Y ). The Algebraic K¨unneth Formula By adding together the various cross products we obtain a map and it is natural to ask whether this is an isomorphism. Example 3B.4 above shows i Hi(X; Z) ⊗ Hn−i(Y ; Z) L ---------→ Hn(X × Y ; Z) that
this is not always the case, though it is true in Example 3B.3. Our main goal in what follows is to show that the map is always injective, and that its cokernel is. More generally, we consider other coeﬃcients besides iTor Hi(X; Z), Hn−i−1(Y ; Z) Z and show in particular that with ﬁeld coeﬃcients the map is an isomorphism. L For CW complexes X and Y, the relationship between the cellular chain complexes C∗(X), C∗(Y ), and C∗(X × Y ) can be expressed nicely in terms of tensor products. Since the n cells of X × Y are the products of i cells of X with (n − i) cells of Y,, with ei × ej corresponding to ei ⊗ ej. UnCi(X) ⊗ Cn−i(Y ) we have Cn(X × Y ) ≈ der this identiﬁcation the boundary formula of Proposition 3B.1 becomes d(ei ⊗ ej) = dei ⊗ ej + (−1)iei ⊗ dej. Our task now is purely algebraic, to compute the homology of the chain complex C∗(X × Y ) from the homology of C∗(X) and C∗(Y ). i L Suppose we are given chain complexes C and C ′ of abelian groups Cn and C ′ n, or more generally R modules over a commutative ring R. The tensor product chain complex C ⊗RC ′ is then deﬁned by (C ⊗RC ′)n = n−i), with boundary maps given by ∂(c ⊗ c′) = ∂c ⊗ c′ + (−1)ic ⊗ ∂c′ for c ∈ Ci and c′ ∈ C ′ n−i. The sign (−1)i guarantees that ∂2 = 0 in C ⊗RC ′, since i(Ci ⊗RC ′ L ∂2(c ⊗ c′) = ∂ ∂c ⊗ c′ + (−1)ic ⊗ ∂c′ = ∂
2c ⊗ c′ + (−1)i−1∂c ⊗ ∂c′ + (−1)i∂c ⊗ ∂c′ + c ⊗ ∂2c′ = 0 From the boundary formula ∂(c ⊗ c′) = ∂c ⊗ c′ + (−1)ic ⊗ ∂c′ it follows that the tensor product of cycles is a cycle, and the tensor product of a cycle and a boundary, in either order, is a boundary, just as for the cross product deﬁned earlier. So there is induced a natural map on homology groups Hi(C) ⊗RHn−i(C ′)→Hn(C ⊗RC ′). Summing over i 274 Chapter 3 Cohomology then gives a map Hi(C) ⊗RHn−i(C ′) i algebraic version of the K¨unneth formula: L →Hn(C ⊗RC ′). This ﬁgures in the following Theorem 3B.5. If R is a principal ideal domain and the R modules Ci are free, then for each n there is a natural short exact sequence 0→ Hi(C) ⊗R Hn−i(C ′) →Hn(C ⊗R C ′)→ iTorR L Hi(C), Hn−i−1(C ′) →0 i L and this sequence splits. This is a generalization of the universal coeﬃcient theorem for homology, which is the case that R = Z and C ′ consists of just the coeﬃcient group G in dimension zero. The proof will also be a natural generalization of the proof of the universal coeﬃcient theorem. Proof: First we do the special case that the boundary maps in C are all zero, so In this case ∂(c ⊗ c′) = (−1)ic ⊗ ∂c′ and the chain complex C ⊗RC ′ is Hi(C) = Ci. simply the direct sum of the complexes Ci ⊗RC ′, each of which is a direct sum of copies of C ′ since Ci is free. Hence Hn(Ci ⊗RC ′) �
� Ci ⊗RHn−i(C ′) = Hi(C) ⊗RHn−i(C ′). Summing over i yields an isomorphism Hn(C ⊗RC ′) ≈, which is the statement of the theorem since there are no Tor terms, Hi(C) = Ci being free. In the general case, let Zi ⊂ Ci and Bi ⊂ Ci denote kernel and image of the boundary homomorphisms for C. These give subchain complexes Z and B of C Hi(C) ⊗RHn−i(C ′) L i with trivial boundary maps. We have a short exact sequence of chain complexes ∂-----→ Bi−1→0 0→Z→C→B→0 made up of the short exact sequences 0→Zi→Ci each of which splits since Bi−1 is free, being a submodule of Ci−1 which is free by assumption. Because of the splitting, when we tensor 0→Z→C→B→0 with C ′ we obtain another short exact sequence of chain complexes, and hence a long exact sequence in homology ··· -→ Hn(Z ⊗RC ′) -→ Hn(C ⊗RC ′) -→ Hn−1(B ⊗RC ′) -→ Hn−1(Z ⊗RC ′) -→ ··· where we have Hn−1(B ⊗RC ′) instead of the expected Hn(B ⊗RC ′) since ∂ : C→B decreases dimension by one. Checking deﬁnitions, one sees that the ‘boundary’ map Hn−1(B ⊗RC ′)→Hn−1(Z ⊗RC ′) in the preceding long exact sequence is just the map induced by the natural map B ⊗RC ′→Z ⊗RC ′ coming from the inclusion B ⊂ Z. Since Z and B are chain complexes with trivial boundary maps, the special case at the beginning of the proof converts the preceding exact sequence into ··· in-----→ Zi ⊗R Hn−i(C ′) i L -→ Hn(C ⊗R C ′) -→ i L So we have short exact sequences
L Bi ⊗R Hn−i−1(C ′) in−1 ------------→ Zi ⊗R Hn−i−1(C ′) i -→ ··· 0 -→ Coker in -→ Hn(C ⊗R C ′) -→ Ker in−1 -→ 0 The General K¨unneth Formula Section 3.B 275 where Coker in = by Lemma 3A.1. It remains to identify Ker in−1 with Hi(C) ⊗RHn−i(C ′) By the deﬁnition of Tor, tensoring the free resolution 0→Bi→Zi→Hi(C)→0 i Hi(C), Hn−i−1(C ′) L / Im in, and this equals iTorR Zi ⊗RHn−i(C ′) L L. i with Hn−i(C ′) yields an exact sequence 0 -→ TorR Hi(C), Hn−i(C ′) -→ Bi ⊗R Hn−i(C ′) -→ Zi ⊗R Hn−i(C ′) -→ Hi(C)⊗R Hn−i(C ′) -→ 0 Hence, summing over i, Ker in = iTorR Hi(C), Hn−i(C ′). Naturality should be obvious, and we leave it for the reader to ﬁll in the details. L We will show that the short exact sequence in the statement of the theorem splits assuming that both C and C ′ are free. This suﬃces for our applications. For the extra argument needed to show splitting when only C is assumed to be free, see the exposition in [Hilton & Stammbach 1970]. i L Hi(C) ⊗RHn−i(C ′) The splitting is via a homomorphism Hn(C ⊗RC ′)→ constructed in the following way. As already noted, the sequence 0→Zi→Ci→Bi−1→0 splits, so the quotient maps Zi→Hi(C) extend to homomorphisms Ci→Hi(C). Simj→Hj(C ′) if C ′ is free. Viewing the sequences of homology
groups ilarly we obtain C ′ Hi(C) and Hj(C ′) as chain complexes H(C) and H(C ′) with trivial boundary maps, we thus have chain maps C→H(C) and C ′→H(C ′), whose tensor product is a chain map C ⊗RC ′→H(C) ⊗RH(C ′). The induced map on homology for this last chain map is the desired splitting map since the chain complex H(C) ⊗RH(C ′) equals its own ⊔⊓ homology, the boundary maps being trivial. The Topological K¨unneth Formula Now we can apply the preceding algebra to obtain the topological statement we are looking for: Theorem 3B.6. If X and Y are CW complexes and R is a principal ideal domain, then there are natural short exact sequences 0 -→ i L Hi(X; R)⊗R Hn−i(Y ; R) -→ Hn(X × Y ; R) -→ iTorR and these sequences split. L Hi(X; R), Hn−i−1(Y ; R) -→ 0 Naturality means that maps X→X ′ and Y →Y ′ induce a map from the short exact sequence for X × Y to the corresponding short exact sequence for X ′× Y ′, with commuting squares. The splitting is not natural, however, as an exercise at the end of this section demonstrates. Proof: When dealing with products of CW complexes there is always the bothersome fact that the compactly generated CW topology may not be the same as the product topology. However, in the present context this is not a real problem. Since the two 276 Chapter 3 Cohomology topologies have the same compact sets, they have the same singular simplices and hence the same singular homology groups. Let C = C∗(X; R) and C ′ = C∗(Y ; R), the cellular chain complexes with coeﬃcients in R. Then C ⊗RC ′ = C∗(X × Y ; R) by Proposition 3B.1, so the algebraic K¨unneth formula gives the desired short exact sequences. Their naturality follows from naturality in the algebraic K¨unneth
formula, since we can homotope arbitrary maps X→X ′ and Y →Y ′ to be cellular by Theorem 4.8, assuring that they induce chain maps of ⊔⊓ cellular chain complexes. With ﬁeld coeﬃcients the K¨unneth formula simpliﬁes because the Tor terms are always zero over a ﬁeld: Corollary 3B.7. If F is a ﬁeld and X and Y are CW complexes, then the cross product -→ Hn(X × Y ; F ) is an isomorphism for all n. ⊔⊓ map h : Hi(X; F ) ⊗F Hn−i(Y ; F ) i L There is also a relative version of the K¨unneth formula for CW pairs (X, A) and (Y, B). This is a split short exact sequence 0 -→ i Hi(X, A; R)⊗R Hn−i(Y, B; R) L -→ Hn(X × Y, A× Y ∪ X × B; R) -→ iTorR Hi(X, A; R), Hn−i−1(Y, B; R) L -→ 0 for R a principal ideal domain. This too follows from the algebraic K¨unneth formula since the isomorphism of cellular chain complexes C∗(X × Y ) ≈ C∗(X) ⊗ C∗(Y ) passes down to a quotient isomorphism C∗(X × Y )/C∗(A× Y ∪ X × B) ≈ C∗(X)/C∗(A) ⊗ C∗(Y )/C∗(B) since bases for these three relative cellular chain complexes correspond bijectively with the cells of (X − A)× (Y − B), X − A, and Y − B, respectively. As a special case, suppose A and B are basepoints x0 ∈ X and y0 ∈ Y. Then the subcomplex A× Y ∪ X × B can be identiﬁed with the wedge sum X ∨ Y and the quotient X × Y /X ∨ Y is the smash product X ∧ Y. Thus we
have a reduced K¨unneth formula 0 -→ i L Hi(X; R)⊗R e Hn−i(Y ; R) e -→ Hn(X ∧ Y ; R) -→ iTorR e Hi(X; R), Hn−i−1(Y ; R) -→ 0 If we take Y = S k for example, then X ∧ S k is the k fold reduced suspension of X, and we obtain isomorphisms Hn+k(X ∧ S k; R). Hn(X; R) ≈ e e L The K¨unneth formula and the universal coeﬃcient theorem can be combined to e give a more concise formula Hn(X × Y ; R) ≈ this isomorphism is somewhat problematic, however, since it uses the splittings in X; Hn−i(Y ; R). The naturality of i Hi L e the K¨unneth formula and the universal coeﬃcient theorem. With a little more algebra the formula can be shown to hold more generally for an arbitrary coeﬃcient group G in place of R ; see [Hilton & Wylie 1967], p. 227. The General K¨unneth Formula Section 3.B 277 Hi Hn(X ∧ Y ; R) ≈ There is an analogous formula Hn−i(Y ; R). As a speHn(X; G) ≈ cial case, when Y is a Moore space M(G, k) we obtain isomorphisms Hn+k(X ∧ M(G, k); Z). Again naturality is an issue, but in this case there is a natural isomorphism obtainable by applying Theorem 4.59 in §4.3, after verifying that the e Hn+k(X ∧ M(G, k); Z) deﬁne a reduced homology theory, which is functors hn(X) = Hn+k(X∧M(G, k); Z) says that homology with not hard. The isomorphism arbitrary coeﬃcients can be obtained from homology with Z coeﬃcients by a topolog- Hn(X; G) ≈ i L X; e e e e e e e ical construction as
well as by the algebra of tensor products. For general homology theories this formula can be used as a deﬁnition of homology with coeﬃcients. One might wonder about a cohomology version of the K¨unneth formula. Taking coeﬃcients in a ﬁeld F and using the natural isomorphism Hom(A ⊗ B, C) ≈ Hom A, Hom(B, C), the K¨unneth formula for homology and the universal coeﬃcient theorem give isomorphisms H n(X × Y ; F ) ≈ HomF (Hn(X × Y ; F ), F ) ≈ iHomF ≈ L Hi(X; F ), HomF (Hn−i(Y ; F ), F ) Hi(X; F ), H n−i(Y ; F ) X; H n−i(Y ; F ) ≈ ≈ L L iHomF iH i iHomF (Hi(X; F )⊗Hn−i(Y ; F ), F ) More generally, there are isomorphisms H n(X × Y ; G) ≈ coeﬃcient group G ; see [Hilton & Wylie 1967], p. 227. However, in practice it usu- X; H n−i(Y ; G) for any i H i L L ally suﬃces to apply the K¨unneth formula for homology and the universal coeﬃcient theorem for cohomology separately. Also, Theorem 3.15 shows that with stronger hypotheses one can draw stronger conclusions using cup products. The Simplicial Cross Product Let us sketch how the cross product Hm(X; R) ⊗ Hn(Y ; R)→Hm+n(X × Y ; R) can be deﬁned directly in terms of singular homology. What one wants is a cross product at the level of singular chains, Cm(X; R) ⊗ Cn(Y ; R)→Cm+n(X × Y ; R). If we are n→Y, then we have the product map given singular simplices f : n into simplices of dimenf × g : sion m + n and then take the sum of the restrictions of f × g to these simplices,
with n→X × Y, and the idea is to subdivide m→X and g : m × m × ∆ ∆ ∆ ∆ appropriate signs. ∆ ∆ m × In the special cases that m or n is 1 we have already seen how to subdivide n into simplices when we constructed prism operators in §2.1. The generalm as ization to ∆ n as w0, w1, ···, wn. Think of the pairs (i, j) with v0, v1, ···, vm and the vertices of ∆ 0 ≤ i ≤ m and 0 ≤ j ≤ n as the vertices of an m× n rectangular grid in R2. Let σ be a path formed by a sequence of m + n horizontal and vertical edges in this grid n is not completely obvious, however. Label the vertices of m × ∆ ∆ ∆ ∆ starting at (0, 0) and ending at (m, n), always moving either to the right or upward. m+n→ n sending the k th To such a path σ we associate a linear map ℓσ : ) where (ik, jk) is the k th vertex of the edgepath σ. Then vertex of, wjk m+n to (vik m × ∆ ∆ ∆ ∆ 278 Chapter 3 Cohomology we deﬁne a simplicial cross product Cm(X; R) ⊗ Cn(Y ; R) ×--------------→ Cm+n(X × Y ; R) by the formula f × g = (−1)\|σ \|(f × g)ℓσ Xσ where \|σ \| is the number of squares in the grid lying below the path σ. Note that the symbol ‘ × ’ means diﬀerent things on the two sides of the equation. From this deﬁnition it is a calculation to show that ∂(f × g) = ∂f × g+(−1)mf × ∂g. This implies that the cross product of two cycles is a cycle, and the cross product of a cycle and a boundary is a boundary, so there is an induced cross product in singular homology. n One can see that the images of the maps ℓσ give a simplicial structure on m × ∆ ∆ m as the subspace of
Rm deﬁned by the inin the following way. We can view ∆ equalities 0 ≤ x1 ≤ ··· ≤ xm ≤ 1, with the vertex vi as the point having coordin ⊂ Rn with coordinates nates m − i zeros followed by i ones. Similarly we have n then consists of (m + n) tuples 0 ≤ y1 ≤ ··· ≤ yn ≤ 1. The product (x1, ···, xm, y1, ···, yn) satisfying both sets of inequalities. The combined inequal∆ n, ities 0 ≤ x1 ≤ ··· ≤ xm ≤ y1 ≤ ··· ≤ yn ≤ 1 deﬁne a simplex n satisﬁes a similar set of inequalities obtained from and every other point of 0 ≤ x1 ≤ ··· ≤ xm ≤ y1 ≤ ··· ≤ yn ≤ 1 by a permutation of the variables ‘shuﬄing’ the yj ’s into the xi ’s. Each such shuﬄe corresponds to an edgepath σ consisting of a rightward edge for each xi and an upward edge for each yj in the shuﬄed seindexed quence. Thus we have n expressed as the union of simplices m+n in +n σ a by the edgepaths σ. One can check that these simplices ﬁt together nicely to form n, which is also a simplicial complex structure. See [Eilenberg & Steenrod 1952], p. 68. In fact this construction is suﬃciently natural to ∆ complex structure on ∆ m × ∆ ∆ ∆ make the product of any two ∆ complexes into a complex. The Cohomology Cross Product ∆ ∆ In §3.2 we deﬁned a cross product H k(X; R)× H ℓ(Y ; R) ×--------------→ H k+ℓ(X × Y ; R) in terms of the cup product. Let us now describe the alternative approach in which the cross product is deﬁned directly via cellular cohomology, and then cup product is deﬁned in terms of this cross product. The cellular deﬁnition of cohomology cross product is very much like the de�
��ni- tion in homology. Given CW complexes X and Y, deﬁne a cross product of cellular cochains ϕ ∈ C k(X; R) and ψ ∈ C ℓ(Y ; R) by setting (ϕ× ψ)(ek α × eℓ β) = ϕ(ek α)ψ(eℓ β) and letting ϕ× ψ take the value 0 on (k + ℓ) cells of X × Y which are not the product of a k cell of X with an ℓ cell of Y. Another way of saying this is to use the convention The General K¨unneth Formula Section 3.B 279 that a cellular cochain in C k(X; R) takes the value 0 on cells of dimension diﬀerent α × en from k, and then we can let (ϕ× ψ)(em β ) for all m and n. β ) = ϕ(em α )ψ(en The cellular coboundary formula δ(ϕ× ψ) = δϕ× ψ + (−1)kϕ× δψ for cellular cochains ϕ ∈ C k(X; R) and ψ ∈ C ℓ(Y ; R) follows easily from the corresponding boundary formula in Proposition 3B.1, namely δ(ϕ× ψ)(em α × en ∂(em β ) = (ϕ× ψ) α × en β ) β + (−1)mem α × en β ) + (−1)mϕ(em = (δϕ× ψ + (−1)kϕ× δψ)(em = (ϕ× ψ)(∂em α )ψ(en = δϕ(em α × ∂en β ) α )δψ(en β ) α × en β ) where the coeﬃcient (−1)m in the next-to-last line can be replaced by (−1)k since α ) = 0 unless k = m. From the formula δ(ϕ× ψ) = δϕ× ψ + (−1)kϕ× δψ ϕ(em it follows just as for homology and for cup product that there is an induced cross product
in cellular cohomology. To show this agrees with the earlier deﬁnition, we can ﬁrst reduce to the case that X has trivial (k − 1) skeleton and Y has trivial (ℓ − 1) skeleton via the commutative diagram The left-hand vertical map is surjective, so by commutativity, if the two deﬁnitions of cross product agree in the upper row, they agree in the lower row. Next, assuming X k−1 and Y ℓ−1 are trivial, consider the commutative diagram The vertical maps here are injective, X k × Y ℓ being the (k + ℓ) skeleton of X × Y, so it suﬃces to see that the two deﬁnitions agree in the lower row. We have X k = α S k α and Y ℓ = β, so by restriction to these wedge summands the question is reduced W ﬁnally to the case of a product S k In this case, taking R = Z, we showed in Theorem 3.15 that the cross product in question is the map Z× Z→Z sending (1, 1) to ±1, with the original deﬁnition of cross product. The same is obviously true using the cellular cross product. So for R = Z the two cross products agree up to sign, and β S ℓ W α × S ℓ β. it follows that this is also true for arbitrary R. We leave it to the reader to sort out the matter of signs. To relate cross product to cup product we use the diagonal map : X→X × X, x ֏ (x, x). If we are given a deﬁnition of cross product, we can deﬁne cup product as the composition ∆ H k(X; R)× H ℓ(X; R) ×--------------→ H k+ℓ(X × X; R) ∗ --------------------→ H k+ℓ(X; R) ∆ 280 Chapter 3 Cohomology This agrees with the original deﬁnition of cup product since we have ` ∗ ∗ = a ` b, as both compositions p1 ∗(a× b) = ∗ 1 (a) ` p∗ p∗ and p2 �
� 2 (b) = are the identity map of X. p∗ 1 (a) p∗ 2 (b) ∆ Unfortunately, the deﬁnition of cellular cross product cannot be combined with ∆ ∆ ∆ to give a deﬁnition of cup product at the level of cellular cochains. This is because ∆ is not a cellular map, so it does not induce a map of cellular cochains. It is possible ∆ to homotope ∆ For example, the diagonal of a square can be pushed across either adjacent triangle. In to a cellular map by Theorem 4.8, but this involves arbitrary choices. particular cases one might hope to understand the geometry well enough to compute ∆ an explicit cellular approximation to the diagonal map, but usually other techniques for computing cup products are preferable. The cohomology cross product satisﬁes the same commutativity relation as for homology, namely T ∗(a× b) = (−1)kℓb× a for T : X × Y →Y × X the transposition map, a ∈ H k(Y ; R), and b ∈ H ℓ(X; R). The proof is the same as for homology. Taking X = Y and noting that T, we obtain a new proof of the commutativity = property of cup product. ∆ ∆ Exercises 1. Compute the groups Hi(RPm × RPn; G) and H i(RPm × RPn; G) for G = Z and Z2 via the cellular chain and cochain complexes. [See Example 3B.4.] 2. Let C and C ′ be chain complexes, and let I be the chain complex consisting of Z in dimension 1 and Z× Z in dimension 0, with the boundary map taking a generator e in dimension 1 to the diﬀerence v1 − v0 of generators vi of the two Z ’s in dimension 0. Show that a chain map f : I ⊗ C→C ′ is precisely the same as a chain homotopy between the two chain maps fi : C→C ′, c ֏ f (vi ⊗ c), i = 0, 1. [The chain homotopy is h(c) = f (e ⊗ c).] 3. Show that the splitting in the top
ological K¨unneth formula cannot be natural by considering the map f × 11 : M(Zm, n)× M(Zm, n)→S n+1 × M(Zm, n) where f collapses the n skeleton of M(Zm, n) = S n ∪ en+1 to a point. 4. Show that the cross product of fundamental classes for closed R orientable manifolds M and N is a fundamental class for M × N. 5. Show that slant products Hn(X × Y ; R)× H j(Y ; R) -→ Hn−j(X; R), H n(X × Y ; R)× Hj(Y ; R) -→ H n−j(X; R), (ei × ej, ϕ) ֏ ϕ(ej )ei (ϕ, ej ) ֏ ei ֏ ϕ(ei × ej) can be deﬁned via the indicated cellular formulas. [These ‘products’ are in some ways more like division than multiplication, and this is reﬂected in the common notation a/b for them, or a\b when the order of the factors is reversed. The ﬁrst of the two slant products is related to cap product in the same way that the cohomology cross product is related to cup product.] H–Spaces and Hopf Algebras Section 3.C 281 Of the three axioms for a group, it would seem that the least subtle is the existence of an identity element. However, we shall see in this section that when topology is added to the picture, the identity axiom becomes much more potent. To give a name to the objects we will be considering, deﬁne a space X to be an H–space, ‘H’ standing for ‘Hopf’, if there is a continuous multiplication map µ : X × X→X and an ‘identity’ element e ∈ X such that the two maps X→X given by x ֏ µ(x, e) and x ֏ µ(e, x) are homotopic to the identity through maps (X, e)→(X, e). In particular, this implies that µ(e, e) = e. In terms of generality, this de
ﬁnition represents something of a middle ground. One could weaken the deﬁnition by dropping the condition that the homotopies pre- serve the basepoint e, or one could strengthen it by requiring that e be a strict identity, without any homotopies. An exercise at the end of the section is to show the three possible deﬁnitions are equivalent if X is a CW complex. An advantage of allowing homotopies in the deﬁnition is that a space homotopy equivalent in the basepointed sense to an H–space is again an H–space. Imposing basepoint conditions is fairly standard in homotopy theory, and is usually not a serious restriction. The most classical examples of H–spaces are topological groups, spaces X with a group structure such that both the multiplication map X × X→X and the inversion map X→X, x ֏ x−1, are continuous. For example, the group GLn(R) of invertible n× n matrices with real entries is a topological group when topologized as a subspace of the n2 dimensional vector space Mn(R) of all n× n matrices over R. It is an open subspace since the invertible matrices are those with nonzero determinant, and the determinant function Mn(R)→R is continuous. Matrix multiplication is certainly continuous, being deﬁned by simple algebraic formulas, and it is not hard to see that matrix inversion is also continuous if one thinks for example of the classical adjoint formula for the inverse matrix. Likewise GLn(C) is a topological group, as is the quaternionic analog GLn(H), though in the latter case one needs a somewhat diﬀerent justiﬁcation since deter- minants of quaternionic matrices do not have the good properties one would like. Since these groups GLn over R, C, and H are open subsets of Euclidean spaces, they are examples of Lie groups, which can be deﬁned as topological groups which are also manifolds. The GLn groups are noncompact, being open subsets of Euclidean spaces, but they have the homotopy types of compact Lie groups O(n), U(n), and Sp(n). This is explained in §
3.D for GLn(R), and the other two cases are similar. Among the simplest H–spaces from a topological viewpoint are the unit spheres S 1 in C, S 3 in the quaternions H, and S 7 in the octonions O. These are H–spaces since the multiplications in these division algebras are continuous, being deﬁned by 282 Chapter 3 Cohomology polynomial formulas, and are norm-preserving, \|ab\| = \|a\|\|b\|, hence restrict to multi- plications on the unit spheres, and the identity element of the division algebra lies in the unit sphere in each case. Both S 1 and S 3 are Lie groups since the multiplications in C and H are associative and inverses exist since aa = \|a\|2 = 1 if \|a\| = 1. However, S 7 is not a group since multiplication of octonions is not associative. Of course S 0 = {±1} is also a topological group, trivially. A famous theorem of J. F. Adams asserts that S 0, S 1, S 3, and S 7 are the only spheres that are H–spaces; see §4.B for a fuller discussion. Let us describe now some associative H–spaces where inverses fail to exist. Multiplication of polynomials provides an H–space structure on CP∞ in the following way. A nonzero polynomial a0 + a1z + ··· + anzn with coeﬃcients ai ∈ C corresponds to a point (a0, ···, an, 0, ···) ∈ C∞ − {0}. Multiplication of two such polynomials determines a multiplication C∞ − {0}× C∞ − {0}→C∞ − {0} which is associative, commutative, and has an identity element (1, 0, ···). Since C is commutative we can factor out by scalar multiplication by nonzero constants and get an induced product CP∞ × CP∞→CP∞ with the same properties. Thus CP∞ is an associative, commutative H–space with a strict identity. Instead of factoring out by all nonzero scalars, we could factor out only by scalars of the form ρe2
π ik/q with ρ an arbitrary positive real, k an arbitrary integer, and q a ﬁxed positive integer. The quotient of C∞ − {0} under this identiﬁcation, an inﬁnite-dimensional lens space L∞ with π1(L∞) ≈ Zq, is therefore also an associative, commutative H–space. This includes RP∞ in particular. The spaces J(X) deﬁned in §3.2 are also H–spaces, with the multiplication given by (x1, ···, xm)(y1, ···, yn) = (x1, ···, xm, y1, ···, yn), which is associative and has an identity element (e) where e is the basepoint of X. One could describe J(X) as the free associative H–space generated by X. There is also a commutative ana- log of J(X) called the inﬁnite symmetric product SP (X) deﬁned in the following way. Let SPn(X) be the quotient space of the n fold product X n obtained by identifying all n tuples (x1, ···, xn) that diﬀer only by a permutation of their coordinates. The inclusion X n ֓ X n+1, (x1, ···, xn)֏ (x1, ···, xn, e) induces an inclusion SPn(X)֓SPn+1(X), and SP (X) is deﬁned to be the union of this increasing sequence of SPn(X) ’s, with the weak topology. Alternatively, SP (X) is the quotient of J(X) obtained by identifying points that diﬀer only by permutation of coordinates. The H–space structure on J(X) induces an H–space structure on SP (X) which is commu- tative in addition to being associative and having a strict identity. The spaces SP (X) are studied in more detail in §4.K. The goal of this section will be to describe the extra structure which the multi- plication in an H–space gives to its homology and cohomology. This is
of particular interest since many of the most important spaces in algebraic topology turn out to be H–spaces. H–Spaces and Hopf Algebras Section 3.C 283 Hopf Algebras Let us look at cohomology ﬁrst. Choosing a commutative ring R as coeﬃcient ring, we can regard the cohomology ring H ∗(X; R) of a space X as an algebra over R rather than merely a ring. Suppose X is an H–space satisfying two conditions: (1) X is path-connected, hence H 0(X; R) ≈ R. (2) H n(X; R) is a ﬁnitely generated free R module for each n, so the cross product H ∗(X; R) ⊗R H ∗(X; R)→H ∗(X × X; R) is an isomorphism. The multiplication µ : X × X→X induces a map µ∗ : H ∗(X; R)→H ∗(X × X; R), and when we combine this with the cross product isomorphism in (2) we get a map H ∗(X; R) --------------→ H ∗(X; R) ⊗R H ∗(X; R) ∆ which is an algebra homomorphism since both µ∗ and the cross product isomorphism are algebra homomorphisms. The key property of turns out to be that for any α ∈ H n(X; R), n > 0, we have ∆ where \|α′ (α′ i ⊗ α′′ i i\| > 0 and \|α′′ i \| > 0 To verify this, let i : X→X × X be the inclusion x ֏ (x, e) for e the identity element of X, and consider the commutative diagram ∆ X The map P is deﬁned by commutativity, and by looking at the lower right triangle we see that P (α ⊗ 1) = α and P (α ⊗ β) = 0 if \|β\| > 0. The H–space property says that (α) in H n(X; R) ⊗R H 0(X; R) µi ≃ 11, so P is α
⊗ 1. A similar argument shows the component in H 0(X; R) ⊗R H n(X; R) is 1 ⊗ α. We can summarize this situation by saying that H ∗(X; R) is a Hopf algebra, that n≥0 An over a commutative base ring R, satisfying the = 11. This implies that the component of is, a graded algebra A = ∆ ∆ following two conditions: (1) There is an identity element 1 ∈ A0 such that the map R→A0, r ֏ r · 1, is an L isomorphism. In this case one says A is connected. (2) There is a diagonal or coproduct gebras satisfying (α→A ⊗ A, a homomorphism of graded ali \| > 0, i\| > 0 and \|α′′ i where \|α′ i ⊗ α′′ i α′ for all α with \|α\| > 0. ∆ ∆ P Here and in what follows we take ⊗ to mean ⊗R. The multiplication in A ⊗ A is given by the standard formula (α ⊗ β)(γ ⊗ δ) = (−1)\|β\|\|γ\|(αγ ⊗ βδ). For a general Hopf algebra the multiplication is not assumed to be either associative or commutative (in the graded sense), though in the example of H ∗(X; R) for X an H–space the algebra structure is of course associative and commutative. 284 Chapter 3 Cohomology ∆ Example 3C.1. One of the simplest Hopf algebras is a polynomial ring R[α]. The (α) must equal α ⊗ 1 + 1 ⊗ α since the only elements of R[α] of lower coproduct dimension than α are the elements of R in dimension zero, so the terms α′ i and α′′ i ∆ (α must be zero. The requirein the coproduct formula completely. To de- be an algebra homomorphism then determines ment that i ⊗ α′′ i α′ P ∆ ∆ ∆ ∆ n i and i P explicitly we distinguish two cases. If the dimension of α is even or if 2 = 0 scribe (αn) =
in R, then the multiplication in R[α] ⊗ R[α] is strictly commutative and αi ⊗ αn−i. In the opposite case that α is odd-dimensional, (α ⊗ 1 + 1 ⊗ α)n = (α2) = (α ⊗ 1 + 1 ⊗ α)2 = α2 ⊗ 1 + 1 ⊗ α2 since (α ⊗ 1)(1 ⊗ α) = α ⊗ α and then (1 ⊗ α)(α ⊗ 1) = −α ⊗ α if α has odd dimension. Thus if we set β = α2, then β βi ⊗ βn−i is even-dimensional and we have (βn) = (β ⊗ 1 + 1 ⊗ β)n = n i Example 3C.2. The exterior algebra ∆ Hopf algebra, with (α) = α ⊗ 1+1 ⊗ α. To verify that i P R[α] on an odd-dimensional generator α is a is an algebra homomorphism (α)2, or in other words, since α2 = 0, we need to see (α)2 = (α ⊗ 1 + 1 ⊗ α)2 = ∆ (α)2 is indeed 0. Note that if α were even-dimensional we would R[α] (α)2 = α2 ⊗ 1 + 2α ⊗ α + 1 ⊗ α2, which would be 0 in ∆ (α)2 = 0. As in the preceding example we have ∆ i βi ⊗ αβn−i. P we must check that ∆ (α2n) = (βn) = ∆ αβi ⊗ βn−i + (α2n+1) = (αβn) = R[α] ⊗ (α2) = i ∆ (α ∆ that α2 ⊗ 1 + 1 ⊗ α2, so ∆ instead have only if 2 = 0 in R. ∆ ∆ An element α of a Hopf algebra is called primitive if Λ (α) = α ⊗ 1 + 1 ⊗ α. As the Λ preceding examples illustrate, if a Hopf
algebra is generated as an algebra by primitive elements, then the coproduct ∆ is uniquely determined by the product. This happens in a number of interesting special cases, but certainly not in general, as we shall see. The existence of the coproduct in a Hopf algebra turns out to restrict the multi- ∆ plicative structure considerably. Here is an important example illustrating this: Example 3C.3. Suppose that the truncated polynomial algebra F [α]/(αn) over a ﬁeld F is a Hopf algebra. Then α is primitive, just as it is in F [α], so if we assume either that α is even-dimensional or that F has characteristic 2, then the relation αn = 0 yields an equation 0 = (αn) = αn ⊗ 1 + 1 ⊗ αn + n i X0<i<n αi ⊗ αn−i = n i αi ⊗ αn−i X0<i<n which implies that ∆ n i = 0 in F for each i in the range 0 < i < n. This is impossible if F has characteristic 0, and if the characteristic of F is p > 0 then it happens only when n is a power of p. For p = 2 this was shown in the proof of Theorem 3.21, and the argument given there works just as well for odd primes. Conversely, it is easy to check that if F has characteristic p then F [α]/(αpi that α is even-dimensional if p is odd. ) is a Hopf algebra, assuming still The characteristic 0 case of this result implies that CPn is not an H–space for ﬁnite n, in contrast with CP∞ which is an H–space as we saw earlier. Similarly, taking H–Spaces and Hopf Algebras Section 3.C 285 F = Z2, we deduce that RPn can be an H–space only if n + 1 is a power of 2. Indeed, RP1 = S 1/±1, RP3 = S 3/±1, and RP7 = S 7/±1 have quotient H–space structures from S 1, S 3 and S 7 since −1 commutes with all elements of S 1, S 3, or S 7. However, these are the only cases when RPn is an H–space since,
by an exercise at the end of this section, the universal cover of an H–space is an H–space, and S 1, S 3, and S 7 are the only spheres that are H–spaces, by the theorem of Adams mentioned earlier. The tensor product A ⊗ B of Hopf algebras A and B is again a Hopf algebra, with ----------------------------→ (A ⊗ A) ⊗ (B ⊗ B)→(A ⊗ B) ⊗ (A ⊗ B) where the second map interchanging the middle two factors includes the usual sign in graded coproduct the composition A ⊗ B ⊗ ∆ ∆ commutativity. Thus the preceding examples yield many other Hopf algebras, tensor products of polynomial, truncated polynomial, and exterior algebras on any number of generators. The following theorem of Hopf is a partial converse: Theorem 3C.4. If A is a commutative, associative Hopf algebra over a ﬁeld F of characteristic 0, and An is ﬁnite-dimensional over F for each n, then A is isomorphic as an algebra to the tensor product of an exterior algebra on odd-dimensional generators and a polynomial algebra on even-dimensional generators. There is an analogous theorem of Borel when F is a ﬁnite ﬁeld of characteris- tic p. In this case A is again isomorphic to a tensor product of single-generator Hopf algebras, of one of the following types: F [α], with α even-dimensional if p ≠ 2. F [α] with α odd-dimensional. F [α]/(αpi Λ ), with α even-dimensional if p ≠ 2. For a proof see [Borel 1953] or [Kane 1988]. Proof of 3C.4: Since An is ﬁnitely generated over F for each n, we may choose algebra generators x1, x2, ··· for A with \|xi\| ≤ \|xi+1\| for all i. Let An be the subalgebra (An) ⊂ An ⊗ An, generated by x1, ···, xn. This is a Hopf subalgebra of A, that is, (
xi) involves only xi and terms of smaller dimension. We may assume xn since does not lie in An−1. Since A is associative and commutative, there is a natural F [xn]→An if \|xn\| is odd. surjection An−1 ⊗ F [xn]→An if \|xn\| is even, or An−1 ⊗ By induction on n it will suﬃce to prove these surjections are injective. Thus in the Λ i αixi n = 0 and α0 + α1xn = 0, ∆ ∆ two cases we must rule out nontrivial relations respectively, with coeﬃcients αi ∈ An−1. P Let I be the ideal in An generated by x2 n and the positive-dimensional elements of i αixi n with coeﬃcients αi ∈ An−1, the ﬁrst An−1, so I consists of the polynomials two coeﬃcients α0 and α1 having trivial components in A0. Note that xn 6∈ I since P elements of I having dimension \|xn\| must lie in An−1. Consider the composition An -----------------→ An ⊗ An q--------------→ An ⊗ (An/I) ∆ 286 Chapter 3 Cohomology with q the natural quotient map. By the deﬁnition of I, this composition q sends α ∈ An−1 to α ⊗ 1 and xn to xn ⊗ 1 + 1 ⊗ xn where xn is the image of xn in An/I. In case \|xn\| is even, applying q to a nontrivial relation n = 0 gives i αixi ∆ 0 = i(αi ⊗ 1)(xn ⊗ 1 + 1 ⊗ xn)i = ∆ i αixi n P ⊗ 1 + i iαixi−1 n ⊗ xn P P i iαixi−1 P n = 0, this implies that P P i iαixi−1 P i iαixi−1 i αixi Since An ⊗ (An/I), hence n ⊗ xn is zero in the tensor product n = 0 since xn
6∈ I implies xn ≠ 0. The relation n = 0 has lower degree than the original relation, and is not the trivial relation since F has characteristic 0, αi ≠ 0 implying iαi ≠ 0 if i > 0. Since we could P assume the original relation had minimum degree, we have reached a contradiction. to a relation α0 + α1xn = 0 gives 0 = α0 ⊗ 1+(α1 ⊗ 1)(xn ⊗ 1+1 ⊗ xn) = (α0 +α1xn) ⊗ 1+α1 ⊗ xn. Since α0 +α1xn = 0, ⊔⊓ we get α1 ⊗ xn = 0, which implies α1 = 0 and hence α0 = 0. The case \|xn\| odd is similar. Applying q ∆ The structure of Hopf algebras over Z is much more complicated than over a ﬁeld. Here is an example that is still fairly simple. Example 3C.5: Divided Polynomial Algebras. We showed in Proposition 3.22 that the H–space J(S n) for n even has H ∗(J(S n); Z) a divided polynomial algebra, the algebra Z[α] with additive generators αi in dimension ni and multiplication given by αk 1 = k!αk, hence αiαj = Z[α] is uniquely determined by Γ k αi the multiplicative structure since i i (αk (αk) = P (αk 1/i!) ⊗ (αk−i ∆ in this case the coproduct has a simpler description than the product. P 1) = (α1 ⊗ 1 + 1 ⊗ α1)k = 1 /(k − i)!), that is, 1 ⊗ αk−i, which i αi ⊗ αk−i. So αi+j. The coproduct in implies that 1/k!) = i(αi i+j i P Γ 1 ∆ ∆ Zp with this, when we change to ﬁeld coeﬃcients. Clearly [α], with multiplication deﬁned by αiαj = isomorphic as an algebra to the
inﬁnite tensor product It is interesting to see what happens to the divided polynomial algebra Z[α] Q[α] is the same as Q[α]. In contrast αi+j, happens to be pi ), as we will show in a moment. However, as Hopf algebras these two objects are diﬀerent since αpi is primitive in coproduct in [α] when i > 0, since the i≥0 Zp[αpi ]/(αp i≥0 Zp[αpi ]/(αp pi ) but not in i αi ⊗ αk−i. [α] is given by (αk) = i+j i N N Zp Γ Γ Γ Zp Now let us show that there is an algebra isomorphism P Γ Γ ∆ [α] ≈ Zp i≥0 Zp[αpi ]/(αp pi ) N Since Z[α] ⊗ Zp, this is equivalent to: Zp [α] = (∗) The element αn0 Γ The product αn0 1 αn1 integer m. The question is whether p divides m. We will show: Γ p ··· αnk pk equals mαn for n = n0 + n1p + ··· + nkpk and some Z[α] is divisible by p iﬀ ni ≥ p for some i. 1 αn1 p ··· αnk pk in Γ Γ (∗∗) αnαpk is divisible by p iﬀ nk = p − 1, assuming that ni < p for each i. H–Spaces and Hopf Algebras Section 3.C 287 This implies (∗) by an inductive argument in which we build up the product in (∗) by repeated multiplication on the right by terms αpi. n+pk n To prove (∗∗) we recall that αnαpk = αn+pk. The mod p value of this binomial coeﬃcient can be computed using Lemma 3C.6 below. Assuming that ni < p for each i and that nk +1 < p, the p adic representations of n+pk and n diﬀer
only in the coeﬃcient of pk, so mod p we have = nk + 1. This conclusion also holds if nk + 1 = p, when the p adic representations of n + pk and n diﬀer also in the coeﬃcient of pk+1. The statement (∗∗) then follows. n+pk n nk+1 nk = Lemma 3C.6. If p is a prime, then k = i nipi and i i kipi with 0 ≤ ni < p and 0 ≤ ki < p are the p adic representations of n Q mod p where n = ni ki P n k ≡ and k. P Here the convention is that n k = 0 if n < k, and n 0 = 1 for all n ≥ 0. Proof: In Zp[x] there is an identity (1 + x)p = 1 + xp since p clearly divides p!/k!(p − k)! for 0 < k < p. By induction it follows that (1 + x)pi if n = i nipi is the p adic representation of n then: = 1 + xpi = p k. Hence = 1 + P (1 + x)n = (1 + x)n0 (1 + xp)n1 (1 + xp2 n0 n0 x2 + ··· + 2 1 xp + 1 + )n2 ··· n0 p−1 x2p + ··· + x2p2 + n1 2 xp2 1 + x + n2 2 n2 1 n1 1 × × h h xp−1 n1 p−1 i x(p−1)p + ··· + n2 p−1 i x(p−1)p2 × ··· When this is multiplied out, one sees that no terms combine, and the coeﬃcient of xk ⊔⊓ is just i kipi is the p adic representation of k. where k = h i ni ki i Q P Pontryagin Product Another special feature of H–spaces is that their homology groups have a prod- uct operation, called the Pontryagin product. For an H–space X with multiplication µ : X × X→X, this is the composition H∗(X; R) ⊗ H∗(X;
R) ×--------------→ H∗(X × X; R) µ∗--------------→ H∗(X; R) where the ﬁrst map is the cross product deﬁned in §3.B. Thus the Pontryagin product consists of bilinear maps Hi(X; R)× Hj(X; R)→Hi+j(X; R). Unlike cup product, the Pontryagin product is not in general associative unless the multiplication µ is associative or at least associative up to homotopy, in the sense that the maps X × X × X→X, (x, y, z) ֏ µ(x, µ(y, z)) and (x, y, z) ֏ µ(µ(x, y), z) are homotopic. Fortunately most H–spaces one meets in practice satisfy this associativity property. Nor is the Pontryagin product generally commutative, even in the graded sense, unless µ is commutative or homotopy-commutative, which is relatively rare for H–spaces. We will give examples shortly where the Pontryagin product is not commutative. 288 Chapter 3 Cohomology In case X is a CW complex and µ is a cellular map the Pontryagin product can be computed using cellular homology via the cellular chain map Ci(X; R)× Cj(X; R) ×--------------→ Ci+j (X × X; R) µ∗--------------→ Ci+j(X; R) where the cross product map sends generators corresponding to cells ei and ej to the generator corresponding to the product cell ei × ej, and then µ∗ is applied to this product cell. Example 3C.7. Let us compute the Pontryagin product for J(S n). Here there is one cell ein for each i ≥ 0, and µ takes the product cell ein × ejn homeomorphically onto the cell e(i+j)n. This means that H∗(J(S n); Z) is simply the polynomial ring Z[x] on an n dimensional generator x. This holds for n odd as well as for n even, so the Pontryagin product need not satisfy the same general commutativity relation as cup product. In this example the Pontryagin product structure is simpler than
the cup product structure, though for some H–spaces it is the other way round. In applications it is often convenient to have the choice of which product structure to use. This calculation immediately generalizes to J(X) where X is any connected CW complex whose cellular boundary maps are all trivial. The cellular boundary maps in the product Xm of m copies of X are then trivial by induction on m using Proposition 3B.1, and therefore the cellular boundary maps in J(X) are all trivial since the quotient map Xm→Jm(X) is cellular and each cell of Jm(X) is the homeomorphic image of a cell of Xm. Thus H∗(J(X); Z) is free with additive basis the products en1 × ··· × enk of positive-dimensional cells of X, and the multiplicative structure is that of polynomials in noncommuting variables corresponding to the positive- dimensional cells of X. T Another way to describe H∗(J(X); Z) in this example is as the tensor algebra H∗(X; Z), where for a graded R module M that is trivial in dimension zero, like the reduced homology of a path-connected space, the tensor algebra T M is the direct sum of the n fold tensor products of M with itself for all n ≥ 1, together with a copy e of R in dimension zero, with the obvious multiplication coming from tensor product and scalar multiplication. Generalizing the preceding example, we have: Proposition 3C.8. If X is a connected CW complex with H∗(X; R) a free R module, then H∗(J(X); R) is isomorphic to the tensor algebra T H∗(X; R). This can be paraphrased as saying that the homology of the free H–space gener- e ated by a space with free homology is the free algebra generated by the homology of the space. Proof: With coeﬃcients in R, let ϕ : T whose restriction to the n fold tensor product H∗(X)→H∗ H∗(X)⊗n ֓ H∗(X)⊗n ×-----→ H∗(X n) -→ H∗ e H∗(X)⊗n is the composition e Jn(
X) -→ H∗ J(X) J(X) be the homomorphism e H–Spaces and Hopf Algebras Section 3.C 289 where the next-to-last map is induced by the quotient map X n→Jn(X). It is clear that ϕ is a ring homomorphism since the product in J(X) is induced from the natural map Xm × X n→Xm+n. To show that ϕ is an isomorphism, consider the following commutative diagram of short exact sequences: e H∗(X) denotes the direct sum of the products e Jn(X), Jn−1(X) H∗(X)⊗k for In the upper row, Tm k ≤ m, so this row is exact. The second row is the homology exact sequence for, with quotient Jn(X)/Jn−1(X) the n fold smash product the pair X ∧n. This long exact sequence breaks up into short exact sequences as indicated, by commutativity of the right-hand square and the fact that the right-hand vertical map is an isomorphism by the K¨unneth formula, using the hypothesis that H∗(X) is free over the given coeﬃcient ring. By induction on n and the ﬁve-lemma we deduce from the diagram that ϕ : Tn go to ∞, this implies that ϕ : T given dimension Tn same is true of H∗ H∗(X) is independent of n when n is suﬃciently large, and the ⊔⊓ Jn(X) e by the second row of the diagram. is an isomorphism for all n. Letting n Jn(X) H∗(X)→H∗ is an isomorphism since in any H∗(X)→H∗ J(X) e e Dual Hopf Algebras There is a close connection between the Pontryagin product in homology and the Hopf algebra structure on cohomology. Suppose that X is an H–space such that, with coeﬃcients in a ﬁeld R, the vector spaces Hn(X; R) are ﬁnite-dimensional for all n. Alternatively, we could take
R = Z and assume Hn(X; Z) is ﬁnitely generated and free for all n. In either case we have H n(X; R) = HomR(Hn(X; R), R), and as a consequence the Pontryagin product H∗(X; R) ⊗ H∗(X; R)→H∗(X; R) and the : H ∗(X; R)→H ∗(X; R) ⊗ H ∗(X; R) are dual to each other, both being incoproduct duced by the H–space product µ : X × X→X. Therefore the coproduct in cohomology determines the Pontryagin product in homology, and vice versa. Speciﬁcally, ∆ is dual to the product ij : H i+j(X; R)→H i(X; R) ⊗ H j(X; R) of the component Hi(X; R) ⊗ Hj(X; R)→Hi+j(X; R). ∆ Example 3C.9. Consider J(S n) with n even, so H ∗(J(S n); Z) is the divided poly(αk) = nomial algebra ij takes αi+j to αi ⊗ αj, so if xi is the generator of Hin(J(S n); Z) dual to αi, then xixj = xi+j. This says that H∗(J(S n); Z) is the polynomial ring Z[x]. P We showed this in Example 3C.7 using the cell structure of J(S n), but the present proof deduces it purely algebraically from the cup product structure. Z[α]. In Example 3C.5 we derived the coproduct formula i αi ⊗ αk−i. Thus ∆ ∆ ∆ Γ Now we wish to show that the relation between H ∗(X; R) and H∗(X; R) is per- fectly symmetric: They are dual Hopf algebras. This is a purely algebraic fact: 290 Chapter 3 Cohomology Proposition 3C.10. Let A be a Hopf algebra over R that is a
ﬁnitely generated free R module in each dimension. Then the product π : A ⊗ A→A and coproduct ∗ : A∗ ⊗ A∗→A∗ that give A∗ the : A→A ⊗ A have duals π ∗ : A∗→A∗ ⊗ A∗ and structure of a Hopf algebra. ∆ ∆ Proof: This will be apparent if we reinterpret the Hopf algebra structure on A formally as a pair of graded R module homomorphisms π : A ⊗ A→A and : A→A ⊗ A together with an element 1 ∈ A0 satisfying: (1) The two compositions A ∆ π-----→ A are the identity, π-----→ A and A where iℓ(a) = a ⊗ 1 and ir (a) = 1 ⊗ a. This says that 1 is a two-sided identity for the multiplication in A. iℓ-----→ A ⊗ A ir-----→ A ⊗ A (2) The two compositions A pℓ-----→ A and A pr-----→ A are the identity, where pℓ(a ⊗ 1) = a = pr (1 ⊗ a), pℓ(a ⊗ b) = 0 if \|b\| > 0, and pr (a ⊗ b) = 0 if \|a\| > 0. This is just the coproduct formula (a) = a ⊗ 1 + 1 ⊗ a + -----→ A ⊗ A -----→ A ⊗ A ∆ ∆ i a′ i ⊗ a′′ i. (3) The diagram at the right commutes, with τ(a ⊗ b ⊗ c ⊗ d) = (−1)\|b\|\|c\|a ⊗ c ⊗ b ⊗ d. is an algeThis is the condition that ∆ P bra homomorphism since if we follow ∆ an element a ⊗ b across the top of the diagram we get j ⊗ b′′ (b) = j = i ⊗ a′′ ⊗ i i \|\|b′ j \|a′ j ⊗ a′′ ib′ P i a′ i,
j(−1)\|a′′ P route gives ﬁrst j b′ i b′′ and π ⊗ π this becomes ∆ (a) ⊗ j which is (a) ∆ (b). (ab), while the lower, then after applying τ ∆ i a′ j ⊗ b′′ i ⊗ a′′ j b′ j i, P P P Condition (1) for A dualizes to (2) for A∗, and similarly (2) for A dualizes to (1) for A∗. Condition (3) for A dualizes to (3) for A∗. ⊔⊓ ∆ ∆ Example 3C.11. Let us compute the dual of a polynomial algebra R[x]. Suppose xi ⊗ xn−i, ﬁrst that x has even dimension. Then (xn) gives the product relation so if αi is dual to xi, the term P n αiαn−i = αn. This is the rule for multiplication in a divided polynomial algebra, i R[α] if the dimension of x is even. This also holds if 2 = 0 so the dual of R[x] is in R, since the even-dimensionality of x was used only to deduce that R[x] ⊗ R[x] xi ⊗ xn−i in (xn) = (x ⊗ 1 + 1 ⊗ x) is strictly commutative. In case x is odd-dimensional, then as we saw in Example 3C.1, if we set y = x2, (y n) = (xy n) = (x) (y n) = (y ⊗ 1 + 1 ⊗ y)n = n i y i ⊗ y n−i and n i i y i ⊗ xy n−i. These formulas for P say that the dual of R[x] ∆ ∆ ∆ we have xy i ⊗ y n−i + n i ∆ R[α] ⊗ i is P i P R[β] where α is dual to x and β is dual to y. ∆ Γ Λ This algebra allows us to deduce the cup product structure on H ∗(
J(S n); R) from the geometric calculation H∗(J(S n); R) ≈ R[x] in Example 3C.7. As another application, recall from earlier in this section that RP∞ and CP∞ are H–spaces, so from their H–Spaces and Hopf Algebras Section 3.C 291 cup product structures we can conclude that the Pontryagin rings H∗(RP∞; Z2) and H∗(CP∞; Z) are divided polynomial algebras. In these examples the Hopf algebra is generated as an algebra by primitive ele- ments, so the product determines the coproduct and hence the dual algebra. This is not true in general, however. For example, we have seen that the Hopf algebra is isomorphic as an algebra to product as the tensor product of the Hopf algebras Zp[αpi ]/(αp N αpi are primitive, though they are not primitive in algebra [α] p pi ), but if we regard the latter tensor pi ) then the elements [α] for i > 0. In fact, the Hopf pi ) is its own dual, according to one of the exercises below, i≥0 Zp[αpi ]/(αp i≥0 Zp[αpi ]/(α Zp Zp Γ Γ but the dual of N Zp [α] is Zp[α]. Exercises Γ 1. Suppose that X is a CW complex with basepoint e ∈ X a 0 cell. Show that X is an H–space if there is a map µ : X × X→X such that the maps X→X, x ֏ µ(x, e) and x ֏ µ(e, x), are homotopic to the identity. [Sometimes this is taken as the deﬁnition of an H–space, rather than the more restrictive condition in the deﬁnition we have given.] With the same hypotheses, show also that µ can be homotoped so that e is a strict two-sided identity. 2. Show that a retract of an H–space is an H–space if it contains the identity element. 3. Show that in a homotopy-associative H–space whose set of path-components is a
group with respect to the multiplication induced by the H–space structure, all the path- components must be homotopy equivalent. [Homotopy-associative means associative up to homotopy.] 4. Show that an H–space or topological group structure on a path-connected, locally path-connected space can be lifted to such a structure on its universal cover. [For the group SO(n) considered in the next section, the universal cover for n > 2 is a 2 sheeted cover, a group called Spin(n).] 5. Show that if (X, e) is an H–space then π1(X, e) is abelian. composition f g of loops with the product µ f (t), g(t) [Compare the usual coming from the H–space multiplication µ.] 6. Show that S n is an H–space iﬀ the attaching map of the 2n cell of J2(S n) is homotopically trivial. 7. What are the primitive elements of the Hopf algebra Zp[x] for p prime? 8. Show that the tensor product of two Hopf algebras is a Hopf algebra. 9. Apply the theorems of Hopf and Borel to show that for an H–space X that is a H∗(X; Z) ≠ 0, the Euler characteristic χ (X) is 0. connected ﬁnite CW complex with 10. Let X be a path-connected H–space with H ∗(X; R) free and ﬁnitely generated in each dimension. For maps f, g : X→X, the product f g : X→X is deﬁned by (f g)(x) = f (x)g(x), using the H–space product. e 292 Chapter 3 Cohomology (a) Show that (f g)∗(α) = f ∗(α) + g∗(α) for primitive elements α ∈ H ∗(X; R). (b) Deduce that the k th power map x ֏ xk induces the map α ֏ kα on primitive elements α. In particular the quaternionic kth power map S 3→S 3 has degree k. (c) Show that every polynomial anxnbn + ·
·· + a1xb1 + a0 of nonzero degree with coeﬃcients in H has a root in H. [See Theorem 1.8.] 11. If T n is the n dimensional torus, the product of n circles, show that the Pontryagin ring H∗(T n; Z) is the exterior algebra Z[x1, ···, xn] with \|xi\| = 1. 12. Compute the Pontryagin product structure in H∗(L; Zp) where L is an inﬁniteΛ dimensional lens space S ∞/Zp, for p an odd prime, using the coproduct in H ∗(L; Zp). 13. Verify that the Hopf algebras R[α] and Zp[α]/(αp) are self-dual. Λ 14. Show that the coproduct in the Hopf algebra H∗(X; R) dual to H ∗(X; R) is induced by the diagonal map X→X × X, x ֏ (x, x). 15. Suppose that X is a path-connected H–space such that H ∗(X; Z) is free and ﬁnitely generated in each dimension, and H ∗(X; Q) is a polynomial ring Q[α]. Show that the Pontryagin ring H∗(X; Z) is commutative and associative, with a structure uniquely determined by the ring H ∗(X; Z). 16. Classify algebraically the Hopf algebras A over Z such that An is free for each n and A ⊗ Q ≈ Q[α]. In particular, determine which Hopf algebras A ⊗ Zp arise from such A ’s. After the general discussion of homological and cohomological properties of H–spaces in the preceding section, we turn now to a family of quite interesting and subtle examples, the orthogonal groups O(n). We will compute their homology and cohomology by constructing very nice CW structures on them, and the results illus- trate the general structure theorems of the last section quite well. After dealing with the orthogonal groups we then describe the straightforward generalization to Stiefel manifolds, which are also fairly
basic objects in algebraic and geometric topology. The orthogonal group O(n) can be deﬁned as the group of isometries of Rn ﬁxing the origin. Equivalently, this is the group of n× n matrices A with entries in R such that AAt = I, where At is the transpose of A. From this viewpoint, O(n) is topologized as a subspace of Rn2, with coordinates the n2 entries of an n× n matrix. Since the columns of a matrix in O(n) are unit vectors, O(n) can also be regarded as a subspace of the product of n copies of S n−1. It is a closed subspace since the conditions that columns be orthogonal are deﬁned by polynomial equations. Hence The Cohomology of SO(n) Section 3.D 293 O(n) is compact. The map O(n)× O(n)→O(n) given by matrix multiplication is continuous since it is deﬁned by polynomials. The inversion map A ֏ A−1 = At is clearly continuous, so O(n) is a topological group, and in particular an H–space. The determinant map O(n)→{±1} is a surjective homomorphism, so its kernel SO(n), the ‘special orthogonal group’, is a subgroup of index two. The two cosets SO(n) and O(n) − SO(n) are homeomorphic to each other since for ﬁxed B ∈ O(n) of determinant −1, the maps A ֏ AB and A ֏ AB−1 are inverse homeomorphisms between these two cosets. The subgroup SO(n) is a union of components of O(n) since the image of the map O(n)→{±1} is discrete. In fact, SO(n) is path-connected since by linear algebra, each A ∈ SO(n) is a rotation, a composition of rotations in a family of orthogonal 2 dimensional subspaces of Rn, with the identity map on the subspace orthogonal to all these planes, and such a rotation can obviously be joined to the identity by a path of rotations of the
same planes through decreasing angles. Another reason why SO(n) is connected is that it has a CW structure with a single 0 cell, as we show in Proposition 3D.1. An exercise at the end of the section is to show that a topological group with a ﬁnite-dimensional CW structure is an orientable manifold, so SO(n) is a closed orientable manifold. From the CW structure it follows that its dimension is n(n − 1)/2. These facts can also be proved using ﬁber bundles. The group O(n) is a subgroup of GLn(R), the ‘general linear group’ of all invertible n× n matrices with entries in R, discussed near the beginning of §3.C. The Gram– Schmidt orthogonalization process applied to the columns of matrices in GLn(R) provides a retraction r : GLn(R)→O(n), continuity of r being evident from the explicit formulas for the Gram–Schmidt process. By inserting appropriate scalar factors into these formulas it is easy to see that O(n) is in fact a deformation retract of GLn(R). Using a bit more linear algebra, namely the polar decomposition, it is possible to show that GLn(R) is actually homeomorphic to O(n)× Rk for k = n(n + 1)/2. The topological structure of SO(n) for small values of n can be described in terms of more familiar spaces: SO(1) is a point. SO(2), the rotations of R2, is both homeomorphic and isomorphic as a group to S 1, thought of as the unit complex numbers. SO(3) is homeomorphic to RP3. To see this, let ϕ : D3→SO(3) send a nonzero vector x to the rotation through angle \|x\|π about the axis formed by the line through the origin in the direction of x. An orientation convention such as the ‘right-hand rule’ is needed to make this unambiguous. By continuity, ϕ then sends 0 to the identity. Antipodal points of S 2 = ∂D3 are sent to the same rotation through angle π, so ϕ induces a map ϕ : RP3→SO(3), regarding RP3 as D3 with antipodal boundary points ident
iﬁed. The map ϕ is clearly injective since the axis of a nontrivial rotation is uniquely determined as its ﬁxed point set, and ϕ is surjective since by easy linear algebra each nonidentity element 294 Chapter 3 Cohomology of SO(3) is a rotation about some axis. It follows that ϕ is a homeomorphism RP3 ≈ SO(3). SO(4) is homeomorphic to S 3 × SO(3). Identifying R4 with the quaternions H and S 3 with the group of unit quaternions, the quaternion multiplication v֏vw for ﬁxed w ∈ S 3 deﬁnes an isometry ρw ∈ O(4) since \|vw\| = \|v\|\|w\| = \|v\| if \|w\| = 1. Points of O(4) are 4 tuples (v1, ···, v4) of orthonormal vectors vi ∈ H = R4, and we view O(3) as the subspace with v1 = 1. A homeomorphism S 3 × O(3)→O(4) is deﬁned by sending to (v, v2v, v3v, v4v) = ρv (1, v2, v3, v4), with inverse (v, v2, v3, v4) ֏ v, (1, v2v −1, v3v −1, v4v −1) = v, ρv −1 (v, v2, v3, v4) morphism S 3 × SO(3) ≈ SO(4). This is not a group isomorphism, however. can be shown, though we will not digress to do so here, that the homomorphism ψ : S 3 × S 3→SO(4) sending a pair (u, v) of unit quaternions to the isometry w ֏ uwv −1 of H is surjective with kernel Z2 = {±(1, 1)}, and that ψ is a covering space projection, representing S 3 × S 3 as a 2 sheeted cover of SO(4), the universal cover. Restricting ψ to the diagonal S 3 = {(u, u)} ⊂
S 3 × S 3 gives the universal cover S 3→SO(3), so SO(3) is isomorphic to the quotient group of S 3 by the normal subgroup {±1}.. Restricting to identity components, we obtain a homeo- v, (1, v2, v3, v4) It Using octonions one can construct in the same way a homeomorphism SO(8) ≈ S 7 × SO(7). But in all other cases SO(n) is only a ‘twisted product’ of SO(n − 1) and S n−1 ; see Example 4.55 and the discussion following Corollary 4D.3. Cell Structure Our ﬁrst task is to construct a CW structure on SO(n). This will come with a very nice cellular map ρ : RPn−1 × RPn−2 × ··· × RP1→SO(n). To simplify notation we will write P i for RPi. To each nonzero vector v ∈ Rn we can associate the reﬂection r (v) ∈ O(n) across the hyperplane consisting of all vectors orthogonal to v. Since r (v) is a reﬂection, it has determinant −1, so to get an element of SO(n) we consider the composition ρ(v) = r (v)r (e1) where e1 is the ﬁrst standard basis vector (1, 0, ···, 0). Since ρ(v) depends only on the line spanned by v, ρ deﬁnes a map P n−1→SO(n). This map is injective since it is the composition of v ֏r (v), which is obviously an injection of P n−1 into O(n)−SO(n), with the homeomorphism O(n)−SO(n)→SO(n) given by right-multiplication by r (e1). Since ρ is injective and P n−1 is compact Hausdorﬀ, we may think of ρ as embedding P n−1 as a subspace of SO(n). More generally, for a sequence I = (i1, ···, im) with each ij < n, we deﬁ
ne a map ρ : P I = P i1 × ··· × P im→SO(n) by letting ρ(v1, ···, vm) be the composition If ϕi : Di→P i is the standard characteristic map for the i cell of ρ(v1) ··· ρ(vm). P i, restricting to the 2 sheeted covering projection ∂Di→P i−1, then the product ϕI : DI→P I of the appropriate ϕij ’s is a characteristic map for the top-dimensional The Cohomology of SO(n) Section 3.D 295 cell of P I. We will be especially interested in the sequences I = (i1, ···, im) satisfying n > i1 > ··· > im > 0. These sequences will be called admissible, as will the sequence consisting of a single 0. Proposition 3D.1. The maps ρϕI : DI→SO(n), for I ranging over all admissible sequences, are the characteristic maps of a CW structure on SO(n) for which the map ρ : P n−1× P n−2 × ··· × P 1→SO(n) is cellular. In particular, there is a single 0 cell e0 = {11}, so SO(n) is path-connected. The other cells eI = ei1 ··· eim are products, via the group operation in SO(n), of the cells ei ⊂ P n−1 ⊂ SO(n). Proof: According to Proposition A.2 in the Appendix, there are three things to show in order to obtain the CW structure: (1) For each decreasing sequence I, ρϕI is a homeomorphism from the interior of DI onto its image. (2) The resulting image cells eI are all disjoint and cover SO(n). (3) For each eI, ρϕI (∂DI ) is contained in a union of cells of lower dimension than eI. To begin the veriﬁcation of these properties, deﬁne p : SO(n)→S n−1 by evaluation at the vector en = (0, ···, 0, 1), p(α) = α(en). Isometries in P n−2 ⊂
P n−1 ⊂ SO(n) ﬁx en, so p(P n−2) = {en}. We claim that p is a homeomorphism from P n−1 − P n−2 onto S n−1 − {en}. This can be seen as follows. Thinking of a point in P n−1 as a vector v, the map p takes this to ρ(v)(en) = r (v)r (e1)(en), which equals r (v)(en) since en is in the hyperplane orthogonal to e1. From the picture at the right it is then clear that p simply stretches the lower half of each meridian circle in S n−1 onto the whole meridian circle, doubling the angle up from the south pole, so P n−1 − P n−2, represented by vectors whose last coordinate is negative, is taken homeomorphically onto S n−1 − {en}. The next statement is that the map → h : P n−1 × SO(n − 1), P n−2 × SO(n − 1) SO(n), SO(n − 1), h(v, α) = ρ(v)α is a homeomorphism from (P n−1 − P n−2)× SO(n − 1) onto SO(n) − SO(n − 1). Here we view SO(n − 1) as the subgroup of SO(n) ﬁxing the vector en. To construct an inverse to this homeomorphism, let β ∈ SO(n) − SO(n − 1) be given. Then β(en) ≠ en so by the preceding paragraph there is a unique vβ ∈ P n−1 − P n−2 with ρ(vβ)(en) = β(en), and vβ depends continuously on β since β(en) does. The composition αβ = ρ(vβ)−1β then ﬁxes en, hence lies in SO(n − 1). Since ρ(vβ)αβ = β, the map β ֏ (vβ, αβ) is an inverse to h on SO(n) − SO(n − 1). Statements (1) and (2) can now be proved by induction on n. The map ρ takes P n
−2 to SO(n − 1), so we may assume inductively that the maps ρϕI for I ranging 296 Chapter 3 Cohomology over admissible sequences with ﬁrst term i1 < n − 1 are the characteristic maps for a CW structure on SO(n − 1), with cells the corresponding products eI. The admissible sequences I with i1 = n − 1 then give disjoint cells eI covering SO(n) − SO(n − 1) by what was shown in the previous paragraph. So (1) and (2) hold for SO(n). To prove (3) it suﬃces to show there is an inclusion P iP i ⊂ P iP i−1 in SO(n) since for an admissible sequence I, the map ρ : P I→SO(n) takes the boundary of the top-dimensional cell of P I to the image of products P J with J obtained from I by decreasing one term ij by 1, yielding a sequence which is admissible except perhaps for having two successive terms equal. As a preliminary to showing that = αr (v)α−1. Hence P iP i ⊂ P iP i−1, observe that for α ∈ O(n) we have r ρ(v)ρ(w) = r (v)r (e1)r (w)r (e1) = r (v)r (w ′) where w ′ = r (e1)w. Thus to show P iP i ⊂ P iP i−1 it suﬃces to ﬁnd for each pair v, w ∈ Ri+1 a pair x ∈ Ri+1, y ∈ Ri with r (v)r (w) = r (x)r (y). α(v) Let V ⊂ Ri+1 be a 2 dimensional subspace containing v and w. Since V ∩ Ri is at least 1 dimensional, we can choose a unit vector y ∈ V ∩ Ri. Let α ∈ O(i + 1) take V to R2 and y to e1. Then the conjugate αr (v)r (w)α−1 = r lies in SO(2), hence has the form ρ(z) = r (z)r (e1) for some z �
� R2 by statement (2) for n = 2. Therefore α(w) α(v) r r (v)r (w) = α−1r (z)r (e1)α = r α−1(z) r for x = α−1(z) ∈ Ri+1 and y ∈ Ri. α−1(e1) = r (x)r (y) It remains to show that the map ρ : P n−1× P n−2 × ··· × P 1→SO(n) is cellular. This follows from the inclusions P iP i ⊂ P iP i−1 derived above, together with another family of inclusions P iP j ⊂ P jP i for i < j. To prove the latter we have the formulas ρ(v)ρ(w) = r (v)r (w ′) where w ′ = r (e1)w, as earlier = r (v)r (w ′)r (v)r (v) = r r (v)w ′ r (v) from r α(v) = αr (v)α−1 = r = ρ r (v)r (e1)w ρ(v)w ρ(v ′) r (v) = r ρ(v)w r (v) where v ′ = r (e1)v, hence v = r (e1)v ′ In particular, taking v ∈ Ri+1 and w ∈ Rj+1 with i < j, we have ρ(v)w ∈ Rj+1, and the product ρ(v)ρ(w) ∈ P iP j equals the product ρ ⊔⊓ ρ(v ′) ∈ P jP i. ρ(v)w Mod 2 Homology and Cohomology Each cell of SO(n) is the homeomorphic image of a cell in P n−1 × P n−2 × ··· × P 1, so the cellular chain map induced by ρ : P n−1 × P n−2× ··· × P 1→SO(n) is surjective. It follows that with Z2 coeﬃcients the cellular boundary maps for SO(n) are all trivial since this is true in
P i and hence in P n−1× P n−2 × ··· × P 1 by Proposition 3B.1. Thus H∗(SO(n); Z2) has a Z2 summand for each cell of SO(n). One can rephrase this The Cohomology of SO(n) Section 3.D 297 as saying that there are isomorphisms Hi(SO(n); Z2) ≈ Hi(S n−1 × S n−2 × ··· × S 1; Z2) for all i since this product of spheres also has cells in one-to-one correspondence with admissible sequences. The full structure of the Z2 homology and cohomology rings is given by: Theorem 3D.2. (a) H ∗(SO(n); Z2) ≈ the smallest power of 2 such that \|βpi (b) The Pontryagin ring H∗(SO(n); Z2) is the exterior algebra i odd Z2[βi]/(β i \| ≥ n. N pi i ) where \|βi\| = i and pi is Z2 [e1, ···, en−1]. Here ei denotes the cellular homology class of the cell ei ⊂ P n−1 ⊂ SO(n), and βi is the dual class to ei, represented by the cellular cochain assigning the value 1 to the cell ei and 0 to all other i cells. Λ Proof: As we noted above, ρ induces a surjection on cellular chains. Since the cellular boundary maps with Z2 coeﬃcients are trivial for both P n−1 × ··· × P 1 and SO(n), it follows that ρ∗ is surjective on H∗(−; Z2) and ρ∗ is injective on H ∗(−; Z2). We know that H ∗(P n−1 × ··· × P 1; Z2) is the polynomial ring Z2[α1, ···, αn−1] truncated by the relations αi+1 i = 0. For βi ∈ H i(SO(n); Z2) the dual class to ei, we have ρ∗(βi) = j, the class assigning 1 to each i cell in a factor P j
of P n−1 × ··· × P 1 and 0 to all other i cells, which are products of lower-dimensional cells and hence map to cells in SO(n) disjoint from ei. j αi P P j αi ··· βim First we will show that the monomials βI = βi1 corresponding to admissible sequences I are linearly independent in H ∗(SO(n); Z2), hence are a vector space basis. Since ρ∗ is injective, we may identify each βi with its image j in the truncated polynomial ring Z2[α1, ···, αn−1]/(α2 1, ···, αn n−1). Suppose we have a linear P I bI βI = 0 with bI ∈ Z2 and I ranging over the admissible sequences. Since relation each βI is a product of distinct βi ’s, we can write the relation in the form xβ1 + y = 0 where neither x nor y has β1 as a factor. Since α1 occurs only in the term β1 of xβ1 + y, where it has exponent 1, we have xβ1 + y = xα1 + z where neither x nor z involves α1. The relation xα1 + z = 0 in Z2[α1, ···, αn−1]/(α2 n−1) then implies x = 0. Thus we may assume the original relation does not involve β1. Now we repeat the argument for β2. Write the relation in the form xβ2 + y = 0 where neither x nor y involves β2 or β1. The variable α2 now occurs only in the term β2 of xβ2 + y, where it has exponent 2, so we have xβ2 + y = xα2 2 + z where x and z do not involve α1 or α2. Then xα2 2 + z = 0 implies x = 0 and we have a relation involving neither β1 nor β2. Continuing inductively, we eventually deduce that all coeﬃcients bI in the original relation I bIβI = 0 must be zero. 1, ···, αn j α2i i = β2i if 2i < n and β2 Observe now that β2 i = 0 if 2i ≥ n, since P
j. The quotient Q of the algebra Z2[β1, β2, ···] by the relations β2 2 = j αi j i = β2i and P βj = 0 for j ≥ n then maps onto H ∗(SO(n); Z2). This map Q→H ∗(SO(n); Z2) P is also injective since the relations deﬁning Q allow every element of Q to be represented as a linear combination of admissible monomials βI, and the admissible 298 Chapter 3 Cohomology monomials are linearly independent in H ∗(SO(n); Z2). The algebra Q can also be described as the tensor product in statement (a) of the theorem since the relations β2 i = β2i allow admissible monomials to be written uniquely as monomials in powers pi i = 0 of the βi ’s with i odd, and the relation βj = 0 for j ≥ n becomes βipi where j = ipi with i odd and pi a power of 2. For a given i, this relation holds iﬀ ipi ≥ n, or in other words, iﬀ \|β pi i \| ≥ n. This ﬁnishes the proof of (a). = β For part (b), note ﬁrst that the group multiplication SO(n)× SO(n)→SO(n) is cellular in view of the inclusions P iP i ⊂ P iP i−1 and P iP j ⊂ P jP i for i < j. So we can compute Pontryagin products at the cellular level. We know that there is at Z2 [e1, ···, en−1] since the products least an additive isomorphism H∗(SO(n); Z2) ≈ eI = ei1 ··· eim with I admissible form a basis for H∗(SO(n); Z2). The inclusion P iP i ⊂ P iP i−1 then implies that the Pontryagin product (ei)2 is 0. It remains only to see the commutativity relation eiej = ej ei. The inclusion P iP j ⊂ P jP i for i < j was obtained from the formula ρ(v)ρ(w) = ρ(ρ
(v)w)ρ(v ′) for v ∈ Ri+1, w ∈ Rj+1, and v ′ = r (e1)v. The map f : P i × P j→P j × P i, f (v, w) = (ρ(v)w, v ′), is a homeomorphism since it is the composition of homeomorphisms (v, w) ֏ (v, ρ(v)w) ֏ (v ′, ρ(v)w) ֏ (ρ(v)w, v ′). The ﬁrst of these maps takes ei × ej homeomorphically onto itself since ρ(v)(ej ) = ej if i < j. Obviously the second map also takes ei × ej homeomorphically onto itself, while the third map simply transposes the two factors. Thus f restricts to a homeomorphism from ei × ej onto ej × ei, and therefore eiej = ej ei in H∗(SO(n); Z2). ⊔⊓ Λ The cup product and Pontryagin product structures in this theorem may seem at Γ If we truncate the polynomial algebra by a relation x2n Z2 [α], and with Z2 coeﬃcients the latter is an exterior algebra ﬁrst glance to be unrelated, but in fact the relationship is fairly direct. As we saw in the previous section, the dual of a polynomial algebra Z2[x] is a divided polynomial alZ2 [α0, α1, ···] gebra where \|αi\| = 2i\|x\|. = 0, then this just eliminates the generators αi for i ≥ n. In view of this, if it were the case that the generators βi for the algebra H ∗(SO(n); Z2) happened to be primitive, then H ∗(SO(n); Z2) would be isomorphic as a Hopf algebra to the tensor product of the single-generator Hopf algebras Z2[βi]/(βpi i ), i = 1, 3, ···, hence the dual algebra H∗(SO(n); Z2) would be the tensor
product of the corresponding truncated divided polynomial algebras, in other words an exterior algebra as just explained. This is in fact the structure of H∗(SO(n); Z2), so since the Pontryagin product in H∗(SO(n); Z2) determines the coproduct in H ∗(SO(n); Z2) uniquely, it follows that the βi ’s must indeed be primitive. Λ It is not diﬃcult to give a direct argument that each βi is primitive. The coprod: H ∗(SO(n); Z2)→H ∗(SO(n); Z2) ⊗ H ∗(SO(n); Z2) is induced by the group mul(βi) on (βi), eI ⊗ eJ i, is the same as the value hβi ⊗ 1+1 ⊗ βi, eI ⊗ eJ i uct tiplication µ : SO(n)× SO(n)→SO(n). We need to show that the value of eI ⊗ eJ, which we denote h ∆ ∆ ∆ The Cohomology of SO(n) Section 3.D 299 = µ∗, we have for all cells eI and eJ whose dimensions add up to i. Since (βi), eI ⊗ eJ i = hβi, µ∗(eI ⊗ eJ )i. Because µ is the multiplication map, µ(eI × eJ ) h is contained in P IP J, and if we use the relations P jP j ⊂ P jP j−1 and P jP k ⊂ P kP j for ∆ j < k to rearrange the factors P j of P I P J so that their dimensions are in decreasing order, then the only way we will end up with a term P i is if we start with P IP J equal to P iP 0 or P 0P i. Thus hβi, µ∗(eI ⊗ eJ )i = 0 unless eI ⊗ eJ equals ei ⊗ e0 or e0 ⊗ ei. Hence (βi) contains no other terms besides βi ⊗ 1 + 1 ⊗ βi, and
βi is primitive. ∆ Integer Homology and Cohomology ∆ With Z coeﬃcients the homology and cohomology of SO(n) turns out to be a good bit more complicated than with Z2 coeﬃcients. One can see a little of this complexity already for small values of n, where the homeomorphisms SO(3) ≈ RP3 and SO(4) ≈ S 3 × RP3 would allow one to compute the additive structure as a direct sum of a certain number of Z ’s and Z2 ’s. For larger values of n the additive structure is qualitatively the same: Proposition 3D.3. H∗(SO(n); Z) is a direct sum of Z ’s and Z2 ’s. Proof: We compute the cellular chain complex of SO(n), showing that it splits as a tensor product of simpler complexes. For a cell ei ⊂ P n−1 ⊂ SO(n) the cellular boundary dei is 2ei−1 for even i > 0 and 0 for odd i. To compute the cellular boundary of a cell ei1 ··· eim we can pull it back to a cell ei1 × ··· × eim of P n−1 × ··· × P 1 whose j(−1)σj ei1 × ··· × deij × ··· × eim where cellular boundary, by Proposition 3B.1, is j(−1)σj ei1 ··· deij ··· eim, where it is unσj = i1 + ··· + ij−1. Hence d(ei1 ··· eim ) = derstood that ei1 ··· deij ··· eim is zero if ij = ij+1 + 1 since P ij −1P ij −1 ⊂ P ij −1P ij −2, P in a lower-dimensional skeleton. P To split the cellular chain complex C∗ as a tensor product of smaller SO(n) chain complexes, let C 2i be the subcomplex of C∗ with basis the cells e0, e2i, e2i−1, and e2ie2i−1. This is a subcomplex since de2i−1 = 0, de2i = 2e2i−1, and,
in P 2i × P 2i−1, d(e2i × e2i−1) = de2i × e2i−1 + e2i × de2i−1 = 2e2i−1× e2i−1, hence d(e2ie2i−1) = 0 since P 2i−1P 2i−1 ⊂ P 2i−1P 2i−2. The claim is that there are chain complex isomorphisms SO(n) C∗ C∗ SO(2k + 1) SO(2k + 2) ≈ C 2⊗C 4⊗ ··· ⊗C 2k ≈ C 2⊗C 4⊗ ··· ⊗C 2k⊗C 2k+1 where C 2k+1 has basis e0 and e2k+1. Certainly these isomorphisms hold for the chain groups themselves, so it is only a matter of checking that the boundary maps agree. For the case of C∗ easily verify. Then the case of C∗ argument. this can be seen by induction on k, as the reader can reduces to the ﬁrst case by a similar SO(2k + 2) SO(2k + 1) Since H∗(C 2i) consists of Z ’s in dimensions 0 and 4i − 1 and a Z2 in dimension 2i − 1, while H∗(C 2k+1) consists of Z ’s in dimensions 0 and 2k + 1, we conclude 300 Chapter 3 Cohomology from the algebraic K¨unneth formula that H∗(SO(n); Z) is a direct sum of Z ’s and Z2 ’s. ⊔⊓ Note that the calculation shows that SO(2k) and SO(2k − 1)× S 2k−1 have iso- morphic homology groups in all dimensions. In view of the preceding proposition, one can get rather complete information about H∗(SO(n); Z) by considering the natural maps to H∗(SO(n); Z2) and to the quotient of H∗(SO(n); Z) by its torsion subgroup. Let us denote this quotient by H f r ee (SO(n); Z).
The same strategy applies equally well to cohomology, and the uni∗ versal coeﬃcient theorem gives an isomorphism H ∗ (SO(n); Z). (SO(n); Z) f r ee(SO(n); Z) ≈ H f r ee The proof of the proposition shows that the additive structure of H f r ee ∗ ∗ is fairly simple: H f r ee ∗ H f r ee ∗ (SO(2k + 1); Z) ≈ H∗(S 3 × S 7 × ··· × S 4k−1) (SO(2k + 2); Z) ≈ H∗(S 3 × S 7 × ··· × S 4k−1 × S 2k+1) The multiplicative structure is also as simple as it could be: Proposition 3D.4. The Pontryagin ring H f r ee ∗ (SO(n); Z) is an exterior algebra, H f r ee ∗ H f r ee ∗ (SO(2k + 1); Z) ≈ (SO(2k + 2); Z) ≈ Z[a3, a7, ···, a4k−1] where \|ai\| = i Z[a3, a7, ···, a4k−1, a′ 2k+1 are primitive, so the dual Hopf algebra H ∗ 2k+1] Λ The generators ai and a′ is an exterior algebra on the dual generators αi and α′ Λ 2k+1. f r ee(SO(n); Z) ∗ Proof: As in the case of Z2 coeﬃcients we can work at the level of cellular chains since the multiplication in SO(n) is cellular. Consider ﬁrst the case n = 2k + 1. Let Ei be the cycle e2ie2i−1 generating a Z summand of H∗(SO(n); Z). By what we have shown above, the products Ei1 ··· Eim with i1 > ··· > im form an additive basis for H f r ee (SO(n); Z), so we need only verify that the multiplication is as in an exterior algebra on the classes Ei. The map f in the proof of Theorem 3
D.2 gives a homeomorphism ei × ej ≈ ej × ei if i < j, and this homeomorphism has local degree (−1)ij+1 since it is the composition (v, w) ֏ (v, ρ(v)w) ֏ (v ′, ρ(v)w) ֏ (ρ(v)w, v ′) of homeomorphisms with local degrees +1, −1, and (−1)ij. Applying this four times to commute EiEj = e2ie2i−1e2je2j−1 to EjEi = e2je2j−1e2ie2i−1, three of the four applications give a sign of −1 and the fourth gives a +1, so we conclude that EiEj = −EjEi if i < j. When i = j we have (Ei)2 = 0 since e2ie2i−1e2ie2i−1 = e2ie2ie2i−1e2i−1, which lies in a lower-dimensional skeleton because of the relation P 2iP 2i ⊂ P 2iP 2i−1. Thus we have shown that H∗(SO(2k + 1); Z) contains Z[E1, ···, Ek] as a subalgebra. The same reasoning shows that H∗(SO(2k + 2); Z) contains the subalgebra Z[E1, ···, Ek, e2k+1]. These exterior subalgebras account for all the nontorsion in Λ H∗(SO(n); Z), so the product structure in H f r ee Λ ∗ (SO(n); Z) is as stated. The Cohomology of SO(n) Section 3.D 301 Now we show that the generators Ei and e2k+1 are primitive in H f r ee ∗ (SO(n); Z). Looking at the formula for the boundary maps in the cellular chain complex of SO(n), we see that this chain complex is the direct sum of the subcomplexes C(m) with basis the m fold products ei1 ··· eim with i1 > ··· > im > 0. We allow m = 0 here
, with C(0) having basis the 0 cell of SO(n). The direct sum C(0)⊕ ··· ⊕ C(m) is the cellular chain complex of the subcomplex of SO(n) consisting of cells that are products of m or fewer cells ei. In particular, taking m = 2 we have a subcomplex X ⊂ SO(n) whose homology, mod torsion, consists of the Z in dimension zero and the Z ’s generated by the cells Ei, together with the cell e2k+1 when n = 2k + 2. The inclusion X ֓ SO(n) induces a commutative diagram ∗ ∆ is the coproduct in H f r ee (SO(n); Z) and the upper where the lower is its analog for X, coming from the diagonal map X→X × X and the K¨unneth formula. The classes Ei in the lower left group pull back to elements we label Ei in the upper left group. Since these have odd dimension and H f r ee (X; Z) vanishes in even positive Ei) can have no components a ⊗ b with both a and b dimensions, the images ( (Ei) by commutativity of the positive-dimensional. The same is therefore true for ∆ diagram, so the classes Ei are primitive. This argument also works for e2k+1 when n = 2k + 2. ∆ ∆ e e ∗ Since the exterior algebra generators of H f r ee ∗ (SO(n); Z) are primitive, this al- gebras Z[ai] (and gebra splits as a Hopf algebra into a tensor product of single-generator exterior al2k+1] ). The dual Hopf algebra H ∗ f r ee(SO(n); Z) therefore 2k+1] ), ⊔⊓ splits as the tensor product of the dual exterior algebras Λ hence H ∗ f r ee(SO(n); Z) is also an exterior algebra. Z[αi] (and Z[α′ Z[a′ Λ Λ Λ The exact ring structure of H ∗(SO(n); Z) can be deduced from these results via Bockstein homomorphisms, as we show in Example 3E.
7, though the process is somewhat laborious and the answer not very neat. Stiefel Manifolds Consider the Stiefel manifold Vn,k, whose points are the orthonormal k frames in Rn, that is, orthonormal k tuples of vectors. Thus Vn,k is a subset of the product of k copies of S n−1, and it is given the subspace topology. As special cases, Vn,n = O(n) and Vn,1 = S n−1. Also, Vn,2 can be identiﬁed with the space of unit tangent vectors to S n−1 since a vector v at the point x ∈ S n−1 is tangent to S n−1 iﬀ it is orthogonal to x. We can also identify Vn,n−1 with SO(n) since there is a unique way of extending an orthonormal (n − 1) frame to a positively oriented orthonormal n frame. 302 Chapter 3 Cohomology There is a natural projection p : O(n)→Vn,k sending α ∈ O(n) to the k frame consisting of the last k columns of α, which are the images under α of the last k standard basis vectors in Rn. This projection is onto, and the preimages of points are precisely the cosets αO(n − k), where we embed O(n − k) in O(n) as the orthogonal transformations of the ﬁrst n − k coordinates of Rn. Thus Vn,k can be viewed as the space O(n)/O(n − k) of such cosets, with the quotient topology from O(n). This is the same as the previously deﬁned topology on Vn,k since the projection O(n)→Vn,k is a surjection of compact Hausdorﬀ spaces. When k < n the projection p : SO(n)→Vn,k is surjective, and Vn,k can also be viewed as the coset space SO(n)/SO(n−k). We can use this to induce a CW structure on Vn,k from the CW structure on SO(n). The cells are the sets of cosets of the form eI SO(n − k) = e
i1 ··· eim SO(n − k) for n > i1 > ··· > im ≥ n − k, together with the coset SO(n − k) itself as a 0 cell of Vn,k. These sets of cosets are unions of cells of SO(n) since SO(n−k) consists of the cells eJ = ej1 ··· ejℓ with n−k > j1 > ··· > jℓ. This implies that Vn,k is the disjoint union of its cells, and the boundary of each cell is contained in cells of lower dimension, so we do have a CW structure. Since the projection SO(n)→Vn,k is a cellular map, the structure of the cellular chain complex of Vn,k can easily be deduced from that of SO(n). For example, the cellular chain complex of V2k+1,2 is just the complex C 2k deﬁned earlier, while for V2k,2 the cellular boundary maps are all trivial. Hence the nonzero homology groups of Vn,2 are Hi(V2k+1,2; Z) = Z Z2 for i = 0, 4k − 1 for i = 2k − 1 Hi(V2k,2; Z) = Z for i = 0, 2k − 2, 2k − 1, 4k − 3 Thus SO(n) has the same homology and cohomology groups as the product space V3,2 × V5,2 × ··· × V2k+1,2 when n = 2k+1, or as V3,2× V5,2 × ··· × V2k+1,2× S 2k+1 when n = 2k + 2. However, our calculations show that SO(n) is distinguished from these products by its cup product structure with Z2 coeﬃcients, at least when n ≥ 5, since β4 1 is nonzero in H 4(SO(n); Z2) if n ≥ 5, while for the product spaces the nontrivial element of H 1(−; Z2) must lie in the factor V3,2, and H 4(V3,2; Z2) = 0. When n = 4 we have SO(4) homeomorphic to SO(3)× S 3
= V3,2× S 3 as we noted at the beginning of this section. Also SO(3) = V3,2 and SO(2) = S 1. Exercises 1. Show that a topological group with a ﬁnite-dimensional CW structure is an orientable manifold. [Consider the homeomorphisms x ֏ gx or x ֏ xg for ﬁxed g and varying x in the group.] 2. Using the CW structure on SO(n), show that π1SO(n) ≈ Z2 for n ≥ 3. Find a loop representing a generator, and describe how twice this loop is nullhomotopic. 3. Compute the Pontryagin ring structure in H∗(SO(5); Z). Bockstein Homomorphisms Section 3.E 303 Homology and cohomology with coeﬃcients in a ﬁeld, particularly Zp with p prime, often have more structure and are easier to compute than with Z coeﬃcients. Of course, passing from Z to Zp coeﬃcients can involve a certain loss of information, a blurring of ﬁner distinctions. For example, a Zpn in integer homology becomes a pair of Zp ’s in Zp homology or cohomology, so the exponent n is lost with Zp coeﬃcients. In this section we introduce Bockstein homomorphisms, which in many interesting cases allow one to recover Z coeﬃcient information from Zp coeﬃcients. Bockstein homomorphisms also provide a small piece of extra internal structure to Zp homology or cohomology itself, which can be quite useful. We will concentrate on cohomology in order to have cup products available, If we take a short exbut the basic constructions work equally well for homology. act sequence 0→G→H→K→0 of abelian groups and apply the covariant functor Hom(Cn(X), −), we obtain 0 -→ C n(X; G) -→ C n(X; H) -→ C n(X; K) -→ 0 which is exact since Cn(X) is free. Letting n vary, we have a short exact sequence of chain complexes, so there is an associated long exact sequence ·
·· -→ H n(X; G) -→ H n(X; H) -→ H n(X; K) -→ H n+1(X; G) -→ ··· whose ‘boundary’ map H n(X; K)→H n+1(X; G) is called a Bockstein homomorphism. We shall be interested primarily in the Bockstein β : H n(X; Zm)→H n+1(X; Zm) asm-----→ Zm2 -→ Zm→0, especially when m is sociated to the coeﬃcient sequence 0→Zm prime, but for the moment we do not need this assumption. Closely related to β is the β : H n(X; Zm)→H n+1(X; Z) associated to 0→Z m-----→ Z -→ Zm→0. From the natural map of the latter short exact sequence onto the former one, we obtain the reβ where ρ : H ∗(X; Z)→H ∗(X; Zm) is the homomorphism induced by lationship β = ρ the map Z→Zm reducing coeﬃcients mod m. Thus we have a commutative triangle in the following diagram, whose upper row is the exact sequence containing Bockstein β. e e e Example 3E.1. Let X be a K(Zm, 1), for example RP∞ when m = 2 or an inﬁnitedimensional lens space with fundamental group Zm for arbitrary m. From the homology calculations in Examples 2.42 and 2.43 together with the universal coeﬃcient theorem or cellular cohomology we have H n(X; Zm) ≈ Zm for all n. Let us show that β : H n(X; Zm)→H n+1(X; Zm) is an isomorphism for n odd and zero for n even. If n is odd the vertical map ρ in the diagram above is surjective for X = K(Zm, 1), as 304 Chapter 3 Cohomology is β since the map m is trivial, so β is surjective, hence an isomorphism. On the other hand, when n is even the ﬁrst map ρ in the diagram
is surjective, so exactness, hence β = 0. e β = 0 by e A useful fact about β is that it satisﬁes the derivation property (∗) β(a ` b) = β(a) ` b + (−1)\|a\|a ` β(b) which comes from the corresponding formula for ordinary coboundary. Namely, let ϕ and ψ be Zm cocycles representing a and b, and let ψ be lifts of these to Zm2 cochains. Concretely, one can view ϕ and ψ as functions on singular simplices with values in {0, 1, ···, m − 1}, and then ψ can be taken to be the same functions, but with {0, 1, ···, m − 1} regarded as a subset of Zm2. Then δ ϕ = mη e ψ = mµ for Zm cocycles η and µ representing β(a) and β(b). Taking cup and δ ϕ and ϕ and e e e products, e ϕ ` ψ is a Zm2 cochain lifting the Zm cocycle ϕ ` ψ, and δ( ϕ ` e e ψ = mη ` ψ ± ϕ ` mµ = m η ` ψ ± ϕ ` µ e where the sign ± is (−1)\|a\|. Hence η ` ψ + (−1)\|a\|ϕ ` µ represents β(a ` b), giving the formula (∗). e e e e e e e e Example 3E.2: Cup Products in Lens Spaces. The cup product structure for lens spaces was computed in Example 3.41 via Poincar´e duality, but using Bocksteins we can deduce it from the cup product structure in CP∞, which was computed in Theorem 3.19 without Poincar´e duality. Consider ﬁrst the inﬁnite-dimensional lens space L = S ∞/Zm where Zm acts on the unit sphere S ∞ in C∞ by scalar multiplication, so the action is generated by the rotation v ֏ e2π i/mv. The quotient map S ∞→CP∞ factors through L, so we have a projection L
→CP∞. Looking at the cell structure on L described in Example 2.43, we see that each even-dimensional cell of L projects homeomorphically onto the corresponding cell of CP∞. Namely, the 2n cell of L is the homeomorphic image of the 2n cell in S 2n+1 ⊂ Cn+1 formed by the points i \|zi\|2 = 1 and 0 < θ ≤ π /2, and the cos θ(z1, ···, zn, 0) + sin θ(0, ···, 0, 1) with same is true for the 2n cell of CP∞. From cellular cohomology it then follows that the map L→CP∞ induces isomorphisms on even-dimensional cohomology with Zm coeﬃcients. Since H ∗(CP∞; Zm) is a polynomial ring, we deduce that if y ∈ H 2(L; Zm) is a generator, then y k generates H 2k(L; Zm) for all k. P By Example 3E.1 there is a generator x ∈ H 1(L; Zm) with β(x) = y. The product formula (∗) gives β(xy k) = β(x)y k − xβ(y k) = y k+1. Thus β takes xy k to a generator, hence xy k must be a generator of H 2k+1(L; Zm). This completely determines the cup product structure in H ∗(L; Zm) if m is odd since the commutativity property of cup product implies that x2 = 0 in this case. The result is that H ∗(L; Zm) ≈ Zm [x] ⊗ Zm[y] for odd m. When m is even this statement needs to be modiﬁed slightly by inserting the relation that x2 is the unique element of order Λ Bockstein Homomorphisms Section 3.E 305 2 in H 2(L; Zm) ≈ Zm, as we showed in Example 3.9 by an explicit calculation in the 2 skeleton of L. The cup product structure in ﬁnite-dimensional lens spaces follows from this since a ﬁnite-dimensional lens space embeds as a skeleton in an
inﬁnite-dimensional lens space, and the homotopy type of an inﬁnite-dimensional lens space is determined by its fundamental group since it is a K(π, 1). It follows that the cup product structure on a lens space S 2n+1/Zm with Zm coeﬃcients is obtained from the preceding calculation by truncating via the relation y n+1 = 0. β implies that β2 = ρ The relation β = ρ sequence containing βρ = 0 in the long exact β = 0 since β. Because β2 = 0, the groups H n(X; Zm) form a chain complex with the Bockstein homomorphisms β as the ‘boundary’ maps. We can then form the associated Bockstein cohomology groups Ker β/ Im β, which we denote BH n(X; Zm) in dimension n. The most interesting case is when m is a prime p, so we shall assume βρ e e e e e this from now on. Proposition 3E.3. If Hn(X; Z) is ﬁnitely generated for all n, then the Bockstein cohomology groups BH n(X; Zp) are determined by the following rules : (a) Each Z summand of H n(X; Z) contributes a Zp summand to BH n(X; Zp). (b) Each Zpk summand of H n(X; Z) with k > 1 contributes Zp summands to both BH n−1(X; Zp) and BH n(X; Zp). (c) A Zp summand of H n(X; Z) gives Zp summands of H n−1(X; Zp) and H n(X; Zp) with β an isomorphism between these two summands, hence there is no contribution to BH ∗(X; Zp). Proof: We will use the algebraic notion of minimal chain complexes. Suppose that C is a chain complex of free abelian groups for which the homology groups Hn(C) are ﬁnitely generated for each n. Choose a splitting of each Hn(C) as a direct sum of cyclic groups. There are countably many of these cyclic groups, so we can list them
as G1, G2, ···. For each Gi choose a generator gi and deﬁne a corresponding chain complex M(gi) by the following prescription. If gi has inﬁnite order in Gi ⊂ Hni (C), let M(gi) consist of just a Z in dimension ni, with generator zi. On the other hand, if (C), let M(gi) consist of Z ’s in dimensions ni and ni + 1, gi has ﬁnite order k in Hni generated by xi and yi respectively, with ∂yi = kxi. Let M be the direct sum of the chain complexes M(gi). Deﬁne a chain map σ : M→C by sending zi and xi to cycles ζi and ξi representing the corresponding homology classes gi, and yi to a chain ηi with ∂ηi = kξi. The chain map σ induces an isomorphism on homology, hence also on cohomology with any coeﬃcients, by Corollary 3.4. The dual cochain complex M ∗ obtained by applying Hom(−, Z) splits as the direct sum of the dual complexes M ∗(gi). So in cohomology with Z coeﬃcients the dual basis element z∗ i generates a Z summand in dimension ni, while y ∗ i generates a Zk summand in dimension i = ky ∗ ni + 1 since δx∗ i gives a Zp summand of i. With Zp coeﬃcients, p prime, z∗ 306 Chapter 3 Cohomology H ni (M; Zp), while x∗ p divides k and otherwise they give nothing. i and y ∗ i give Zp summands of H ni (M; Zp) and H ni+1(M; Zp) if The map σ induces an isomorphism between the associated Bockstein long exact sequences of cohomology groups, with commuting squares, so we can use M ∗ to β, and we can do the calculation separately on each summand M ∗(gi). compute β and Obviously β and i. When p divides k we have the class e x∗ i ∈ H ni (M;
Zp), and from the deﬁnition of Bockstein homomorphisms it follows i ∈ H ni+1(M; Zp). The that latter element is nonzero iﬀ k is not divisible by p2. ⊔⊓ i ∈ H ni+1(M; Z) and β(x∗ β are zero on y ∗ i ) = (k/p)y ∗ i ) = (k/p)y ∗ i and z∗ β(x∗ e e Corollary 3E.4. In the situation of the preceding proposition, H ∗(X; Z) contains no elements of order p2 iﬀ the dimension of BH n(X; Zp) as a vector space over Zp equals the rank of H n(X; Z) for all n. In this case ρ : H ∗(X; Z)→H ∗(X; Zp) is injective on the p torsion, and the image of this p torsion under ρ is equal to Im β. Proof: The ﬁrst statement is evident from the proposition. The injectivity of ρ on p torsion is in fact equivalent to there being no elements of order p2. The equality Im ρ = Im β follows from the fact that Im β = ρ(Im β) = ρ(Ker m) in the commutative diagram near the beginning of this section, and the fact that for m = p the kernel of m is exactly the p torsion when there are no elements of order p2. ⊔⊓ e Example 3E.5. Let us use Bocksteins to compute H ∗(RP∞ × RP∞; Z). This could instead be done by ﬁrst computing the homology via the general K¨unneth formula, then applying the universal coeﬃcient theorem, but with Bocksteins we will only need the simpler K¨unneth formula for ﬁeld coeﬃcients in Theorem 3.15. The cup product structure in H ∗(RP∞ × RP∞; Z) will also be easy to determine via Bocksteins. For p an odd prime we have H ∗(RP∞; Zp
) = 0, hence Theorem 3.15. The universal coeﬃcient theorem then implies that H ∗(RP∞ × RP∞; Zp) = 0 by H ∗(RP∞ × RP∞; Z) consists entirely of elements of order a power of 2. From Example 3E.1 we know that Bockstein homomorphisms in H ∗(RP∞; Z2) ≈ Z2[x] are given by β(x2k−1) = x2k and β(x2k) = 0. In H ∗(RP∞ × RP∞; Z2) ≈ Z2[x, y] we can then compute β via the product formula β(xmy n) = (βxm)y n + xm(βy n). The answer can be represented graphically by the ﬁg- e e e ure to the right. Here the dot, diamond, or circle in the (m, n) position represents the monomial xmy n and line segments indicate nontrivial Bocksteins. For example, the lower left square records the formulas β(xy) = x2y + xy 2, β(x2y) = x2y 2 = β(xy 2), and β(x2y 2) = 0. Thus in this square we see that Ker β = Im β, with generators the ‘diagonal’ sum x2y + xy 2 and x2y 2. The Bockstein Homomorphisms Section 3.E 307 same thing happens in all the other squares, so it is apparent that Ker β = Im β ex- cept for the zero-dimensional class ‘ 1 ’. By the preceding corollary this says that all H ∗(RP∞ × RP∞; Z) have order 2. Furthermore, Im β consists nontrivial elements of of the subring Z2[x2, y 2], indicated by the circles in the ﬁgure, together with the multiples of x2y + xy 2 by elements of Z2[x2, y 2]. It follows that there is a ring isomorphism e H ∗(RP∞ × RP∞; Z) ≈ Z[λ, µ, ν]/(2λ, 2µ, 2ν
, ν 2 + λ2µ + λµ2) where ρ(λ) = x2, ρ(µ) = y 2, ρ(ν) = x2y + xy 2, and the relation ν 2 + λ2µ + λµ2 = 0 holds since (x2y + xy 2)2 = x4y 2 + x2y 4. This calculation illustrates the general principle that cup product structures with Z coeﬃcients tend to be considerably more complicated than with ﬁeld coeﬃcients. One can see even more striking evidence of this by computing H ∗(RP∞ × RP∞ × RP∞; Z) by the same technique. Example 3E.6. Let us construct ﬁnite CW complexes X1, X2, and Y such that the rings H ∗(X1; Z) and H ∗(X2; Z) are isomorphic but H ∗(X1 × Y ; Z) and H ∗(X2 × Y ; Z) are isomorphic only as groups, not as rings. According to Theorem 3.15 this can happen only if all three of X1, X2, and Y have torsion in their Z cohomology. The space X1 is obtained from S 2 × S 2 by attaching a 3 cell e3 to the second S 2 factor 2, e3, e4 with by a map of degree 2. Thus X1 has a CW structure with cells e0, e2 e3 attached to the 2 sphere e0 ∪ e2 2. The space X2 is obtained from S 2 ∨ S 2 ∨ S 4 by attaching a 3 cell to the second S 2 summand by a map of degree 2, so it has a CW structure with the same collection of ﬁve cells, the only diﬀerence being that in X2 the 4 cell is attached trivially. For the space Y we choose a Moore space M(Z2, 2), with cells labeled f 0, f 2, f 3, the 3 cell being attached by a map of degree 2. 1, e2 From cellular cohomology we see that both H ∗(X1; Z) and H ∗(X2; Z) consist of Z ’s in dimensions 0, 2
, and 4, and a Z2 in dimension 3. In both cases all cup products of positive-dimensional classes are zero since for dimension reasons the only possible nontrivial product is the square of the 2 dimensional class, but this is zero as one sees by restricting to the subcomplex S 2 × S 2 or S 2 ∨ S 2 ∨ S 4. For the space Y we have H ∗(Y ; Z) consisting of a Z in dimension 0 and a Z2 in dimension 3, so the cup product structure here is trivial as well. With Z2 coeﬃcients the cellular cochain complexes for Xi, Y, and Xi × Y are all trivial, so we can identify the cells with a basis for Z2 cohomology. In Xi and Y the only nontrivial Z2 Bocksteins are β(e2 2) = e3 and β(f 2) = f 3. The Bocksteins in Xi × Y can then be computed using the product formula for β, which applies to cross product as well as cup product since cross product is deﬁned in terms of cup product. The results are shown in the following table, where an arrow denotes a nontrivial Bockstein. 308 Chapter 3 Cohomology 2 × f 2) = e3 × f 2 + e2 2 × f 2 mean that β(e2 The two arrows from e2 2 × f 3. It is evident that BH ∗(Xi × Y ; Z2) consists of Z2 ’s in dimensions 0, 2, and 4, so Proposition 3E.3 implies that the nontorsion in H ∗(Xi × Y ; Z) consists of Z ’s in these dimensions. Furthermore, by Corollary 3E.4 the 2 torsion in H ∗(Xi × Y ; Z) corresponds to the image of β and consists of Z2 × Z2 ’s in dimensions 3 and 5 together with Z2 ’s in dimensions 6 and 7. In particular, there is a Z2 corresponding to e3 × f 2 +e2 2 × f 3 in dimension 5. There is no p torsion for odd primes p since H ∗(Xi × Y ; Zp) ≈ H ∗(Xi; Zp) ⊗ H ∗(Y ; Zp) is nonzero only in even dimensions. We can see now
that with Z coeﬃcients, the cup product H 2 × H 5→H 7 is nontrivial for X1 × Y but trivial for X2 × Y. For in H ∗(Xi × Y ; Z2) we have, using the relation (a× b) ` (c × d) = (a ` c)× (b ` d) which follows immediately from the deﬁnition of cross product, (1) e2 1 × f 0 ` e2 1 × f 3 = (e2 1 × f 0 ` (e3 × f 2 + e2 (2) e2 2)× f 3 since e2 (e2 1 ` e2 ` e2 2 × f 3) = (e2 ` e3 = 0 1 1 1 1)× (f 0 ` f 3) = 0 since e2 1 ` e2 1 = 0 ` e3)× (f 0 ` f 2) + (e2 1 ` e2 2)× (f 0 ` f 3) = and in H 7(Xi × Y ; Z2) ≈ H 7(Xi × Y ; Z) we have (e2 2)× f 3 = 0× f 3 = 0 for i = 2. but (e2 1 ` e2 1 ` e2 2)× f 3 = e4 × f 3 ≠ 0 for i = 1 Thus the cohomology ring of a product space is not always determined by the cohomology rings of the factors. j α2i−1 Example 3E.7. Bockstein homomorphisms can be used to get a more complete picture of the structure of H ∗(SO(n); Z) than we obtained in the preceding section. Continuing the notation employed there, we know from the calculation for RP∞ in = 0, hence β(β2i−1) = β2i j and β Example 3E.1 that β and β(β2i) = 0. Taking the case n = 5 as an example, we have H ∗(SO(5); Z2) ≈ P Z2[β1, β3]/(β8 3). The upper part of the table at the top of the next page shows the nontrivial Bocksteins. Once again two arrows from an element mean ‘sum’, for example β(β1β3) = β
(β1)β3 + β1β(β3) = β2β3 + β1β4 = β2 1. This Bockstein data allows us to calculate H i(SO(5); Z) modulo odd torsion, with the results indicated in the remainder of the table, where the vertical arrows denote the map ρ. As 1β3 + β5 j α2i j α2i 1, β2 P P = j j we showed in Proposition 3D.3, there is no odd torsion, so this in fact gives the full calculation of H i(SO(5); Z). Bockstein Homomorphisms Section 3.E 309 It is interesting that the generator y ∈ H 3(SO(5); Z) ≈ Z has y 2 nontrivial, since this implies that the ring structures of H ∗(SO(5); Z) and H ∗(RP7 × S 3; Z) are not isomorphic, even though the cohomology groups and the Z2 cohomology rings of these two spaces are the same. An exercise at the end of the section is to show that in fact SO(5) is not homotopy equivalent to the product of any two CW complexes with nontrivial cohomology. A natural way to describe H ∗(SO(5); Z) would be as a quotient of a free graded commutative associative algebra F [x, y, z] over Z with \|x\| = 2, \|y\| = 3, and \|z\| = 7. Elements of F [x, y, z] are representable as polynomials p(x, y, z), subject only to the relations imposed by commutativity. In particular, since y and z are odd-dimensional we have yz = −zy, and y 2 and z2 are nonzero elements of order 2 in F [x, y, z]. Any monomial containing y 2 or z2 as a factor also has order 2. In these terms, the calculation of H ∗(SO(5); Z) can be written H ∗(SO(5); Z) ≈ F [x, y, z]/(2x, x4, y 4, z2, xz, x3 − y 2) The next ﬁgure shows the nontrivial Bocksteins for H
∗(SO(7); Z2). Here the numbers across the top indicate dimension, stopping with 21, the dimension of SO(7). The labels on the dots refer to the basis of products of distinct βi ’s. For example, the dot labeled 135 is β1β3β5. The left-right symmetry of the ﬁgure displays Poincar´e duality quite graphically. Note that the corresponding diagram for SO(5), drawn in a slightly diﬀerent way from 310 Chapter 3 Cohomology the preceding ﬁgure, occurs in the upper left corner as the subdiagram with labels 1 through 4. This subdiagram has the symmetry of Poincar´e duality as well. From the diagram one can with some eﬀort work out the cup product structure in H ∗(SO(7); Z), but the answer is rather complicated, just as the diagram is: F [x, y, z, v, w]/(2x, 2v, x4, y 4, z2, v 2, w 2,xz, vz, vw, y 2w, x3y 2v, y 2z − x3v, xw − y 2v − x3v) where x, y, z, v, w have dimensions 2, 3, 7, 7, 11, respectively. It is curious that the relation x3 = y 2 in H ∗(SO(5); Z) no longer holds in H ∗(SO(7); Z). Exercises 1. Show that H ∗(K(Zm, 1); Zk) is isomorphic as a ring to H ∗(K(Zm, 1); Zm) ⊗ Zk if k divides m. In particular, if m/k is even, this is [x] ⊗ Zk[y]. Zk Λ 1 ··· ℓ′ In this problem we will derive one half of the classiﬁcation of lens spaces up n) then nkn mod m for some integer k. The converse is Exercise 29 2. to homotopy equivalence, by showing that if Lm(ℓ1, ···, ℓn) ≃ Lm(ℓ′ ℓ1 ··· ℓn ≡
±ℓ′ for §4.2. (a) Let L = Lm(ℓ1, ···, ℓn) and let Z∗ m be the multiplicative group of invertible elements of Zm. Deﬁne t ∈ Z∗ m by the equation xy n−1 = tz where x is a generator of H 1(L; Zm), y = β(x), and z ∈ H 2n−1(L; Zm) is the image of a generator of H 2n−1(L; Z). Show that the image τ(L) of t in the quotient group Z∗ m)n depends only on the homotopy type of L. 1, ···, ℓ′ m/±(Z∗ (b) Given nonzero integers k1, ···, kn, deﬁne a map f : S 2n−1→S 2n−1 sending the unit vector (r1eiθ1, ···, rneiθn ) in Cn to (r1eik1θ1, ···, rneiknθn ). Show: (i) e (ii) f has degree k1 ··· kn. f induces a quotient map f : L→L′ for L′ = Lm(ℓ′ e kjℓj ≡ ℓ′ e j mod m for each j. 1, ···, ℓ′ n) provided that (iii) f induces an isomorphism on π1, hence on H 1(−; Zm). (iv) f has degree k1 ··· kn, i.e., f∗ is multiplication by k1 ··· kn on H2n−1(−; Z). (c) Using the f in (b), show that τ(L) = k1 ··· knτ(L′). (d) Deduce that if Lm(ℓ1, ···, ℓn) ≃ Lm(ℓ′ 1, ···, ℓ′ n), then ℓ1 ··· ℓn ≡ ±ℓ′ 1 ··· ℓ′ nkn mod m for some integer k. 3. Let X be the
smash product of k copies of a Moore space M(Zp, n) with p prime. Compute the Bockstein homomorphisms in H ∗(X; Zp) and use this to describe H ∗(X; Z). 4. Using the cup product structure in H ∗(SO(5); Z), show that SO(5) is not homotopy equivalent to the product of any two CW complexes with nontrivial cohomology. Limits and Ext Section 3.F 311 It often happens that one has a CW complex X expressed as a union of an increasing sequence of subcomplexes X0 ⊂ X1 ⊂ X2 ⊂ ···. For example, Xi could be the i skeleton of X, or the Xi ’s could be ﬁnite complexes whose union is X. In situations of this sort, Proposition 3.33 says that Hn(X; G) is the direct limit lim--→Hn(Xi; G). Our goal in this section is to show this holds more generally for any homology the- ory, and to derive the corresponding formula for cohomology theories, which is a bit more complicated even for ordinary cohomology with Z coeﬃcients. For ordinary homology and cohomology the results apply somewhat more generally than just to CW complexes, since if a space X is the union of an increasing sequence of subspaces Xi with the property that each compact set in X is contained in some Xi, then the singular complex of X is the union of the singular complexes of the Xi ’s, and so this gives a reduction to the CW case. Passing to limits can often result in nonﬁnitely generated homology and cohomol- ogy groups. At the end of this section we describe some of the rather subtle behavior of Ext for nonﬁnitely generated groups. Direct and Inverse Limits L α1---------→ G2 As a special case of the general deﬁnition in §3.3, the direct limit lim--→Gi of a α2---------→ G3 ----→ ··· is deﬁned sequence of homomorphisms of abelian groups G1 i Gi by the subgroup consisting of elements of to be the quotient of the direct sum the form (g1, g2 − α1(g1), g3 − α2(g2
), ···). It is easy to see from this deﬁnition that every element of lim--→Gi is represented by an element gi ∈ Gi for some i, and two such representatives gi ∈ Gi and gj ∈ Gj deﬁne the same element of lim--→Gi iﬀ they have the same image in some Gk under the appropriate composition of αℓ ’s. If all the αi ’s are injective and are viewed as inclusions of subgroups, lim--→Gi is just i Gi. p-----→ Z -→ ··· with all Example 3F.1. For a prime p, consider the sequence Z maps multiplication by p. Then lim--→Gi can be identiﬁed with the subgroup Z[1/p] of Q consisting of rational numbers with denominator a power of p. More generally, we can realize any subgroup of Q as the direct limit of a sequence Z -→ Z -→ Z -→ ··· with an appropriate choice of maps. For example, if the nth map is multiplication by n, then the direct limit is Q itself. p-----→ Z S p-----→ Zp3 -→ ···, with p prime, Example 3F.2. The sequence of injections Zp has direct limit a group we denote Zp∞. This is isomorphic to Z[1/p]/Z, the subgroup of Q/Z represented by fractions with denominator a power of p. In fact Q/Z is isomorphic to the direct sum of the subgroups Z[1/p]/Z ≈ Zp∞ for all primes p. It is not hard to determine all the subgroups of Q/Z and see that each one can be realized p-----→ Zp2 as a direct limit of ﬁnite cyclic groups with injective maps between them. Conversely, every such direct limit is isomorphic to a subgroup of Q/Z. 312 Chapter 3 Cohomology We can realize these algebraic examples topologically by the following construcf1---------→ X2 ----→ ··· and Mfi−1 tion. The mapping telescope of a sequence of maps X0 is the union of the mapping cylinders Mfi identiﬁed for all i. Thus the mapping tele- with the copies of Xi in Mfi f0---------→ X
1 ` scope is the quotient space of the disjoint i (Xi × [i, i + 1]) in which each point union (xi, i + 1) ∈ Xi × [i, i + 1] is identiﬁed with (fi(xi), i + 1) ∈ Xi+1 × [i + 1, i + 2]. In the mapping telescope T, let Ti be the union of the ﬁrst i mapping cylinders. This deformation retracts onto Xi by deformation retracting each mapping cylinder onto its right end in turn. If the maps fi are cellular, each mapping cylinder is a CW complex and the telescope T is the increasing union of the subcomplexes Ti ≃ Xi. Then Proposition 3.33, or Theorem 3F.8 below, implies that Hn(T ; G) ≈ lim--→Hn(Xi; G). Example 3F.3. Suppose each fi is a map S n→S n of degree p for a ﬁxed prime p. p-----→ Z -→ ··· considered in p-----→ Z Then Hn(T ) is the direct limit of the sequence Z Hk(T ) = 0 for k ≠ n, so T is a Moore space M(Z[1/p], n). Example 3F.1 above, and Example 3F.4. In the preceding example, if we attach a cell en+1 to the ﬁrst S n in T via the identity map of S n, we obtain a space X which is a Moore space M(Zp∞, n) since X is the union of its subspaces Xi = Ti ∪ en+1, which are M(Zpi, n) ’s, and the inclusion Xi ⊂ Xi+1 induces the inclusion Zpi ⊂ Zpi+1 on Hn. e Generalizing these two examples, we can obtain Moore spaces M(G, n) for arbitrary subgroups G of Q or Q/Z by choosing maps fi : S n→S n of suitable degrees. The behavior of cohomology groups is more complicated. If X is the increasing union of subcomplexes Xi, then the cohomology groups H n(Xi; G), for ﬁxed n and G, form a sequence of homomorphisms ··· ------→ G2 Given such a sequence of
group homomorphisms, the inverse limit lim ←-- Gi is deﬁned i Gi consisting of sequences (gi) with αi(gi) = gi−1 for all i. to be the subgroup of There is a natural map λ : H n(X; G)→ lim ←-- H n(Xi; G) sending an element of H n(X; G) to its sequence of images in H n(Xi; G) under the maps H n(X; G)→H n(Xi; G) induced by inclusion. One might hope that λ is an isomorphism, but this is not true in general, α2------------→ G1 α1------------→ G0 Q as we shall see. However, for some choices of G it is: Proposition 3F.5. If the CW complex X is the union of an increasing sequence of subcomplexes Xi and if G is one of the ﬁelds Q or Zp, then λ : H n(X; G)→ lim ←-- H n(Xi; G) is an isomorphism for all n. Proof: First we have an easy algebraic fact: Given a sequence of homomorphisms ←-- Hom(Gi, G) of abelian groups G1 α2-----→ G3 -→ ···, then Hom(lim--→Gi, G) = lim α1-----→ G2 Limits and Ext Section 3.F 313 for any G. Namely, it follows from the deﬁnition of lim--→Gi that a homomorphism ϕ : lim--→ Gi→G is the same thing as a sequence of homomorphisms ϕi : Gi→G with ϕi = ϕi+1αi for all i. Such a sequence (ϕi) is exactly an element of lim←-- Hom(Gi, G). Now if G is a ﬁeld Q or Zp we have H n(X; G) = Hom(Hn(X; G), G) = Hom(lim--→Hn(Xi; G), G) = lim ←-- Hom(Hn(Xi; G), G) ←-- H n(Xi; G) = lim ⊔⊓ Let us analyze what happens for cohomology with an arbitrary coeﬃcient group,
or more generally for any cohomology theory. Given a sequence of homomorphisms of abelian groups ··· ------→ G2 α2------------→ G1 α1------------→ G0 i Gi→ deﬁne a map δ : lim ←-- Gi is the kernel of δ. Denoting the cokernel of δ by lim ←-sequence i Gi by δ(···, gi, ···) = (···, gi − αi+1(gi+1), ···), so that 1Gi, we have then an exact Q Q 0 -→ lim ←-- Gi -→ i Gi δ-----→ i Gi -→ lim ←-- 1Gi -→ 0 This may be compared with the corresponding situation for the direct limit of a sequence G1 α2---------→ G3 ----→ ···. In this case one has a short exact sequence α1---------→ G2 Q Q 0 -→ i Gi δ-----→ i Gi -→ lim--→Gi -→ 0 L L where δ(···, gi, ···) = (···, gi −αi−1(gi−1), ···), so δ is injective and there is no term lim--→ 1Gi analogous to lim 1Gi. ←-Here are a few simple observations about lim 1 : ←-- and lim ←-- 1Gi = 0. 1Gi = 0 if each αi is surjective, for to realize a given element (hi) ∈ If all the αi ’s are isomorphisms then lim←-- Gi ≈ G0 and lim←-In fact, lim i Gi as ←-δ(gi) we can take g0 = 0 and then solve α1(g1) = −h0, α2(g2) = g1 − h1, ···. If all the αi ’s are zero then lim Deleting a ﬁnite number of terms from the end of the sequence ··· →G1→G0 1Gi are undoes not aﬀect lim changed if we replace the sequence ··· →G1→G0 by a subsequence, with the appropriate compositions of αj ’s as the maps. 1Gi. More generally, lim ←-- Gi and lim ←-- �
�-- Gi or lim ←-- ←-- Gi = lim ←-- 1Gi = 0. Q Example 3F.6. Consider the sequence of natural surjections ··· →Zp3→Zp2→Zp with p a prime. The inverse limit of this sequence is a famous object in number theory, Zp. It is actually a commutative called the p adic integers. Our notation for it will be ring, not just a group, since the projections Zpi+1→Zpi are ring homomorphisms, but Zp are inﬁnite we will be interested only in the additive group structure. Elements of sequences (···, a2, a1) with ai ∈ Zpi such that ai is the mod pi reduction of ai+1. b b 314 Chapter 3 Cohomology b For each choice of ai there are exactly p choices for ai+1, so There is a natural inclusion Z ⊂ Zp is uncountable. Zp as the constant sequences ai = n ∈ Z. It is easy to see that Zp is torsionfree by checking that it has no elements of prime order. b b b b Zp. An element of There is another way of looking at Zp has a unique representation as a sequence (···, a2, a1) of integers ai with 0 ≤ ai < pi for each i. We can write each ai uniquely in the form bi−1pi−1 + ··· + b1p + b0 with 0 ≤ bj < p. The fact that ai+1 reduces mod pi to ai means that the numbers bj depend only on the Zp as the ‘base p inﬁnite Zp, so we can view the elements of element (···, a2, a1) ∈ numbers’ ··· b1b0 with 0 ≤ bi < p for all i, with the familiar rule for addition in base p notation. The ﬁnite expressions bn ··· b1b0 represent the nonnegative integers, but negative integers have inﬁnite expansions. For example, −1 has bi = p − 1 for all i, as one can see by adding 1 to this number. Since the maps Zpi+1→Zpi are surjective, lim ←-- 1Zpi = 0.
The next example shows b b how p adic integers can also give rise to a nonvanishing lim ←-- 1 term. p-----→ Z ∞ Z→ 1 term is the cokernel of the map δ : p-----→ Z for p prime. In this case Example 3F.7. Consider the sequence ··· -→ Z the inverse limit is zero since a nonzero integer can only be divided by p ﬁnitely often. The lim←-∞Z given by δ(y1, y2, ···) = Zp/Z→ Coker δ sending a p adic (y1 − py2, y2 − py3, ···). We claim that the map number ··· b1b0 as in the preceding example to (b0, b1, ···) is an isomorphism. To see this, note that the image of δ consists of the sums y1(1, 0, ···)+y2(−p, 1, 0, ···)+ y3(0, −p, 1, 0, ···) + ···. The terms after y1(1, 0, ···) give exactly the relations that hold among the p adic numbers ··· b1b0, and in particular allow one to reduce an arbitrary sequence (b0, b1, ···) to a unique sequence with 0 ≤ bi < p for all i. The term y1(1, 0, ···) corresponds to the subgroup Z ⊂ Zp. Q Q b We come now to the main result of this section: b Theorem 3F.8. For a CW complex X which is the union of an increasing sequence of subcomplexes X0 ⊂ X1 ⊂ ··· there is an exact sequence 0 -→ lim ←-- 1hn−1(Xi) -→ hn(X) λ-----→ lim ←-- hn(Xi) -→ 0 where h∗ is any reduced or unreduced cohomology theory. For any homology theory h∗, reduced or unreduced, the natural maps lim--→hn(Xi)→hn(X) are isomorphisms. Proof: Let T be the mapping telescope of the inclusion sequence X0֓X1֓···. This is a subcomplex of X × [0, ∞
→ ···. Letting Ti be the union of the ﬁrst i mapping cylinders in the telescope, the inclusions T1 ֓ T2 ֓ ··· induce on H n(−; Z) the sequence p-----→ Z in Example 3F.7. From the theorem we deduce that H n+1(T ; Z) ≈ ··· -→ Z Zp/Z H k(T ; Z) = 0 for k ≠ n+1. Thus we have the rather strange situation that the CW and complex T is the union of subcomplexes Ti each having cohomology consisting only of a Z in dimension n, but T itself has no cohomology in dimension n and instead Zp/Z in dimension n + 1. This contrasts sharply with has a huge uncountable group what happens for homology, where the groups Hn(Ti) ≈ Z ﬁt together nicely to give Hn(T ) ≈ Z[1/p]. b e b Example 3F.10. A more reasonable behavior is exhibited if we consider the space X = M(Zp∞, n) in Example 3F.4 expressed as the union of its subspaces Xi. By the universal coeﬃcient theorem, the reduced cohomology of Xi with Z coeﬃcients consists of a Zpi = Ext(Zpi, Z) in dimension n + 1. The inclusion Xi ֓ Xi+1 induces the inclusion Zpi ֓ Zpi+1 on Hn, and on Ext this induced map is a surjection Zpi+1→Zpi as one can see by looking at the diagram of free resolutions on the left: Applying Hom(−, Z) to this diagram, we get the diagram on the right, with exact rows, and the left-hand vertical map is a surjection since the vertical map to the right of it is surjective. Thus the sequence ··· →H n+1(X2; Z)→H n+1(X1; Z) is the 316 Chapter 3 Cohomology sequence in Example 3F.6, and we deduce that H n+1(X; Z) ≈ and H k(X; Z) = 0 for k ≠ n + 1. This example can be related to the e preceding one. If we view X as the mapping cone of the
inclusion S n֓T of one end of the telescope, then the long exact sequences of homology and cohomology groups for the pair (T, S n) reduce to the short exact sequences at the right. Zp, the p adic integers, b From these examples and the universal coeﬃcient theorem we obtain isomorZp/Z. These can also be derived phisms Ext(Zp∞, Z) ≈ directly from the deﬁnition of Ext. A free resolution of Zp∞ is Zp and Ext(Z[1/p], Z) ≈ b b 0 -→ Z∞ ϕ-----→ Z∞ -→ Zp∞ -→ 0 where Z∞ is the direct sum of an inﬁnite number of Z ’s, the sequences (x1, x2, ···) of integers all but ﬁnitely many of which are zero, and ϕ sends (x1, x2, ···) to (px1 − x2, px2 − x3, ···). We can view ϕ as the linear map corresponding to the inﬁnite matrix with p ’s on the diagonal, −1 ’s just above the diagonal, and 0 ’s everywhere else. Clearly Ker ϕ = 0 since integers cannot be divided by p inﬁnitely often. The image of ϕ is generated by the vectors (p, 0, ···), (−1, p, 0, ···), (0, −1, p, 0, ···), ··· so Coker ϕ ≈ Zp∞. Dualizing by taking Hom(−, Z), we have Hom(Z∞, Z) the inﬁnite direct product of Z ’s, and ϕ∗(y1, y2, ···) = (py1, py2 −y1, py3 −y2, ···), corresponding to the transpose of the matrix of ϕ. By deﬁnition, Ext(Zp∞, Z) = Coker ϕ∗. The image of ϕ∗ consists of the inﬁnite sums y1(p, −1, 0 ···) + y2(0, p, −1, 0
, ···) + ···, so Coker ϕ∗ can be identiﬁed with Zp by rewriting a sequence (z1, z2, ···) as the p adic number ··· z2z1. b The calculation Ext(Z[1/p], Z) ≈ b Zp/Z is quite similar. A free resolution of Z[1/p] can be obtained from the free resolution of Zp∞ by omitting the ﬁrst column of the matrix of ϕ and, for convenience, changing sign. This gives the formula ϕ(x1, x2, ···) = (x1, x2 − px1, x3 − px2, ···), with the image of ϕ generated by the elements (1, −p, 0, ···), (0, 1, −p, 0, ···), ···. The dual map ϕ∗ is given by ϕ∗(y1, y2, ···) = (y1 − py2, y2 − py3, ···), and this has image consisting of the sums y1(1, 0 ···) + y2(−p, 1, 0, ···) + y3(0, −p, 1, 0, ···) + ···, so we get Ext(Z[1/p], Z) = Zp/Z. Note that ϕ∗ is exactly the map δ in Example 3F.7. Coker ϕ∗ ≈ b It is interesting to note also that the map ϕ : Z∞→Z∞ in the two cases Zp∞ and Z[1/p] is precisely the cellular boundary map Hn+1(X n+1, X n)→Hn(X n, X n−1) for the Moore space M(Zp∞, n) or M(Z[1/p], n) constructed as the mapping telescope of the sequence of degree p maps S n→S n→ ···, with a cell en+1 attached to the ﬁrst S n in the case of Zp∞. Limits and Ext Section 3.F 317 More About Ext The functors Hom and Ext behave fairly simply for ﬁnitely generated groups, when cohomology and homology
are essentially the same except for a dimension shift in the torsion. But matters are more complicated in the nonﬁnitely generated case. A useful tool for getting a handle on this complication is the following: Proposition 3F.11. Given an abelian group G and a short exact sequence of abelian groups 0→A→B→C→0, there are exact sequences 0→Hom(G, A)→Hom(G, B)→Hom(G, C)→Ext(G, A)→Ext(G, B)→Ext(G, C)→0 0→Hom(C, G)→Hom(B, G)→Hom(A, G)→Ext(C, G)→Ext(B, G)→Ext(A, G)→0 Proof: A free resolution 0→F1→F0→G→0 gives rise to a commutative diagram Since F0 and F1 are free, the two rows are exact, as they are simply direct products of copies of the exact sequence 0→A→B→C→0, in view of the general fact that i Hom(Gi, H). Enlarging the diagram by zeros above and below, Hom( it becomes a short exact sequence of chain complexes, and the associated long exact iGi, H) = Q L sequence of homology groups is the ﬁrst of the two six-term exact sequences in the proposition. To obtain the other exact sequence we will construct the commutative diagram at the right, where the columns are free resolutions and the 0 →C rows are exact. To start, let F0→A and F ′′ be surjections from free abelian groups onto A and C. Then let F ′ 0 = F0 ⊕ F ′′ 0, with the obvious 0→B is deﬁned on the maps in the second row, inclusion and projection. The map F ′ summand F0 to make the lower left square commute, and on the summand F ′′ 0 it is deﬁned by sending basis elements of F ′′ 0 to elements of B mapping to the images of these basis elements in C, so the lower right square also commutes. Now we have the bottom two rows of the diagram, and we can regard these two rows as a short exact sequence of two-term chain complexes.
The associated long exact sequence of homology groups has six terms, the ﬁrst three being the kernels of the three vertical maps to A, B, and C, and the last three being the cokernels of these maps. Since the vertical maps to A and C are surjective, the fourth and sixth of the six homology groups vanish, hence also the ﬁfth, which says the vertical map to B is surjective. The ﬁrst three of the original six homology groups form a short exact sequence, and we let this be the top row of the diagram, formed by the kernels of the vertical maps to A, B, and C. These kernels are subgroups of free abelian groups, hence are also free. 318 Chapter 3 Cohomology Thus the three columns are free resolutions. The upper two squares automatically commute, so the construction of the diagram is complete. The ﬁrst two rows of the diagram split by freeness, so applying Hom(−, G) yields a diagram with exact rows. Again viewing this as a short exact sequence of chain complexes, the associated long exact sequence of homology groups is the second six-term exact sequence in the statement of the proposition. ⊔⊓ The second sequence in the proposition says in particular that an injection A→B induces a surjection Ext(B, C)→Ext(A, C) for any C. For example, if A has torsion, this says Ext(A, Z) is nonzero since it maps onto Ext(Zn, Z) ≈ Zn for some n > 1. The calculation Ext(Zp∞, Z) ≈ Zp earlier in this section shows that torsion in A does not necessarily yield torsion in Ext(A, Z), however. Two other useful formulas whose proofs we leave as exercises are: b Ext( i Ai, B) ≈ i Ext(Ai, B) Ext(A, i Bi) ≈ i Ext(A, Bi) L For example, since Q/Z = tion Ext(Zp∞, Z) ≈ Ext(Q, Z) ≈ ( Q Zp from the calculaZp. Then from the exact sequence 0→Z→Q→Q/Z→0 we get p Zp∞ we obtain Ext(Q/Z, Z) ≈ L L L p Q
b Zp)/Z using the second exact sequence in the proposition. In these examples the groups Ext(A, Z) are rather large, and the next result says p Q b b this is part of a general pattern: Proposition 3F.12. If A is not ﬁnitely generated then either Hom(A, Z) or Ext(A, Z) is uncountable. Hence if Hn(X; Z) is not ﬁnitely generated then either H n(X; Z) or H n+1(X; Z) is uncountable. Both possibilities can occur, as we see from the examples Hom( ∞Z, Z) ≈ ∞ Z and Ext(Zp∞, Z) ≈ Zp. L Q e b e that if a space X has H ∗(X; Z) = 0, then This proposition has some interesting topological consequences. First, it implies H∗(X; Z) = 0, since the case of ﬁnitely generated homology groups follows from our earlier results. And second, it says that one cannot always construct a space X with prescribed cohomology groups H n(X; Z), as one can for homology. For example there is no space whose only nonvanishing H n(X; Z) is a countable nonﬁnitely generated group such as Q or Q/Z. Even in the ﬁnitely generated case the dimension n = 1 is somewhat special since the group e H 1(X; Z) ≈ Hom(H1(X), Z) is always torsionfree. Proof: We begin with two consequences of Proposition 3F.11: (a) An inclusion B ֓ A induces a surjection Ext(A, Z)→Ext(B, Z). Hence Ext(A, Z) is uncountable if Ext(B, Z) is. Limits and Ext Section 3.F 319 (b) If A→A/B is a quotient map with B ﬁnitely generated, then the ﬁrst term in the exact sequence Hom(B, Z)→Ext(A/B, Z)→Ext(A, Z) is countable, so Ext(A, Z) is uncountable if Ext(A/B, Z) is. There are two explicit calculations that will be used in the
proof: (c) If A is a direct sum of inﬁnitely many nontrivial ﬁnite cyclic groups, then Ext(A, Z) is uncountable, the product of inﬁnitely many nontrivial groups Ext(Zn, Z) ≈ Zn. (d) For p prime, Example 3F.10 gives Ext(Zp∞, Z) ≈ Zp which is uncountable. Consider now the map A→A given by a ֏ pa for a ﬁxed prime p. Denote the kernel, image, and cokernel of this map by pA, pA, and Ap, respectively. The functor A ֏ Ap is the same as A ֏ A ⊗ Zp. We call the dimension of Ap as a vector space over Zp the p-rank of A. b Suppose the p -rank of A is inﬁnite. Then Ext(Ap, Z) is uncountable by (c). There is an exact sequence 0→pA→A→Ap→0, so Hom(pA, Z)→Ext(Ap, Z)→Ext(A, Z) is exact, hence either Hom(pA, Z) or Ext(A, Z) is uncountable. Also, we have an isomorphism Hom(pA, Z) ≈ Hom(A, Z) since the exact sequence 0→pA→A→pA→0 gives an exact sequence 0→Hom(pA, Z)→Hom(A, Z)→Hom(pA, Z) whose last term is 0 since pA is a torsion group. Thus we have shown that either Hom(A, Z) or Ext(A, Z) is uncountable if A has inﬁnite p -rank for some p. In the remainder of the proof we will show that Ext(A, Z) is uncountable if A has ﬁnite p -rank for all p and A is not ﬁnitely generated. Let C be a nontrivial cyclic subgroup of A, either ﬁnite or inﬁnite. If there is no maximal cyclic subgroup of A containing C then there is an inﬁnite ascending chain of cyclic subgroups C
= C1 ⊂ C2 ⊂ ···. If the indices [Ci : Ci−1] involve inﬁnitely ∞ Zp for these many distinct prime factors p then A/C contains an inﬁnite sum p so Ext(A/C, Z) is uncountable by (a) and (c) and hence also Ext(A, Z) by (b). If only ﬁnitely many primes are factors of the indices [Ci : Ci−1] then A/C contains a subgroup Zp∞ so Ext(A/C, Z) and hence Ext(A, Z) is uncountable in this case as well by (a), (b), and (d). Thus we may assume that each nonzero element of A lies in a maximal cyclic subgroup. L If A has positive ﬁnite p -rank we can choose a cyclic subgroup mapping nontrivially to Ap and then a maximal cyclic subgroup C containing this one will also map nontrivially to Ap. The quotient A/C has smaller p -rank since C→A→A/C→0 exact implies Cp→Ap→(A/C)p→0 exact, as tensoring with Zp preserves exactness to this extent. By (b) and induction on p -rank this gives a reduction to the case Ap = 0, so A = pA. If A is torsionfree, the maximality of the cyclic subgroup C in the preceding paragraph implies that A/C is also torsionfree, so by induction on p -rank we reduce to the case that A is torsionfree and A = pA. But in this case A has no maximal cyclic subgroups so this case has already been covered. If A has torsion, its torsion subgroup T is the direct sum of the p -torsion subgroups T (p) for all primes p. Only ﬁnitely many of these T (p) ’s can be nonzero, otherwise A contains ﬁnite cyclic subgroups not contained in maximal cyclic subgroups. If some T (p) is not ﬁnitely generated then by (a) we can assume A = T (p). In this case the reduction from ﬁnite p -rank to
∞, Zp) ≈ Zp. 7. Show that for a short exact sequence of abelian groups 0→A→B→C→0, a Moore space M(C, n) can be realized as a quotient M(B, n)/M(A, n). Applying the long exact Zp for p prime. b sequence of cohomology for the pair M(B, n), M(A, n) with any coeﬃcient group G, deduce an exact sequence 0→Hom(C, G)→Hom(B, G)→Hom(A, G)→Ext(C, G)→Ext(B, G)→Ext(A, G)→0 8. Show that for a Moore space M(G, n) the Bockstein long exact sequence in cohomology associated to the short exact sequence of coeﬃcient groups 0→A→B→C→0 reduces to an exact sequence 0→Hom(G, A)→Hom(G, B)→Hom(G, C)→Ext(G, A)→Ext(G, B)→Ext(G, C)→0 9. For an abelian group A let p : A→A be multiplication by p, and let pA = Ker p, pA = Im p, and Ap = Coker p as in the proof of Proposition 3F.12. Show that the sixterm exact sequences involving Hom(−, Z) and Ext(−, Z) associated to the short exact sequences 0→pA→A→pA→0 and 0→pA→A→Ap→0 can be spliced together to yield the exact sequence across the top of the following diagram Transfer Homomorphisms Section 3.G 321 where the map labeled ‘ p ’ is multiplication by p. Use this to show: (a) Ext(A, Z) is divisible iﬀ A is torsionfree. (b) Ext(A, Z) is torsionfree if A is divisible, and the converse holds if Hom(A, Z) = 0. There is a simple construction called ‘transfer’ that provides very useful informa- tion about homology and cohomology of ﬁnite-sheeted covering spaces. After giving the deﬁnition and proving a few elementary

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.