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�s for 1 ≤ i ≤ n form an additive basis for H ∗(Fn(C∞); Z), hence this ring is the polynomial ring on the xi ’s. There is a corresponding result for Fn(Ck), that H ∗(Fn(Ck); Z) is free with basis the monomials xi1 n with ij ≤ k−j for each j. This is proved in exactly the same way, using induction on n and the ﬁber bundle CPk−n→Fn(Ck)→Fn−1(Ck). Thus the cohomology groups of Fn(Ck) are isomorphic to those of CPk−1 × ··· × CPk−n. 1 ··· xin After these preliminaries we can start the main argument, using the ﬁber bundle p-----→ Gn(C∞). The preceding calculations show that the Leray– Fn(Cn) -→ Fn(C∞) Hirsch theorem applies, so H ∗(Fn(C∞); Z) is a free module over H ∗(Gn(C∞); Z) with basis the monomials xi1 n with ij ≤ n − j for each j. In particular, since 1 is among the basis elements, the homomorphism p∗ is injective and its image is a direct summand of H ∗(Fn(C∞); Z). It remains to show that the image of p∗ is exactly the symmetric polynomials. 1 ··· xin To show that the image of p∗ is contained in the symmetric polynomials, consider a map π : Fn(C∞)→Fn(C∞) permuting the lines in each n ﬂag according to a given Cohomology of Fiber Bundles Section 4.D 437 permutation of the numbers 1, ···, n. The induced map π ∗ on H ∗(Fn(C∞); Z) ≈ Z[x1, ···, xn] is the corresponding permutation of the variables xi. Since permuting the lines in an n ﬂag has no eﬀect on the n plane they span, we have pπ = p, hence π ∗
p∗ = p∗, which says that polynomials in the image of p∗ are invariant under permutations of the variables. As remarked earlier, the symmetric polynomials in Z[x1, ···, xn] form a polynomial ring Z[σ1, ···, σn] where σi has degree i. We have shown that the image of p∗ is a direct summand, so to show that p∗ maps onto the symmetric polynomials it will suﬃce to show that the graded rings H ∗(Gn(C∞); Z) and Z[σ1, ···, σn] have the same rank in each dimension, where the rank of a ﬁnitely generated free abelian group is the number of Z summands. For a graded free Z module A = i Ai, deﬁne its Poincar´e series to be the formal i aiti where ai is the rank of Ai, which we assume to be ﬁnite power series pA(t) = for all i. The basic formula we need is that pA⊗B(t) = pA(t) pB(t), which is immediate from the deﬁnition of the graded tensor product. L P In the case at hand all nonzero cohomology is in even dimensions, so let us simplify notation by taking Ai to be the 2i dimensional cohomology of the space in i ti = (1 − t)−1, the Poincar´e series of question. Since the Poincar´e series of Z[x] is H ∗(Fn(C∞); Z) is (1 − t)−n. For H ∗(Fn(Cn); Z) the Poincar´e series is P (1 + t)(1 + t + t2) ··· (1 + t + ··· + tn−1) = 1 − ti 1 − t n Yi=1 n = (1 − t)−n (1 − ti) Yi=1 From the additive isomorphism H ∗(Fn(C∞); Z) ≈ H ∗(Gn(C∞); Z) ⊗ H ∗(Fn(Cn);
Z) we see that the Poincar´e series p(t) of H ∗(Gn(C∞); Z) satisﬁes p(t)(1 − t)−n (1 − ti) = (1 − t)−n and hence p(t) = n Yi=1 (1 − ti)−1 n Yi=1 This is exactly the Poincar´e series of Z[σ1, ···, σn] since σi has degree i. As noted before, this implies that the image of p∗ is all the symmetric polynomials. This ﬁnishes the proof for Gn(C∞). The same arguments apply in the other two cases, using Z2 coeﬃcients throughout in the real case and replacing ‘rank’ by ‘dimension’ for Z2 vector spaces. ⊔⊓ These calculations show that the isomorphism H ∗(E; R) ≈ H ∗(B; R) ⊗R H ∗(F ; R) of the Leray–Hirsch theorem is not generally a ring isomorphism, for if it were, then the polynomial ring H ∗(Fn(C∞); Z) would contain a copy of H ∗(Fn(Cn); Z) as a subring, but in the latter ring some power of every positive-dimensional element is zero since H k(Fn(Cn); Z) = 0 for suﬃciently large k. The Gysin Sequence Besides the Leray–Hirsch theorem, which deals with ﬁber bundles that are coho- mologically like products, there is another special class of ﬁber bundles for which an 438 Chapter 4 Homotopy Theory elementary analysis of their cohomology structure is possible. These are ﬁber bunp-----→ B satisfying an orientability hypothesis that will always hold if B dles S n−1 -→ E is simply-connected or if we take cohomology with Z2 coeﬃcients. For such bundles we will show there is an exact sequence, called the Gysin sequence, ··· -→ H i−n(B; R) `e -----------------→ H i(B; R) p∗------------→ H i(E;
R) -→ H i−n+1(B; R) -→ ··· where e is a certain ‘Euler class’ in H n(B; R). Since H i(B; R) = 0 for i < 0, the ≈-----→ H i(E; R) initial portion of the Gysin sequence gives isomorphisms p∗ : H i(B; R) for i < n − 1, and the more interesting part of the sequence begins p∗------------→ H n−1(E; R) -→ H 0(B; R) p∗------------→ H n(E; R) -→ ··· `e --------------→ H n(B; R) 0 -→ H n−1(B; R) In the case of a product bundle E = S n−1 × B there is a section, a map s : B→E with ps = 11, so the Gysin sequence breaks up into split short exact sequences 0 -→ H i(B; R) p∗------------→ H i(S n−1 × B; R) -→ H i−n+1(B; R) -→ 0 which agrees with the K¨unneth formula H ∗(S n−1 × B; R) ≈ H ∗(S n−1; R) ⊗R H ∗(B; R). The splitting holds whenever the bundle has a section, even if it is not a product. For example, consider the bundle S n−1→V2(Rn+1) p-----→ S n. Points of V2(Rn+1) are pairs (v1, v2) of orthogonal unit vectors in Rn+1, and p(v1, v2) = v1. If we think of v1 as a point of S n and v2 as a unit vector tangent to S n at v1, then V2(Rn+1) is exactly the bundle of unit tangent vectors to S n. A section of this bundle is a ﬁeld of unit tangent vectors to S n, and such a vector ﬁeld exists iﬀ n is odd by Theorem 2.28. The fact that the Gysin sequence splits when there is a section then says that V2(Rn+
1) has the same cohomology as the product S n−1 × S n if n is odd, at least when n > 1 so that the base space S n is simply-connected and the orientability hypothesis is satisﬁed. When n is even, the calculations at the end of §3.D show that H ∗(V2(Rn+1); Z) consists of Z ’s in dimensions 0 and 2n − 1 and a Z2 in dimension n. The latter group appears in the Gysin sequence as hence the Euler class e must be twice a generator of H n(S n) in the case that n is even. When n is odd it must be zero in order for the Gysin sequence to split. This example illustrates a theorem in diﬀerential topology that explains why the Euler class has this name: The Euler class of the unit tangent bundle of a closed orientable smooth n manifold M is equal to the Euler characteristic χ (M) times a generator of H n(M; Z). Whenever a bundle S n−1→E p-----→ B has a section, the Euler class e must be zero p∗---------→ H n(E) since p∗ is injective if there is a from exactness of H 0(B) section. Thus the Euler class can be viewed as an obstruction to the existence of `e------------→ H n(B) a section: If the Euler class is nonzero, there can be no section. This qualitative Cohomology of Fiber Bundles Section 4.D 439 statement can be made more precise by bringing in the machinery of obstruction theory, as explained in [Milnor & Stasheﬀ 1974] or [VBKT]. Before deriving the Gysin sequence let us look at some examples of how it can be used to compute cup products. p-----→ B with E contractible, for examExample 4D.5. Consider a bundle S n−1 -→ E ple the bundle S 1→S ∞→CP∞ or its real or quaternionic analogs. The long exact sequence of homotopy groups for the bundle shows that B is (n − 1) connected. Thus if n > 1, B is simply-connected and we have a Gysin sequence for cohomology with Z coeﬃcients. For n =
1 we take Z2 coeﬃcients. If n > 1 then since E is contractible, the Gysin sequence implies that H i(B; Z) = 0 for 0 < i < n and that `e : H i(B; Z)→H i+n(B; Z) is an isomorphism for i ≥ 0. It follows that H ∗(B; Z) is the polynomial ring Z[e]. When n = 1 the map p∗ : H n−1(B; Z2)→H n−1(E; Z2) in the Gysin sequence is surjective, so we see that `e : H i(B; Z2)→H i+n(B; Z2) is again an isomorphism for all i ≥ 0, and hence H ∗(B; Z2) ≈ Z2[e]. Thus the Gysin sequence gives a new derivation of the cup product structure in projective spaces. Also, since polynomial rings Z[e] are realizable as H ∗(X; Z) only when e has dimension 2 or 4, as we show in Corollary 4L.10, we can conclude that there exist bundles S n−1→E→B with E contractible only when n is 1, 2, or 4. e Example 4D.6. For the Grassmann manifold Gn = Gn(R∞) we have π1(Gn) ≈ Gn→Gn. One can π0O(n) ≈ Z2, so the universal cover of Gn gives a bundle S 0→ Gn as the space of oriented n planes in R∞, which is obviously a 2 sheeted view covering space of Gn, hence the universal cover since it is path-connected, being the quotient Vn(R∞)/SO(n) of the contractible space Vn(R∞). A portion of the Gysin Gn→Gn is H 0(Gn; Z2) sequence for the bundle S 0→ Gn; Z2). Gn is simply-connected, and H 1(Gn; Z2) ≈ Z2 since This last group is zero since e H ∗(Gn; Z2) ≈ Z2[w1, ···, wn] as we showed earlier in
this section, so e = w1 and the map `e : H ∗(Gn; Z2)→H ∗(Gn; Z2) is injective. The Gysin sequence then breaks `e------------→ H i+1(Gn; Z2)→H i+1( up into short exact sequences 0→H i(Gn; Z2) Gn; Z2)→0, from which it follows that H ∗( Gn; Z2) is the quotient ring Z2[w1, ···, wn]/(w1) ≈ Z2[w2, ···, wn]. `e------------→ H 1(Gn; Z2) -→ H 1( e e e e e Gn(C∞)→Gn(C∞) with Example 4D.7. The complex analog of the bundle in the preceding example is a bundle S 1→ Gn(C∞) 2 connected. This can be constructed in the following way. There is a determinant homomorphism U(n)→S 1 with kernel SU(n), the unitary matrices of determinant 1, so S 1 is the coset space U(n)/SU(n), and by restricting the action of U(n) on Vn(C∞) to SU(n) we obtain the second row of the e e commutative diagram at the right. The second row is a ﬁber bundle by the usual argument of choosing continuously varying orthonormal bases in n planes near a 440 Chapter 4 Homotopy Theory given n plane. One sees that the space Gn(C∞) = Vn(C∞)/SU(n) is 2 connected by looking at the relevant portion of the diagram of homotopy groups associated to e these two bundles: The Gysin sequence for S 1→ boundary map in the lower row, and then exactness implies that The second vertical map is an isomorphism since S 1 embeds in U(n) as the subgroup U(1). Since the boundary map in the upper row is an isomorphism, so also is the Gn is 2 connected. Gn(C∞)→Gn(C∞) can be analyzed just as in the preceding example. Part of the sequence is H 0(Gn; Z) Gn; Z), e Gn is
2 connected, so e must be a generator of and this last group is zero since H 2(Gn; Z) ≈ Z. Since H ∗(Gn; Z) is a polynomial algebra Z[c1, ···, cn], we must have e = ±c1, so the map `e : H ∗(Gn; Z)→H ∗(Gn; Z) is injective, the Gysin sequence breaks up into short exact sequences, and H ∗( Gn; Z) is the quotient ring Z[c1, ···, cn]/(c1) ≈ Z[c2, ···, cn]. e `e------------→ H 2(Gn; Z) -→ H 2( e e e The spaces Gn in the last two examples are often denoted BSO(n) and BSU(n), expressing the fact that they are related to the groups SO(n) and SU(n) via bundles SO(n)→Vn(R∞)→BSO(n) and SU(n)→Vn(C∞)→BSU(n) with contractible total spaces Vn. There is no quaternion analog of BSO(n) and BSU(n) since for n = 2 this would give a space with cohomology ring Z[x] on an 8 dimensional generator, e which is impossible by Corollary 4L.10. Now we turn to the derivation of the Gysin sequence, which follows a rather roundabout route: (1) Deduce a relative version of the Leray–Hirsch theorem from the absolute case. (2) Specialize this to the case of bundles with ﬁber a disk, yielding a basic result called the Thom isomorphism. (3) Show this applies to all orientable disk bundles. (4) Deduce the Gysin sequence by plugging the Thom isomorphism into the long exact sequence of cohomology groups for the pair consisting of a disk bundle and its boundary sphere bundle. (1) A ﬁber bundle pair consists of a ﬁber bundle p : E→B with ﬁber F, together with a subspace E′ ⊂ E such that p : E′→B is a bundle with ﬁber a subspace F ′ �
� F, with local trivializations for E′ obtained by restricting local trivializations for E. For example, if E→B is a bundle with ﬁber Dn and E′ ⊂ E is the union of the boundary spheres of the ﬁbers, then (E, E′) is a ﬁber bundle pair since local trivializations of E restrict to local trivializations of E′, in view of the fact that homeomorphisms from an n disk to an n disk restrict to homeomorphisms between their boundary spheres, boundary and interior points of Dn being distinguished by the local homology groups Hn(Dn, Dn − {x}; Z). Cohomology of Fiber Bundles Section 4.D 441 p-----→ B is a ﬁber bundle pair such that Theorem 4D.8. Suppose that (F, F ′)→(E, E′) H ∗(F, F ′; R) is a free R module, ﬁnitely generated in each dimension. If there exist classes cj ∈ H ∗(E, E′; R) whose restrictions form a basis for H ∗(F, F ′; R) in each ﬁber (F, F ′), then H ∗(E, E′; R), as a module over H ∗(B; R), is free with basis {cj}. The module structure is deﬁned just as in the absolute case by bc = p∗(b) ` c, but now we use the relative cup product H ∗(E; R)× H ∗(E, E′; R)→H ∗(E, E′; R). b b E, M; R) ≈ H ∗( b E, M; R) ≈ H ∗( E→B from E by attaching the mapping cylinder M of Proof: Construct a bundle p : E′→B to E by identifying the subspaces E′ ⊂ E and E′ ⊂ M. Thus the ﬁbers E are obtained from the ﬁbers F by attaching cones CF ′ on the subspaces F of F ′ ⊂ F. Regarding B as the subspace of E at one end of the mapping cylinder M, we b E − B, M − B
; R) ≈ H ∗(E, E′; R) via excision and the obvious have H ∗( E − B onto E. The long exact sequence of a triple gives deformation retraction of b H ∗( b are H ∗(B; R) module isomorphisms. Since B is a retract of E→B, we have a splitting H ∗( Let b The classes b F ∪ CF ′, so the absolute form of the Leray–Hirsch theorem implies that H ∗( a free H ∗(B; R) module with basis {1, H ∗(B; R) module H ∗(E, E′; R). E; R) ≈ H ∗( b E; R) correspond to cj ∈ H ∗(E, E′; R) ≈ H ∗( cj together with 1 restrict to a basis for H ∗( F ; R) in each ﬁber b b E; R) is b b cj}. It follows that {cj} is a basis for the free ⊔⊓ E via the bundle projection H ∗(B; R) as H ∗(B; R) modules. E, B; R) in this splitting. F = E, B; R) since M deformation retracts to B. All these isomorphisms cj ∈ H ∗( E, B; R-----→ B. (2) Now we specialize to the case of a ﬁber bundle pair (Dn, S n−1) -→ (E, E′) An element c ∈ H n(E, E′; R) whose restriction to each ﬁber (Dn, S n−1) is a generator of H n(Dn, S n−1; R) ≈ R is called a Thom class for the bundle. We are mainly interested in the cases R = Z and Z2, but R could be any commutative ring with identity, in which case a ‘generator’ is an element with a multiplicative inverse, so all elements of R are multiples of the generator. A Thom class with Z coeﬃcients gives rise to a Thom class with any other coeﬃcient ring R under the homomorphism H n(E, E′
; Z)→H n(E, E′; R) induced by the homomorphism Z→R sending 1 to the identity element of R. Corollary 4D.9. If the disk bundle (Dn, S n−1) -→ (E, E′) : H i(B; R)→H i+n(E, E′; R), c ∈ H n(E, E′; R), then the map an isomorphism for all i ≥ 0, and H i(E, E′; R) = 0 for i < n. p-----→ B has a Thom class (b) = p∗(b) ` c, is ⊔⊓ Φ Φ The isomorphism is called the Thom isomorphism. The corollary can be made into a statement about absolute cohomology by deﬁning the Thom space T (E) to be the quotient E/E′. Each disk ﬁber Dn of E becomes a sphere S n in T (E), and all these spheres coming from diﬀerent ﬁbers are disjoint except for the common basepoint x0 = E′/E′. A Thom class can be regarded as an element of H n(T (E), x0; R) ≈ Φ 442 Chapter 4 Homotopy Theory H n(T (E); R) that restricts to a generator of H n(S n; R) in each ‘ﬁber’ S n in T (E), and the Thom isomorphism becomes H i(B; R) ≈ H n+i(T (E); R). e (3) The major remaining step in the derivation of the Gysin sequence is to relate the existence of a Thom class for a disk bundle Dn→E→B to a notion of orientability of the bundle. First we deﬁne orientability for a sphere bundle S n−1→E′→B. In the proof of Proposition 4.61 we described a procedure for lifting paths γ in B to homotopy equivalences Lγ between the ﬁbers above the endpoints of γ. We did this for ﬁbrations rather than ﬁber bundles, but the method applies equally well to ﬁber bundles whose
ﬁber is a CW complex since the homotopy lifting property was used only for the ﬁber and for the product of the ﬁber with I. In the case of a sphere bundle S n−1→E′→B, if γ is a loop in B then Lγ is a homotopy equivalence from the ﬁber S n−1 over the basepoint of γ to itself, and we deﬁne the sphere bundle to be orientable if Lγ induces the identity map on H n−1(S n−1; Z) for each loop γ in B. For example, the Klein bottle, regarded as a bundle over S 1 with ﬁber S 1, is nonorientable since as we follow a path looping once around the base circle, the cor- responding ﬁber circles sweep out the full Klein bottle, ending up where they started but with orientation reversed. The same reasoning shows that the torus, viewed as a circle bundle over S 1, is orientable. More generally, any sphere bundle that is a product is orientable since the maps Lγ can be taken to be the identity for all loops γ. Also, sphere bundles over simply-connected base spaces are orientable since γ ≃ η implies Lγ ≃ Lη, hence all Lγ ’s are homotopic to the identity when all loops γ are nullhomotopic. One could deﬁne orientability for a disk bundle Dn→E→B by relativizing the previous deﬁnition, constructing lifts Lγ which are homotopy equivalences of the ﬁber pairs (Dn, S n−1). However, since H n(Dn, S n−1; Z) is canonically isomorphic to H n−1(S n−1; Z) via the coboundary map in the long exact sequence of the pair, it is simpler and amounts to the same thing just to deﬁne E to be orientable if its boundary sphere subbundle E′ is orientable. Theorem 4D.10. Every disk bundle has a Thom class with Z2 coeﬃcients, and orientable disk bundles have Thom classes with Z coeﬃcients. An exercise at the end of the section is to show that the converse of the last statement
is also true: A disk bundle is orientable if it has a Thom class with Z coeﬃcients. Proof: The case of a non-CW base space B reduces to the CW case by pulling back over a CW approximation to B, as in the Leray–Hirsch theorem, applying the ﬁve-lemma to say that the pullback bundle has isomorphic homotopy groups, hence isomorphic absolute and relative cohomology groups. From the deﬁnition of the pullback bundle it is immediate that the pullback of an orientable sphere bundle is orientable. There is also no harm in assuming the base CW complex B is connected. We will show: Cohomology of Fiber Bundles Section 4.D 443 (∗) If the disk bundle Dn→E→B is orientable and B is a connected CW complex, then the restriction map H i(E, E′; Z)→H i(Dn ; Z) is an isomorphism for all ﬁbers Dn x, x ∈ B, and for all i ≤ n. x, S n−1 x For Z2 coeﬃcients we will see that (∗) holds without any orientability hypothesis. Hence with either Z or Z2 coeﬃcients, a generator of H n(E, E′) ≈ H n(Dn ) is a Thom class. x, S n−1 x x ; Z) ≈ Z for one ﬁber Dn If the disk bundle Dn→E→B is orientable, then if we choose an isomorphism x, S n−1 H n(Dn x, this determines such isomorphisms for all ﬁbers by composing with the isomorphisms L∗ γ, which depend only on the endpoints of γ. Having made such a choice, then if (∗) is true, we have a preferred isomorphism H n(E, E′; Z) ≈ Z which restricts to the chosen isomorphism H n(Dn x, S n−1 ; Z) ≈ Z for )֓(E, E′) x, S n−1 each ﬁber. This is because for a path γ from x to y, the inclusion (Dn ) ֓ (E, E′). We y, S n
−1 is homotopic to the composition of Lγ with the inclusion (Dn will use this preferred isomorphism H n(E, E′; Z) ≈ Z in the inductive proof of (∗) In the case of Z2 coeﬃcients, there can be only one isomorphism of given below. a group with Z2 so no choices are necessary and orientability is irrelevant. We will prove (∗) in the Z coeﬃcient case, leaving it to the reader to replace all Z ’s in the proof by Z2 ’s to obtain a proof in the Z2 case. y x x Suppose ﬁrst that the CW complex B has ﬁnite dimension k. Let U ⊂ B be the subspace obtained by deleting one point from the interior of each k cell of B, and let V ⊂ B be the union of the open k cells. Thus B = U ∪ V. For a subspace A ⊂ B let EA→A and E′ A→A be the disk and sphere bundles obtained by taking the subspaces of E and E′ projecting to A. Consider the following portion of a Mayer–Vietoris sequence, with Z coeﬃcients implicit from now on: H n(E, E′) -→ H n(EU, E′ U ) H n(EV, E′ V ) L ------------→ H n(EU ∩V, E′ Ψ U ∩V ) The ﬁrst map is injective since the preceding term in the sequence is zero by induction on k, since U ∩ V deformation retracts onto a disjoint union of (k − 1) spheres and we can apply Lemma 4D.2 to replace EU ∩V by the part of E over this union of (k − 1) spheres. By exactness we then have an isomorphism H n(E, E′) ≈ Ker. Similarly, by Lemma 4D.2 and induction each of the terms H n(EU, E′ U ), H n(EV, E′ V ), Ψ and H n(EU ∩V, E′ U ∩V ) is a product of Z ’s, with one Z factor for each component of the spaces involved, projection onto the Z factor being given by restriction to any
�)−1j∗(c), c being a Thom class. The square containing the map `e commutes since for b ∈ H i−n(B; R) we have j∗ (b) = j∗(p∗(b) ` c) = p∗(b) ` j∗(c), which equals p∗(b ` e) = p∗(b) ` p∗(e) since p∗(e) = j∗(c). Another way of deﬁning e Φ is as the class corresponding to c ` c under the Thom isomorphism, since (e) = p∗(e) ` c = j∗(c) ` c = c ` c. Finally, the lower row of the diagram is by deﬁnition the Gysin sequence. Φ ⊔⊓ To conclude this section we will use the following rather specialized application of the Gysin sequence to compute a few more examples of spaces with polynomial cohomology. p-----→ B is an orientable sphere bundle Proposition 4D.11. Suppose that S 2k−1 -→ E such that H ∗(E; R) is a polynomial ring R[x1, ···, xℓ] on even-dimensional generators xi. Then H ∗(B; R) = R[y1, ···, yℓ, e] where e is the Euler class of the bundle and p∗(yi) = xi for each i. `e------------→ H i+2k(B; R) -----→ H i+2k(E; R) of the Proof: Consider the three terms H i(B; R) Gysin sequence. If i is odd, the third term is zero since E has no odd-dimensional cohomology. Hence the map `e is surjective, and by induction on dimension this implies that H ∗(B; R) is zero in odd dimensions. This means the Gysin sequence reduces to short exact sequences 0 -→ H 2i(B; R) `e -----------------→ H 2i+2k(B; R) p∗------------→ H 2i+2k(E; R) -→ 0 Cohomology of Fiber Bundles Section
4.D 445 Since p∗ is surjective, we can choose elements yj ∈ H ∗(B; R) with p∗(yj ) = xj. It remains to check that H ∗(B; R) = R[y1, ···, yℓ, e], which is elementary algebra: Given b ∈ H ∗(B; R), p∗(b) must be a polynomial f (x1, ···, xℓ), so b − f (y1, ···, yℓ) is in the kernel of p∗ and exactness gives an equation b − f (y1, ···, yℓ) = b′ ` e for some b′ ∈ H ∗(B; R). Since b′ has lower dimension than b, we may assume by induction that b′ is a polynomial in y1, ···, yℓ, e. Hence b = f (y1, ···, yℓ) + b′ ` e is also a polynomial in y1, ···, yℓ, e. Thus the natural map R[y1, ···, yℓ, e]→H ∗(B; R) is surjective. To see that it is injective, suppose there is a polynomial relation f (y1, ···, yℓ, e) = 0 in H ∗(B; R). Applying p∗, we get f (x1, ···, xℓ, 0) = 0 since p∗(yi) = xi and p∗(e) = 0 from the short exact sequence. The relation f (x1, ···, xℓ, 0) = 0 takes place in the polynomial ring R[x1, ···, xℓ], so f (y1, ···, yℓ, 0) = 0 in R[y1, ···, yℓ, e], hence f (y1, ···, yℓ, e) must be divisible by e, say f = ge for some polynomial g. The relation f (y1, ···, yℓ, e) =
0 in H ∗(B; R) then has the form g(y1, ···, yℓ, e) ` e = 0. Since `e is injective, this gives a polynomial relation g(y1, ···, yℓ, e) = 0 with g having lower degree than f. By induction we deduce that g must be the zero polynomial, hence also f. ⊔⊓ Example 4D.12. Let us apply this to give another proof that H ∗(Gn(C∞); Z) is a polynomial ring Z[c1, ···, cn] with \|ci\| = 2i. We use two ﬁber bundles: S 2n−1 -→ E -→ Gn(C∞) S ∞ -→ E -→ Gn−1(C∞) The total space E in both cases is the space of pairs (P, v) where P is an n plane in C∞ and v is a unit vector in P. In the ﬁrst bundle the map E→Gn(C∞) is (P, v)֏P, with ﬁber S 2n−1, and for the second bundle the map E→Gn−1(C∞) sends (P, v) to the (n − 1) plane in P orthogonal to v, with ﬁber S ∞ consisting of all the unit vectors in C∞ orthogonal to a given (n − 1) plane. Local triviality for the two bundles is veriﬁed in the usual way. Since S ∞ is contractible, the map E→Gn−1(C∞) induces isomorphisms on homotopy groups, hence also on cohomology. By induction on n we then have H ∗(E; Z) ≈ Z[c1, ···, cn−1]. The ﬁrst bundle is orientable since Gn(C∞) is simply-connected, so the proposition gives H ∗(Gn(C∞); Z) ≈ Z[c1, ···, cn] for cn = e. The same argument works in the quaternionic case. For a version of this argument in the real case see §3.3
of [VBKT]. Before giving our next example, let us observe that the Gysin sequence with a p¬-----→ B whose ﬁxed coeﬃcient ring R is valid for any orientable ﬁber bundle F -→ E ﬁber is a CW complex F with H ∗(F ; R) ≈ H ∗(S n−1; R). Orientability is deﬁned just γ : H n−1(F ; R)→H n−1(F ; R). No changes are as before in terms of induced maps L∗ needed in the derivation of the Gysin sequence to get this more general case, if the associated ‘disk’ bundle is again taken to be the mapping cylinder CF→Mp→B. Gn(R∞) with Z2 coeﬃcients, Example 4D.13. We have computed the cohomology of ﬁnding it to be a polynomial ring on generators in dimensions 2 through n, and now e 446 Chapter 4 Homotopy Theory we compute the cohomology with Zp coeﬃcients for p an odd prime. The answer will again be a polynomial algebra, but this time on even-dimensional generators, depending on the parity of n. Consider ﬁrst the case that n is odd, say n = 2k + 1. There are two ﬁber bundles V2(R2k+1) -→ E -→ G2k+1(R∞) V2(R∞) -→ E -→ G2k−1(R∞) e where E is the space of triples (P, v1, v2) with P an oriented (2k + 1) plane in R∞ and v1 and v2 two orthogonal unit vectors in P. The projection map in the ﬁrst bundle is (P, v1, v2) ֏ P, and for the second bundle the projection sends (P, v1, v2) to the oriented (2k − 1) plane in P orthogonal to v1 and v2, with the orientation speciﬁed by saying for example that v1, v2 followed by a positively oriented basis for the orthogonal (2k − 1) plane is
a positively oriented basis for P. Both bundles Gn(R∞) are simply-connected, from the bundle e are orientable since their base spaces SO(n)→Vn(R∞)→ Gn(R∞). e e The ﬁber V2(R∞) of the second bundle is contractible, so E has the same cohoG2k−1(R∞). The ﬁber of the ﬁrst bundle has the same Zp cohomology as mology as S 4k−1 if p is odd, by the calculation at the end of §3.D. So if we assume inductively that H ∗( G2k−1(R∞); Zp) ≈ Zp[p1, ···, pk−1] with \|pi\| = 4i, then Proposition 4D.11 G2k+1(R∞); Zp) ≈ Zp[p1, ···, pk] where pk = e has dimension above implies that H ∗( G1(R∞) which is just S ∞ since an oriented line in 4k. The induction can start with R∞ contains a unique unit vector in the positive direction. e e e To handle the case of Gn(R∞) with n = 2k even, we proceed just as in Exam- e ple 4D.12, considering the bundles e S 2k−1 -→ E -→ G2k(R∞) S ∞ -→ E -→ G2k−1(R∞) By the case n odd we have H ∗( G2k−1(R∞); Zp) ≈ Zp[p1, ···, pk−1] with \|pi\| = 4i, so e e G2k(R∞); Zp) is a polynomial ring on these generators the corollary implies that H ∗( and also a generator in dimension 2k. e Summarizing, for p an odd prime we have shown: e H ∗( H ∗( G2k+1(R∞); Zp) ≈ Zp[p1, ···, pk], G2k(R∞); Zp) ≈ Zp[p1, ···, pk
−1, e], e \|pi\| = 4i \|pi\| = 4i, \|e\| = 2k The same result holds also with Q coeﬃcients. e In fact, our proof applies for any coeﬃcient ring in which 2 has a multiplicative inverse, since all that is needed is that Gn(R∞) H ∗(V2(R2k+1); R) ≈ H ∗(S 4k−1; R). For a calculation of the cohomology of with Z coeﬃcients, see [VBKT]. It turns out that all torsion elements have order 2, and modulo this torsion the integral cohomology is again a polynomial ring on the generators pi and e. Similar results hold also for the cohomology of the unoriented Grassmann manifold Gn(R∞), but with the generator e replaced by pk when n = 2k. e Cohomology of Fiber Bundles Section 4.D 447 Exercises 1. By Exercise 35 in §4.2 there is a bundle S 2→CP3→S 4. Let S 2→Ek→S 4 be the pullback of this bundle via a degree k map S 4→S 4, k > 1. Use the Leray-Hirsch theorem to show that H ∗(Ek; Z) is additively isomorphic to H ∗(CP3; Z) but has a diﬀerent cup product structure in which the square of a generator of H 2(Ek; Z) is k times a generator of H 4(Ek; Z). 2. Apply the Leray–Hirsch theorem to the bundle S 1→S ∞/Zp→CP∞ to compute H ∗(K(Zp, 1); Zp) from H ∗(CP∞; Zp). 3. Use the Leray–Hirsch theorem as in Corollary 4D.3 to compute H ∗(Vn(Ck); Z) ≈ Z[x2k−2n+1, x2k−2n+3, ···, x2k−1] and similarly in the quaternionic case. 4. For the ﬂag space Fn(Cn) show that H ∗(Fn(Cn); Z)
≈ Z[x1, ···, xn]/(σ1, ···, σn) Λ where σi is the ith elementary symmetric polynomial. 5. Use the Gysin sequence to show that for a ﬁber bundle S k→Sm p-----→ S n we must have k = n − 1 and m = 2n − 1. Then use the Thom isomorphism to show that the Hopf invariant of p must be ±1. [Hence n = 1, 2, 4, 8 by Adams’ theorem.] 6. Show that if M is a manifold of dimension 2n for which there exists a ﬁber bundle S 1→S 2n+1→M, then M is simply-connected and H ∗(M; Z) ≈ H ∗(CPn; Z) as rings. Conversely, if M is simply-connected and H ∗(M; Z) ≈ H ∗(CPn; Z) as rings, show there is a bundle S 1→E→M where E ≃ S 2n+1. [When n > 1 there are examples where M is not homeomorphic to CPn.] 7. Show that if a disk bundle Dn→E→B has a Thom class with Z coeﬃcients, then it is orientable. H i(B; R) ≈ 8. If E is the product bundle B × Dn with B a CW complex, show that the Thom space n(B+), where B+ is the union of B with T (E) is the n fold reduced suspension a disjoint basepoint, and that the Thom isomorphism specializes to the suspension Σ nB; R) given by the reduced cross product in §3.2. isomorphism 9. Show that the inclusion T n ֓ U(n) of the n torus of diagonal matrices is homotopic to the map T n→U(1) ֓ U(n) sending an n tuple of unit complex numbers (z1, ···, zn) to the 1× 1 matrix (z1 ··· zn). Do the same for the diagonal subgroup of Sp(n). [Hint: Diagonal matrices in U(n) are compositions of scalar multiplication in n lines in Cn, and CPn−1
is connected.] H n+i( Σ e e 10. Fill in the details of the following argument to show that every n× n matrix (The usual argument over C involvA with entries in H has an eigenvalue in H. ing roots of the characteristic polynomial does not work due to the lack of a good quaternionic determinant function.) For t ∈ [0, 1] and λ ∈ S 3 ⊂ H, consider the matrix tλI + (1 − t)A. If A has no eigenvalues, this is invertible for all t. Thus the map S 3→GLn(H), λ ֏ λ I, is nullhomotopic. But by the preceding problem and Exercise 10(b) in §3.C, this map represents n times a generator of π3GLn(H). 448 Chapter 4 Homotopy Theory In Theorem 4.58 in §4.3 we showed that spectra deﬁne cohomology theories, and now we will prove the converse statement that all cohomology theories on the CW category arise in this way from spectra. Ω Theorem 4E.1. Every reduced cohomology theory on the category of basepointed CW complexes and basepoint-preserving maps has the form hn(X) = hX, Kni for some spectrum {Kn}. Ω Ω We will also see that the spaces Kn are unique up to homotopy equivalence. This theorem gives another proof that ordinary cohomology is representable as maps into Eilenberg–MacLane spaces, since for the spaces Kn in an resenting H ∗(−; G) we have πi(Kn) = hS i, Kni = H n(S i; G), so Kn is a K(G, n). spectrum rep- Before getting into the proof of the theorem let us observe that coﬁbration se- Ω e e quences, as constructed in §4.3, allow us to recast the deﬁnition of a reduced coho- mology theory in a slightly more concise form: A reduced cohomology theory on the category C whose objects are CW complexes with a chosen basepoint 0 cell and whose morphisms are basepoint-preserving maps is a sequence of
functors hn, n ∈ Z, from C to abelian groups, together with natural isomorphisms hn(X) ≈ hn+1( X) for all X in C, such that the following axioms hold for each hn : (i) If f ≃ g : X→Y in the basepointed sense, then f ∗ = g∗ : hn(Y )→hn(X). (ii) For each inclusion A֓X in C the sequence hn(X/A)→hn(X)→hn(A) is exact. α Xα with inclusions iα : Xα ֓ X, the product map (iii) For a wedge sum X = α : hn(X)→ W α hn(Xα) is an isomorphism. α i∗ Σ Q Q To see that these axioms suﬃce to deﬁne a cohomology theory, the main thing to A→ ··· allows us to construct note is that the coﬁbration sequence A→X→X/A→ the long exact sequence of a pair, just as we did in the case of the functors hn(X) = hX, Kni. In the converse direction, if we have natural long exact sequences of pairs, then by applying these to pairs of the form (CX, X) we get natural isomorphisms hn(X) ≈ hn+1( maps of pairs (CX, X) uniquely determine the coboundary maps for all pairs (X, A) X). Note that these natural isomorphisms coming from coboundary Σ Σ via the diagram at the right, where the maps from hn(A) are coboundary maps of pairs and the diagram commutes by naturality of these coboundary maps. The isomorphism comes from a deforma- tion retraction of CX onto CA. It is easy to check that these processes for converting one deﬁnition of a cohomology theory into the other are inverses of each other. Most of the work in representing cohomology theories by spectra will be in realizing a single functor hn of a cohomology theory as h−, Kni for some space Kn. Ω The Brown Representability Theorem Section 4.
E 449 So let us consider what properties the functor h(X) = hX, Ki has, where K is a ﬁxed space with basepoint. First of all, it is a contravariant functor from the category of basepointed CW complexes to the category of pointed sets, that is, sets with a distinguished element, the homotopy class of the constant map in the present case. Morphisms in the category of pointed sets are maps preserving the distinguished element. We have already seen in §4.3 that h(X) satisﬁes the three axioms (i)–(iii). A further property is the following Mayer–Vietoris axiom: Suppose the CW complex X is the union of subcomplexes A and B containing the basepoint. Then if a ∈ h(A) and b ∈ h(B) restrict to the same element of h(A ∩ B), there exists an element x ∈ h(X) whose restrictions to A and B are the given elements a and b. Here and in what follows we use the term ‘restriction’ to mean the map induced by inclusion. In the case that h(X) = hX, Ki, this axiom is an immediate consequence of the homotopy extension property. The functors hn in any cohomology theory also satisfy this axiom since there are Mayer–Vietoris exact sequences in any cohomology theory, as we observed in §2.3 in the analogous setting of homology theories. Theorem 4E.2. If h is a contravariant functor from the category of connected basepointed CW complexes to the category of pointed sets, satisfying the homotopy ax- iom (i), the Mayer–Vietoris axiom, and the wedge axiom (iii), then there exists a connected CW complex K and an element u ∈ h(K) such that the transformation Tu : hX, Ki→h(X), Tu(f ) = f ∗(u), is a bijection for all X. Such a pair (K, u) is called universal for the functor h. It is automatic from the deﬁnition that the space K in a universal pair (K, u) is unique up to homotopy equivalence. For if (K′, u′) is also universal for
h, then, using the notation f : (K, u)→(K′, u′) to mean f : K→K′ with f ∗(u′) = u, universality implies that there are maps f : (K, u)→(K′, u′) and g : (K′, u′)→(K, u) that are unique up to homotopy. Likewise the compositions gf : (K, u)→(K, u) and f g : (K′, u′)→(K′, u′) are unique up to homotopy, hence are homotopic to the identity maps. Before starting the proof of this theorem we make a few preliminary comments on the axioms. (1) The wedge axiom implies that h(point) is trivial. To see this, just use the fact that for any X we have X ∨ point = X, so the map h(X)× h(point)→h(X) induced by inclusion of the ﬁrst summand is a bijection, but this map is the projection (a, b)֏a, hence h(point) must have only one element. (2) Axioms (i), (iii), and the Mayer–Vietoris axiom imply axiom (ii). Namely, (ii) is equivalent to exactness of h(A)← h(X)← h(X ∪ CA), where CA is the reduced cone since we are in the basepointed category. The inclusion Im ⊂ Ker holds since the composition A→X ∪ CA is nullhomotopic, so the induced map factors through h(point) = 0. 450 Chapter 4 Homotopy Theory To obtain the opposite inclusion Ker ⊂ Im, decompose X ∪ CA into two subspaces Y and Z by cutting along a copy of A halfway up the cone CA, so Y is a smaller copy of CA and Z is the reduced mapping cylinder of the inclusion A ֓ X. Given an element x ∈ h(X), this extends to an element z ∈ h(Z) since Z deformation retracts to X. If x restricts to the trivial element of h(A), then z restricts to the trivial element of h(Y ∩ Z). The latter element extends to the trivial element of h(Y
), so the Mayer–Vietoris axiom implies there is an element of h(X ∪ CA) restricting to z in h(Z) and hence to x in h(X). (3) If h satisﬁes axioms (i) and (iii) then h( Y ) is a group and Tu : h Y, Ki→h( Y ) is a homomorphism for all suspensions Y and all pairs (K, u). [See the Corrections.] Σ Σ Σ The proof of Theorem 4E.2 will use two lemmas. To state the ﬁrst, consider Σ pairs (K, u) with K a basepointed connected CW complex and u ∈ h(K), where h satisﬁes the hypotheses of the theorem. Call such a pair (K, u) n universal if the homomorphism Tu : πi(K)→h(S i), Tu(f ) = f ∗(u), is an isomorphism for i < n and surjective for i = n. Call (K, u) π∗ universal if it is n universal for all n. Lemma 4E.3. Given any pair (Z, z) with Z a connected CW complex and z ∈ h(Z), there exists a π∗ universal pair (K, u) with Z a subcomplex of K and u \|\| Z = z. Proof: We construct K from Z by an inductive process of attaching cells. To begin, α where α ranges over the elements of h(S 1). By the wedge axiom let K1 = Z there exists u1 ∈ h(K1) with u1 \|\| Z = z and u1 \|\| S 1 α = α, so (K1, u1) is 1 universal. α S 1 W : πn(Kn)→h(S n) by a map fα : S n→Kn. Let f = For the inductive step, suppose we have already constructed (Kn, un) with un n universal, Z ⊂ Kn, and un \|\| Z = z. Represent each element α in the kernel of α→Kn. The reduced α S n Tun mapping cylinder Mf deformation retracts to Kn, so we can regard un as an element W of h(Mf ), and
this element restricts to the trivial element of h( α ) by the deﬁnition of f. The exactness property of h then implies that for the reduced mapping cone Cf = Mf / α there is an element w ∈ h(Cf ) restricting to un on Kn. Note that Cf is obtained from Kn by attaching cells en+1 by the maps fα. To ﬁnish the construction where β ranges over h(S n+1). By the wedge axiom, of Kn+1, set Kn+1 = Cf there exists un+1 ∈ h(Kn+1) restricting to w on Cf and β on S n+1 β S n+ fα : W α β β. To see that (Kn+1, un+1) is (n + 1) universal, consider the commutative diagram displayed at the right. Since Kn+1 is obtained from Kn by attaching (n + 1) cells, the upper map is an isomorphism, hence it is also true for Tun+1 for i < n and a surjection for i = n. By induction the same is true for Tun is trivial for i = n since an element of this kernel pulls back to kernel of Tun+1 ⊂ πn(Kn), by surjectivity of the upper map when i = n, and we attached Ker Tun cells to Kn by maps representing all elements of Ker Tun. Also, Tun+1 is surjective for i = n + 1 by construction.. The The Brown Representability Theorem Section 4.E 451 S S Now let K = n Kn. We apply a mapping telescope argument as in the proofs of Lemma 2.34 and Theorem 3F.8 to show there is an element u ∈ h(K) restricting to un on Kn, for all n. The mapping telescope of the inclusions K1 ֓ K2 ֓ ··· is the i Ki × [i, i + 1] of K × [1, ∞). We take ‘ × ’ to be the reduced product subcomplex T = here, with basepoint × interval collapsed to a point. The natural projection T→K is a homotopy equivalence since K × [1, ∞) deformation retracts onto T, as we showed in the proof of Lemma 2.34. Let A ⊂ T be the union of the sub
complexes Ki × [i, i+1] for i odd and let B be the corresponding union for i even. Thus Ki, A ≃ i K2i. By the wedge axiom there exist a ∈ h(A) and b ∈ h(B) restricting to ui on each Ki. Then using the fact that ui+1 \|\| Ki = ui, the Mayer–Vietoris axiom implies that a and b are the restrictions of an element t ∈ h(T ). Under the isomorphism h(T ) ≈ h(K), t corresponds to an element u ∈ h(K) restricting to un on Kn for all n. i K2i−1, and B ≃ W W W To verify that (K, u) is π∗ universal we use the commutative diagram at the right. For n > i + 1 the upper map is an isomorphism and Tun kernel, so the same is true of Tu. is surjective with trivial ⊔⊓ Lemma 4E.4. Let (K, u) be a π∗ universal pair and let (X, A) be a basepointed CW pair. Then for each x ∈ h(X) and each map f : A→K with f ∗(u) = x \|\| A there exists a map g : X→K extending f with g∗(u) = x. Schematically, this is saying that the diagonal arrow in the diagram at the right always exists, where the map i is inclusion. Proof: Replacing K by the reduced mapping cylinder of f reduces us to the case that f is the inclusion of a subcomplex. Let Z be the union of X and K with the two copies of A identiﬁed. By the Mayer–Vietoris axiom, there exists z ∈ h(Z) with z \|\|X = x and z \|\|K = u. By the previous lemma, we can embed (Z, z) in a π∗ universal pair (K′, u′). The inclusion (K, u) ֓ (K′, u′) induces an isomorphism on homotopy groups since both u and u′ are π∗ universal, so K′ deformation retracts onto K. This deformation retraction induces a homotopy rel A of the inclusion X
֓ K′ to a map g : X→K. The relation g∗(u) = x holds since u′ \|\| K = u and u′ \|\| X = x. ⊔⊓ Proof of Theorem 4E.2: It suﬃces to show that a π∗ universal pair (K, u) is universal. Applying the preceding lemma with A a point shows that Tu : hX, Ki→h(X) is surjective. To show injectivity, suppose Tu(f0) = Tu(f1), that is, f ∗ 1 (u). We apply the preceding lemma with (X × I, X × ∂I) playing the role of (X, A), using the maps f0 and f1 on X × ∂I and taking x to be p∗f ∗ 1 (u) where p is the projection X × I→X. Here X × I should be the reduced product, with basepoint × I ⊔⊓ collapsed to a point. The lemma then gives a homotopy from f0 to f1. 0 (u) = p∗f ∗ 0 (u) = f ∗ 452 Chapter 4 Homotopy Theory Kn+1. The natural isomorphism hn(X) ≈ hn+1( Proof of Theorem 4E.1: Since suspension is an isomorphism in any reduced cohomology theory, and the suspension of any CW complex is connected, it suﬃces to restrict attention to connected CW complexes. Each functor hn satisﬁes the homotopy, wedge, and Mayer–Vietoris axioms, as we noted earlier, so the preceding theorem gives CW complexes Kn with hn(X) = hX, Kni. ral isomorphisms hn(X) ≈ hn+1( Kn→ ural bijection hX, Kni ≈ h of this bijection gives, for any map f : X→Kn, a commutative diagram as at the right. Let εn = (11) : Kn→ Kn+1. Then using commutativity (11) = f ∗(εn) = f ∗(11) = f ∗ we have Φ : hX, Kni→hX, ε
nf, which says that the map Φ Kn+1i is composition with εn. Since is a bijection, if we take X to be S i, we see that εn induces an isomorphism on πi X) correspond to weak homotopy equivalences It remains to show that the natu- Σ X, Kn+1i = hX, X) corresponds to a nat- Kn+1i which we call. The naturality ( for all i, so εn is a weak homotopy equivalence and we have an Φ Ω Kn+1i = h There is one ﬁnal thing to verify, that the bijection hn(X) = hX, Kni is a group isomorphism, where hX, Kni has the group structure that comes from identifying it with hX, X) this is equivalent to showing the bijection hn+1( X, Kn+1i preserves group Σ X→K, the relation Tu(f + g) = Tu(f ) + Tu(g) means structure. For maps f, g : (f +g)∗(u) = f ∗(u)+g∗(u), and this holds since (f +g)∗ = f ∗ +g∗ : h(K)→h( X) by Lemma 4.60. X, Kn+1i. Via the natural isomorphism hn(X) ≈ hn+1( X) = h Ω Ω ⊔⊓ Σ Σ Σ Σ spectrum. Σ We have seen in §4.3 and the preceding section how cohomology theories have a homotopy-theoretic interpretation in terms of spectra, and it is natural to look for a corresponding description of homology theories. In this case we do not already have a homotopy-theoretic description of ordinary homology to serve as a starting Ω point. But there is another homology theory we have encountered which does have a very homotopy-theoretic ﬂavor: Proposition 4F.1. Stable homotopy groups π s on the category of basepointed CW complexes and basepoint-preserving maps. n(X) deﬁne a reduced homology theory Proof: In
the preceding section we reformulated the axioms for a cohomology theory so that the exactness axiom asserts just the exactness of hn(X/A)→hn(X)→hn(A) for CW pairs (X, A). In order to derive long exact sequences, the reformulated axioms Spectra and Homology Theories Section 4.F 453 Σ require also that natural suspension isomorphisms hn(X) ≈ hn+1( as part of the cohomology theory. The analogous reformulation of the axioms for a X) be speciﬁed homology theory is valid as well, by the same argument, and we shall use this in what follows. For stable homotopy groups, suspension isomorphisms π s n(X) ≈ π s n+1( X) are Σ n(A)→π s n(X)→π s automatic, so it remains to verify the three axioms. The homotopy axiom is apparent. The exactness of a sequence π s n(X/A) follows from exactness of πn(A)→πn(X)→πn(X, A) together with the isomorphism πn(X, A) ≈ πn(X/A) which holds under connectivity assumptions that are achieved after suﬃciently many suspensions. The wedge sum axiom π s n(Xα) reduces to the case of n( ﬁnitely many summands by the usual compactness argument, and the case of ﬁnitely W many summands reduces to the case of two summands by induction. Then we have iY ), the ﬁrst iX × isomorphisms πn+i( πn+i( iY is iX ∨ of these isomorphisms holding when n + i < 2i − 1, or i > n + 1, since Σ iY. Passing to the limit over increasing i, we get the the (2i − 1) skeleton of desired isomorphism π s ⊔⊓ iY ) ≈ πn+i( iY ) ≈ πn+i( α Xα) ≈ Σ iX × απ s iX ∨ iX(Y ). n
((X) L A modest generalization of this homology theory can be obtained by deﬁning hn(X) = π s n(X ∧ K) for a ﬁxed complex K. Verifying the homology axioms reduces to the case of stable homotopy groups themselves by basic properties of smash product: X) since (X ∧ K) = ( hn(X) ≈ hn+1( The exactness axiom holds since (X ∧ K)/(A ∧ K) = (X/A) ∧ K, both spaces being Σ quotients of X × K with A× K ∪ X × {k0} collapsed to a point. The wedge axiom follows from distributivity: ( X) ∧ K, both spaces being Xα) ∧ K = α (Xα ∧ K). The coeﬃcients of this homology theory are hn(S 0) = π s n(K). Suppose for example that K is an Eilenberg–MacLane space K(G, n). Because K(G, n) is (n − 1) connected, its stable homotopy groups are the same as its unstable homotopy groups below dimension 2n. Thus if we shift dimensions by deﬁning hi(X) = π s we obtain a homology theory whose coeﬃcient groups below dimension n are the same as ordinary homology with coeﬃcients in G. It follows as W n(S 0 ∧ K) = π s X ∧ K(G, n) i+n W in Theorem 4.59 that this homology theory agrees with ordinary homology for CW complexes of dimension less than n − 1. This dimension restriction could be removed if there were a ‘stable Eilenberg– MacLane space’ whose stable homotopy groups were zero except in one dimension. i+n X ∧ K(G, n) as n goes to inﬁnity. The spaces K(G, n) for varying n are related However, this is a lot to ask for, so instead one seeks to form a limit of the groups π s by weak homotopy equivalences K(G, n)→ a large role in the current discussion, let us consider instead the corresponding map K(
G, n)→K(G, n + 1), or to write this more concisely, Kn→Kn+1 This induces a map π s i+n+1(X ∧ Kn+1). Via these maps, it then i+n+1(X ∧ Σ K(G, n+1). Since suspension plays such i+n(X ∧ Kn) = π s Kn)→π s Ω Σ Σ 454 Chapter 4 Homotopy Theory makes sense to consider the direct limit as n goes to inﬁnity, the group hi(X) = lim--→π s i+n(X ∧ Kn). This gives a homology theory since direct limits preserve exact sequences so the exactness axiom holds, and direct limits preserve isomorphisms so the suspension isomorphism and the wedge axiom hold. The coeﬃcient groups of this homology theory are the same as for ordinary homology with G coeﬃcients since hi(S 0) = lim--→π s i+n(Kn) is zero unless i = 0, when it is G. Hence this homology theory coincides with ordinary homology by Theorem 4.59. To place this result in its natural generality, deﬁne a spectrum to be a sequence Kn→Kn+1. This of CW complexes Kn together with basepoint-preserving maps Kn→Kn+1 come from generalizes the notion of an weak homotopy equivalences Kn→ Kn+1. Another obvious family of examples is Σ suspension spectra, where one starts with an arbitrary CW complex X and deﬁnes Kn = Kn→Kn+1 the identity map. spectrum, where the maps nX with Ω Ω Σ The homotopy groups of a spectrum K are deﬁned to be πi(K) = lim--→πi+n(Kn) Σ Σ where the direct limit is computed using the compositions πi+n(Kn) ----------→ πi+n+1( Kn) ------→ πi+n+1(Kn+1) Σ with the latter map induced by the given map Kn→Kn+1. Thus in the case of the suspension spectrum of a space X, the homotopy groups of the spectrum are the Σ
same as the stable homotopy groups of X. For a general spectrum K we could also i+n(Kn) since the composition πi+n(Kn)→πi+n+j(Kn+j) facdescribe πi(K) as lim--→π s jKn). So the homotopy groups of a spectrum are ‘stable homotors through πi+n+j( topy groups’ essentially by deﬁnition. Σ Returning now to the context of homology theories, if we are given a spectrum K and a CW complex X, then we have a spectrum X ∧ K with (X ∧ K)n = X ∧ Kn, Kn→X ∧ Kn+1. The groups πi(X ∧ K) are using the obvious maps the groups lim--→π s i+n(X ∧ Kn) considered earlier in the case of an Eilenberg–MacLane spectrum, and the arguments given there show: (X ∧ Kn) = X ∧ Σ Σ Σ Proposition 4F.2. For a spectrum K, the groups hi(X) = πi(X ∧ K) form a reduced homology theory. When K is the Eilenberg–MacLane spectrum with Kn = K(G, n), ⊔⊓ this homology theory is ordinary homology, so πi(X ∧ K) ≈ Hi(X; G). If one wanted to associate a cohomology theory to an arbitrary spectrum K, one’s nX, Kn+ii, the direct limit with respect e ﬁrst inclination would be to set hi(X) = lim--→h to the compositions Σ h nX, Kn+ii ----------→ h n+1X, Kn+ii ------→ h n+1X, Kn+i+1i Σ Σ For example, in the case of the sphere spectrum S = {S n} this deﬁnition yields the Σ nX, S n+ii. Unfortunately the deﬁnition stable cohomotopy groups π i s (X) = lim--→h nX, Kn+ii runs into problems with the wedge sum axiom since the direct hi(X) = lim--→h Σ Σ �
� Σ Spectra and Homology Theories Section 4.F 455 limit of a product need not equal the product of the direct limits. For ﬁnite wedge sums there is no diﬃculty, so we do have a cohomology theory for ﬁnite CW complexes. But for general CW complexes a diﬀerent deﬁnition is needed. The simplest thing to do ni. We iKn+i, the mapping telescope of the sequence Ω is to associate to each spectrum K an obtain K′ from K by setting K′ n = lim--→ Kn→ spectrum K′ and let hn(X) = hX, K′ Kn+1→ K′ n = lim--→ Ω 2Kn+2→ ···. The iKn+i ≃ lim--→ Ω spectrum structure is given by equivalences Ω i+1Kn+i+1 Ω iKn+i+1 = κ------------→ K′ n+1 lim--→ Ω Ω Kn+1→ The ﬁrst homotopy equivalence comes from deleting the ﬁrst term of the sequence Ω 2Kn+2→ ···, which has negligible eﬀect on the mapping telescope. Kn→ lim--→Zn The next map κ is a special case of the natural weak equivalence lim--→ that holds for any sequence Z1→Z2→ ···. Strictly speaking, we should let K′ n be a Ω iKn+i in order to obtain a spectrum CW approximation to the mapping telescope lim--→ consisting of CW complexes, in accordance with our deﬁnition of a spectrum. Zn→ Ω Ω Ω Ω Ω In case one starts with a suspension spectrum Kn = to take mapping telescopes since one can just set K′ the union with respect to the natural inclusions Ω nK it is not necessary i+nK = iKn, i i+1Kn. The union i+1 S Σ Ω ∞X. Another common notation for this union Ω i (X), so Q is a functor converting stable homotopy groups n = i Σ iKn ⊂ iX is usually abbreviated to i is QX. Thus πi
(QX) = π s S into ordinary homotopy groups. Ω Σ Ω Σ It follows routinely from the deﬁnitions that the homology theory deﬁned by a spectrum is the same as the homology theory deﬁned by the associated spectrum. One may ask whether every homology theory is deﬁned by a spectrum, as we showed for cohomology. The answer is yes if one replaces the wedge axiom by a stronger direct limit axiom: hi(X) = lim--→hi(Xα), the direct limit over the ﬁnite subcomplexes Xα of X. The homology theory deﬁned by a spectrum satisﬁes this axiom, and the converse is proved in [Adams 1971]. Ω Spectra have become the preferred language for describing many stable phenom- ena in algebraic topology. The increased ﬂexibility of spectra is not without its price, however, since a number of concepts that are elementary for spaces become quite a bit more subtle for spectra, such as the proper deﬁnition of a map between spectra, or the smash product of two spectra. For the reader who wants to learn more about this language a good starting point is [Adams 1974]. Exercises 1. Assuming the ﬁrst two axioms for a homology theory on the CW category, show that the direct limit axiom implies the wedge sum axiom. Show that the converse also holds for countable CW complexes. 2. For CW complexes X and Y consider the suspension sequence hX, Y i -----→ h Σ X, Y i -----→ h 2X, 2Y i -----→ ··· Σ Σ Σ Σ Σ 456 Chapter 4 Homotopy Theory Show that if X is a ﬁnite complex, these maps eventually become isomorphisms. [Use induction on the number of cells of X and the ﬁve-lemma.] 3. Show that for any sequence Z1→Z2→ ···, the natural map lim--→ a weak homotopy equivalence, where the direct limits mean mapping telescopes. Zn→ lim--→Zn is Ω Ω It is a common practice in algebraic topology to glue spaces together to form more complicated spaces
be the original topology on X. An action of a group G on a space X determines a diagram of spaces XG, with X itself as the only space and with maps the homeomorphisms g : X→X, g ∈ G, given by the action. In this case XG is the orbit space X/G. Gluing Constructions Section 4.G 457 complex X can be viewed as a diagram of spaces X A where each simplex of X gives a vertex space Xv which is a simplex of the same dimension, and the edge maps are the inclusions of faces into the simplices that contain them. Then ∆ ∆ X = X. It can very easily happen that for a diagram of spaces X the amalgamation X ∆ is rather useless because so much collapsing has occurred that little of the original diagram remains. For example, consider a diagram X of the form X0← X0 × X1→X1 whose maps are the projections onto the two factors. In this case X is simply a point. To correct for problems like this, and to get a notion with nicer homotopy-theoretic properties, we introduce the homotopy version of X, which we shall denote X and call the realization of X. Here we again start with the disjoint union of all the vertex spaces Xv, but instead of passing to a quotient space of this disjoint union, we enlarge it by ﬁlling in a mapping cylinder Mf for each map f of the diagram, identifying the two ends of this cylinder with the appropriate Xv ’s. In the case of the projection diagram X0← X0 × X1→X1, the union of the two mapping cylinders is the same as the quotient of X0 × X1 × I with X0 × X1 × {0} collapsed to X0 and X0 × X1 × {1} collapsed to X1. Thus X is the join X0 ∗ X1 deﬁned in Chapter 0. ∆ ∆ f←------ X1 We have seen a number of other special cases of the construction X. For a diagram consisting of just one map f : X0→X1 one gets of course the mapping cylinder Mf itself. For a diagram X0 X is a double mapping cylinder. In case X2 is a point this is the mapping cone of f. When the diagram has ∆ just one space and
one map from this space to itself, then X is the mapping torus. For a diagram consisting of two maps f, g : X0→X1 the space X was studied in Ex∆ ample 2.48. Mapping telescopes are the case of a sequence of maps X0→X1→ ···. In §1.B we considered general diagrams in which the spaces are K(G, 1) ’s. g-----→ X2 the realization ∆ ∆ There is a natural generalization of X in which one starts with a complex and a diagram of spaces associated to the 1 skeleton of ∆ such that the maps ∆ Γ f1-----→ X1, n > 1, form a commutative di- corresponding to the edges of each n simplex of Γ agram. We call this data a complex of spaces. Γ f2-----→ ··· If X is a complex of spaces, then Γ fn-----→ Xn, for each n simplex of we have a sequence of maps X0 and we deﬁne the iterated mapping cylinder M(f1, ···, fn) to be the usual mapping cylinder for n = 1, and inductively for n > 1, the mapping cylinder of the fn-----→ Xn where the ﬁrst map is the canonical composition M(f1, ···, fn−1)→Xn−1 projection of a mapping cylinder onto its target end. There is a natural projection M(f1, ···, fn)→ n one has the iterated mapping cylinder for the maps associated to the edges in this face. For example when n = 2 one has the three mapping cylinders M(f1), M(f2), and M(f2f1) over the three edges thus ﬁt of 2. All these iterated mapping cylinders over the various simplices of n, and over each face of ∆ ∆ together to form a space X with a canonical projection X→. We again call X the realization of the complex of spaces X, and we call ∆ the base of X or ∆ Γ X. ∆ ∆ Γ Γ ∆ 458 Chapter 4 Homotopy Theory Some of our earlier examples of diagrams of spaces can be regarded in a natural way as complexes of spaces: For a cover U = {Xi} of a space X the diagram of spaces XU whose vertices
are the ﬁnite intersections of Xi ’s and whose edges are inclusions is a complex of for this complex of spaces with n simplices the n fold inclusions. The base spaces is the barycentric subdivision of the nerve of the cover. Recall from the end of §3.3 that the nerve of a cover is the simplicial complex with n simplices the nonempty (n + 1) fold intersections of sets in the cover. Γ The diagram of spaces XG associated to an action of a group G on a space X is a complex of spaces, with n simplices corresponding to the n fold compositions X g1-----→ X g2-----→ ··· gn-----→ X. The base complex is the K(G, 1) called BG in §1.B. Checking through the deﬁnitions, one sees that the space This was the orbit space of a free action of G on a contractible complex EG. XG in this case can be regarded as the quotient of X × EG under the diagonal action of G, g(x, y) = (g(x), g(y)). This is the space we called the Borel construction in §3.G, with the notation X ×G EG. ∆ ∆ ∆ Γ By a map f : X→Y of complexes of spaces over the same base we mean a collection of maps fv : Xv→Yv for all the vertices of f : over all edges of. There is then an induced map Γ, with commutative squares X→ Y. Γ Proposition 4G.1. If all the maps fv making up a map of complexes of spaces f : X→Y are homotopy equivalences, then so is the map X→ f : Y. ∆ ∆ ∆ Γ Proof: The mapping cylinders M(fv ) form a complex of spaces M(f ) over the same f ). This deformation M(f ) is the mapping cylinder M( base, and the space ∆ ∆ ∆ Y, so it will suﬃce to show that it also deformation retracts onto X. retracts onto Γ Let M n( f ) ∪ = ∆ f ) be the part of M( ∆ X deformation retracts onto M n−1( ∆ n. ∆ M n( when ∆ Corollary 0.20
it suﬃces to show that the inclusion M n−1( homotopy equivalence and the pair (M( X. It is enough to show this In this case f is a map from X0→ ··· →Xn to Y0→ ··· →Yn. By ∆ f ) is a X ֓ M( X) satisﬁes the homotopy ∆ extension property. The latter assertion is evident from Example 0.15 since a mapping f ), M n−1 ) lying over the n skeleton of f ) ∪. We claim that ∆ ∆ ∆ ∆ ∆ cylinder neighborhood is easily constructed for this pair. For the other condition, note that by induction on the dimension of we may assume that M n−1( f ) deformation retracts onto the part of f ) is a homotopy equivalence since it is equivalent to the map Xn→Yn, which is a homotopy ∆ equivalence by hypothesis. So Corollary 0.20 applies, and the claim that M n( X deformation retracts onto M n−1( f ) ∪ n. Also, the inclusion Γ X ֓ M( ∆ X is proved. X over ∂ f )∪ ∆ ∆ ∆ Letting n vary, the inﬁnite concatenation of these deformation retractions in the t intervals [1/2n+1, 1/2n] gives a deformation retraction of M( f ) onto X. ⊔⊓ ∆ ∆ ∆ ∆ ∆ ∆ Gluing Constructions Section 4.G 459 ∆ There is a canonical map X→ X induced by retracting each mapping cylinder onto its target end. In some cases this is a homotopy equivalence, for example, for a diagram X0← A ֓ X1 where the pair (X1, A) has the homotopy extension property. Another example is a sequence of inclusions X0 ֓ X1 ֓ ··· for which the pairs (Xn, Xn−1) satisfy the homotopy extension property, by the argument involving mapping telescopes in the proof of Lemma 2.34. However, without some conditions on X→ X is a homotopy equivalence, as the earlier the maps it need not be true that example of
section s embeds X as a retract of is a deformation retract since points in ﬁbers p−1(x) can move ⊔⊓ linearly along line segments to s(x). ∆ Corollary 4G.3. If U is an open cover of a paracompact space X such that every nonempty intersection of ﬁnitely many sets in U is contractible, then X is homotopy equivalent to the nerve NU. Proof: The proposition gives a homotopy equivalence X ≃ XU. Since the nonempty ﬁnite intersections of sets in U are contractible, the earlier proposition implies that induced by sending each intersection to a point is a homotopy the map ∆ is the barycentric subdivision of NU, the result follows. ⊔⊓ Γ XU→ equivalence. Since Γ ∆ 460 Chapter 4 Homotopy Theory Let us conclude this section with a few comments about terminology. For some diagrams of spaces such as sequences X1→X2→ ··· the amalgamation X can be regarded as the direct limit of the vertex spaces Xv with respect to the edge maps fe. Following this cue, the space X is commonly called the direct limit for arbitrary diagrams, even ﬁnite ones. If one views X as a direct limit, then X becomes a sort of homotopy direct limit. For reasons that are explained in the next section, direct limits are often called ‘colimits’. This has given rise to the rather unfortunate name of ∆ ‘hocolim’ for X, short for ‘homotopy colimit’. In preference to this we have chosen the term ‘realization’, both for its intrinsic merits and because to what is called the geometric realization of a simplicial space. ∆ X is closely related ∆ Exercises f1-----→ X1 f2-----→ ···, the inﬁnite iterated mapping 1. Show that for a sequence of maps X0 cylinder M(f1, f2, ···), which is the union of the ﬁnite iterated mapping cylinders M(f1, ···, fn), deformation retracts onto the mapping telescope. 2. Show that if X is a complex of spaces in which all the maps are homeomorphisms, then the projection X
→ is a ﬁber bundle. 3. What is the nerve of the cover of a simplicial complex by the open stars of its Γ vertices? [See Lemma 2C.2.] ∆ 4. Show that Proposition 4G.2 and its corollary hold also for CW complexes and covers by families of subcomplexes. [CW complexes are paracompact; see [VBKT].] There is a very nice duality principle in homotopy theory, called Eckmann–Hilton duality in its more reﬁned and systematic aspects, but which in its most basic form involves the simple idea of reversing the direction of all arrows in a given construc- tion. For example, if in the deﬁnition of a ﬁbration as a map satisfying the homotopy lifting property we reverse the direction of all the arrows, we obtain the dual notion of a coﬁbration. This is a map i : A→B satisfying the following propg0 : B→X and a homotopy gt : A→X such that g0i = g0, erty: Given gt : B→X such that gti = gt. In the special there exists a homotopy e case that i is the inclusion of a subspace, this is the homotopy extension property, e and the next proposition says that this is indeed the general case. So a coﬁbration is e e the same as an inclusion satisfying the homotopy extension property. Proposition 4H.1. If i : A→B is a coﬁbration, then i is injective, and in fact a homeomorphism onto its image. Eckmann–Hilton Duality Section 4.H 461 Proof: Consider the mapping cylinder Mi, the quotient of A× I ∐ B in which (a, 1) is identiﬁed with i(a). Let gt : A→Mi be the homotopy mapping a ∈ A to the image of g0 be the inclusion B ֓ Mi. The coﬁbration property (a, 1 − t) ∈ A× I in Mi, and let gti = gt. Restricting to a ﬁxed t > 0, this implies i is injective gives since gt is. Furthermore, since gt is a home
omorphism onto its image A× {1 − t}, e gt : i(A)→A is a continuous inverse of the relation i : A→i(A). ⊔⊓ gti = gt implies that the map g−1 gt : B→Mi with e e t e e Many constructions for ﬁbrations have analogs for coﬁbrations, and vice versa. For example, for an arbitrary map f : A→B the inclusion A ֓ Mf is readily seen to be a coﬁbration, so the analog of the factorization A ֓ Ef →B of f into a homotopy equivalence followed by a ﬁbration is the factorization A֓ Mf →B into a coﬁbration followed by a homotopy equivalence. Even the deﬁnition of Mf is in some way dual to the deﬁnition of Ef, since Ef can be deﬁned as a pullback and Mf can be deﬁned as a dual pushout. In general, the pushout of maps f : Z→X and g : Z→Y is deﬁned as the quotient of X ∐ Y under the identiﬁcations f (z) ∼ g(z). Thus the pushout is a quotient of X ∐Y, while the pullback of maps X→Z and Y →Z is a subobject of X × Y, so we see here two instances of duality: a duality between disjoint union and product, and a duality between subobjects and quotients. The ﬁrst of these is easily explained, since a collection of maps Xα→X is equivalent to a map α Xα. The. If we were Q αXα→X, while a collection of maps X→Xα is equivalent to a map X→ for the ‘coproduct’ was chosen to indicate that it is dual to notation ` dealing with basepointed spaces and maps, the coproduct would be wedge sum. In ` Q the category of abelian groups the coproduct is direct sum. The duality between subobjects and quotient objects is clear for abelian groups, where subobjects are kernels and quotient objects cokernels. The
strict topological analog of a kernel is a ﬁber of a ﬁbration. Dually, the topological analog of a cokernel is the coﬁber B/A of a coﬁbration A֓ B. If we make an arbitrary map f : A→B into a coﬁbration A ֓ Mf, the coﬁber is the mapping cone Cf = Mf /(A× {0}). In the diagram showing Ef and Mf as pullback and pushout, there also appears to be some sort of duality involving the terms A× I and BI. This leads us to ask whether X × I and X I are in some way dual. Indeed, if we ignore topology and just think settheoretically, this is an instance of the familiar product–coproduct duality since the product of copies of X indexed by I is X I, all functions I→X, while the coproduct of copies of X indexed by I is X × I, the disjoint union of the sets X × {t} for t ∈ I. Switching back from the category of sets to the topological category, we can view X I as a ‘continuous product’ of copies of X and X × I as a ‘continuous coproduct’. On a less abstract level, the fact that maps A× I→B are the same as maps A→BI indicates a certain duality between A× I and BI. This leads to a duality between 462 Chapter 4 Homotopy Theory suspension and loopspace, since of BI. This duality is expressed in the adjoint relation h Combining this duality between Σ and A is a quotient of A× I and X, Y i = hX, Ω with the duality between ﬁbers and coﬁbers, B is a subspace Y i from §4.3. we see a duality relationship between the ﬁbration and coﬁbration sequences of §4.3: Σ Ω Σ F -→ ··· -→ A→X→X/A→ Ω Ω Ω E -→ A→ Ω B -→ F -→ E -→ B X→ (X/A)→ ··· Σ Σ Pushout
and pullback constructions can be generalized to arbitrary diagrams. In Σ the case of pushouts, this was done in §4.G where we associated a space X to a diav Xv, with v ranging over gram of spaces X. This was the quotient of the coproduct vertices of the diagram, under the identiﬁcations x ∼ fe(x) for all maps fe associated to edges e of the diagram. The dual construction X would be the subspace of the v Xv consisting of tuples (xv ) with fe(xv ) = xw for all maps fe : Xv→Xw in the diagram. Perhaps more useful in algebraic topology is the homotopy variant of product ` Q this notion obtained by dualizing the deﬁnition of X in the previous section. This is the space ∇X consisting of all choices of a point xv in each Xv and a path γe in the target space of each edge map fe : Xv→Xw, with γe(0) = f (xv ) and γe(1) = xw. The subspace with all paths constant is X. In the case of a diagram ··· →X2→X1 such as a Postnikov tower this construction gives something slightly diﬀerent from ∆ simply turning each successive map into a ﬁbration via the usual pathspace construction, starting with X2→X1 and proceeding up the tower, as we did in §4.3. The latter construction is rather the dual of an iterated mapping cylinder, involving spaces of n→Xv instead of simply pathspaces. One could use such mapping spaces to maps generalize the deﬁnition of ∇X from diagrams of spaces to complexes of spaces. As special cases of the constructions X and X we have direct limits and inverse limits for diagrams X0→X1→ ··· and ··· →X1→X0, respectively. Since inverse limit is related to product and direct limit to coproduct, it is common practice ∆ in some circles to use reverse logic and call inverse limit simply ‘limit’ and direct limit ‘colimit’. The homotopy versions are then called ‘holim’ for ∇X and ‘hocolim’ for X. This terminology is frequently used for more
general diagrams as well. Homotopy Groups with Coeﬃcients ∆ There is a somewhat deeper duality between homotopy groups and cohomology, which one can see in the fact that cohomology groups are homotopy classes of maps into a space with a single nonzero homotopy group, while homotopy groups are ho- motopy classes of maps from a space with a single nonzero cohomology group. This duality is in one respect incomplete, however, in that the cohomology statement holds for an arbitrary coeﬃcient group, but we have not yet deﬁned homotopy groups with coeﬃcients. In view of the duality, one would be tempted to deﬁne πn(X; G) to be the set of basepoint-preserving homotopy classes of maps from the cohomology analog of a Moore space M(G, n) to X. The cohomology analog of M(G, n) would be a space Eckmann–Hilton Duality Section 4.H 463 e H i(Y ; Z) is G for i = n. Unfortunately, Y whose only nonzero cohomology group such a space does not exist for arbitrary G, for example for G = Q, since we showed in Proposition 3F.12 that if the cohomology groups of a space are all countable, then they are all ﬁnitely generated. As a ﬁrst approximation to πn(X; G) let us consider hM(G, n), Xi, the set of basepoint-preserving homotopy classes of maps M(G, n)→X. To give this set a more suggestive name, let us call it µn(X; G). We should assume n > 1 to guarantee that the homotopy type of M(G, n) is well-deﬁned, as shown in Example 4.34. For n > 1, µn(X; G) is a group since we can choose M(G, n) to be the suspension of an M(G, n − 1). And if n > 2 then µn(X; G) is abelian since we can choose M(G, n) to be a double suspension. There is something like a universal coeﬃcient theorem
for these groups µn(X; G) : Proposition 4H.2. For n > 1 there are natural short exact sequences 0 -→ Ext(G, πn+1(X)) -→ µn(X; G) -→ Hom(G, πn(X)) -→ 0. The similarity with the universal coeﬃcient theorem for cohomology is apparent, but with a reversal of the variables in the Ext and Hom terms, reﬂecting the fact that µn(X; G) is covariant as a functor of X and contravariant as a functor of G, just like the Ext and Hom terms. i-----→ F -→ G→0 be a free resolution of G. The inclusion map i is Proof: Let 0→R realized by a map M(R, n)→M(F, n), where M(R, n) and M(F, n) are wedges of S n ’s corresponding to bases for F and R. Converting this map into a coﬁbration via the mapping cylinder, the coﬁber is an M(G, n), as one sees from the long exact sequence of homology groups. As in §4.3, the coﬁbration sequence M(R, n)→M(F, n)→M(G, n)→M(R, n + 1)→M(F, n + 1) gives rise to the exact sequence across the top of the following diagram: The four outer terms of the exact sequence can be identiﬁed with the indicated Hom terms since mapping a wedge sum of S n ’s into X amounts to choosing an element of πn(X) for each wedge summand. The kernel and cokernel of i∗ are Hom(G, −) and Ext(G, −) by deﬁnition, and so we obtain the short exact sequence we are looking for. ⊔⊓ Naturality will be left for the reader to verify. Unlike in the universal coeﬃcient theorems for homology and cohomology, the short exact sequence in this proposition does not split in general. For an example, take G = Z2 and X = M(Z2, n) for n ≥ 2, where the identity map of
M(Z2, n) 464 Chapter 4 Homotopy Theory deﬁnes an element of µn(M(Z2, n); Z2) = hM(Z2, n), M(Z2, n)i having order 4, as we show in Example 4L.7, whereas the two outer terms in the short exact sequence can only contain elements of order 2 since G = Z2. This example shows also that µn(X; Zm) need not be a module over Zm, as homology and cohomology groups with Zm coeﬃcients are. The proposition implies that the ﬁrst nonzero µi(S n; Zm) is µn−1(S n; Zm) = Zm, from the Ext term. This result would look more reasonable if we changed notation to replace the subscript n − 1 by n. So let us make the deﬁnition πn(X; Zm) = hM(Zm, n − 1), Xi = µn−1(X; Zm) There are two good reasons to expect this to be the right deﬁnition. The ﬁrst is formal: M(Zm, n − 1) is a ‘cohomology M(Zm, n) ’ since its only nontrivial cohomology group H i(M(Zm, n − 1); Z) is Zm in dimension n. The second reason is more geometric: Elements of πn(X; Zm) should be homotopy classes of ‘homotopy-theoretic cycles e mod m ’, meaning maps Dn→X whose boundary is not necessarily a constant map as would be the case for πn(X), but rather whose boundary is m times a cycle S n−1→X. This is precisely what a map M(Zm, n − 1)→X is, if we choose M(Zm, n − 1) to be S n−1 with a cell en attached by a map of degree m. Besides the calculation πn(S n; Zm) ≈ Zm, the proposition also yields an isomorphism πn(M(Zm, n); Zm) ≈ Ext(Zm, Zm) = Zm. Both these results are
in fact special cases of a Hurewicz-type theorem relating πn(X; Zm) and Hn(X; Zm), which is proved in [Neisendorfer 1980]. Along with Z and Zm, another extremely useful coeﬃcient group for homology and cohomology is Q. We pointed out above the diﬃculty that there is no cohomology analog of M(Q, n). The groups µn(X; Q) are also problematic. For example the proposition gives µn−1(S n; Q) ≈ Ext(Q, Z), which is a somewhat complicated uncountable group as we showed in §3.F. However, there is an alternative approach that turns out to work rather well. One deﬁnes rational homotopy groups simply as πn(X) ⊗ Q, analogous to the isomorphism Hn(X; Q) ≈ Hn(X; Z) ⊗ Q from the universal coeﬃcient theorem for homology. See [SSAT] for more on this. Homology Decompositions Eckmann–Hilton duality can be extremely helpful as an organizational principle, reducing signiﬁcantly what one has to remember, and providing valuable hints on how to proceed in various situations. To illustrate, let us consider what would happen if we dualized the notion of a Postnikov tower of principal ﬁbrations, where a space is represented as an inverse limit of a sequence of ﬁbers of maps to Eilenberg–MacLane spaces. In the dual representation, a space would be realized as a direct limit of a sequence of coﬁbers of maps from Moore spaces. In more detail, suppose we are given a sequence of abelian groups Gn, n ≥ 1, and we build a CW complex X with Hn(X) ≈ Gn for all n by constructing inductively Eckmann–Hilton Duality Section 4.H 465 an increasing sequence of subcomplexes X1 ⊂ X2 ⊂ ··· with Hi(Xn) ≈ Gi for i ≤ n and Hi(Xn) = 0 for i > n, where: (1) X1 is a Moore space M(G1, 1). (2) Xn+1 is the mapping cone of a cellular map
hn : M(Gn+1, n)→Xn such that the induced map hn∗ : Hn (3) X = n Xn. M(Gn+1, n) →Hn(Xn) is trivial. S One sees inductively that Xn+1 has the desired homology groups by comparing the long exact sequences of the pairs (Xn+1, Xn) and (CM, M) where M = M(Gn+1, n) and CM is the cone M × I/M × {0} : The assumption that hn∗ is trivial means that the boundary map in the upper row is zero, hence Hn+1(Xn+1) ≈ Gn+1. The other homology groups of Xn+1 are the same as those of Xn since Hi(Xn+1, Xn) ≈ Hi(CM, M) for all i by excision, and Hi(CM, M) ≈ Hi−1(M) since CM is contractible. In case all the maps hn are trivial, X is the wedge sum of the Moore spaces M(Gn, n) since in this case the mapping cone construction in (2) produces a wedge sum with the suspension of M(Gn+1, n), a Moore space M(Gn+1, n + 1). e For a space Y, a homotopy equivalence f : X→Y where X is constructed as in (1)–(3) is called a homology decomposition of Y. Theorem 4H.3. Every simply-connected CW complex has a homology decomposition. Proof: Given a simply-connected CW complex Y, let Gn = Hn(Y ). Suppose inductively that we have constructed Xn via maps hi as in (2), together with a map f : Xn→Y inducing an isomorphism on Hi for i ≤ n. The induction can start with X1 a point since Y is simply-connected. To construct Xn+1 we ﬁrst replace Y by the mapping cylinder of f : Xn→Y, converting f into an inclusion. By the Hurewicz theorem and the homology exact sequence of the pair (Y, Xn) we have πn+1(Y, Xn) ≈ Hn+1(Y, Xn) �
� Hn+1(Y ) = Gn+1. We will use this isomorphism to construct a map hn : M(Gn+1, n)→Xn and an extension f : Xn+1→Y. The standard construction of an M(Gn+1, n) consists of a wedge of spheres α corresponding to generators gα of Gn+1, with cells en+1 S n attached according to certain linear combinations rβ = α nαβgα that are zero in Gn+1. Under the isomorphism Gn+1 ≈ πn+1(Y, Xn) each gα corresponds to a basepoint-preserving map fα : (CS n, S n)→(Y, Xn) where CS n is the cone on S n. The restrictions of these fα ’s α→Xn, and the maps fα : CS n→Y themselves give an extento S n deﬁne hn : sion of f : Xn→Y to the mapping cone of hn : α→Xn. Each relation rβ gives a homotopy Fβ : (CS n, S n)× I→(Y, Xn) from α nαβfα to the constant map. We use 466 Chapter 4 Homotopy Theory \|\| S n × {0} to attach en+1 Fβ extension of f over the cone on en+1, and then Fβ β. β \|\| S n × I gives hn on en+1 β and Fβ gives an This construction assures that f∗ : Hn+1(Xn+1, Xn)→Hn+1(Y, Xn) is an isomorphism, so from the ﬁve-lemma applied to the long exact sequences of these pairs we deduce that f∗ : Hi(Xn+1)→Hi(Y ) is an isomorphism for i ≤ n + 1. This ﬁnishes the induction step. We may assume the maps fα and Fβ are cellular, so X = n Xn is a CW complex with subcomplexes Xn. Since f : X→Y is a homology isomorphism ⊔⊓ between simply-connected CW complexes, it is a homotopy equivalence. S As an example, suppose that Y is a simply-connected
CW complex having all its homology groups free. Then the Moore spaces used in the construction of X can be taken to be wedges of spheres, and so Xn is obtained from Xn−1 by attaching an n cell for each Z summand of Hn(Y ). The attaching maps may be taken to be cellular, making X into a CW complex whose cellular chain complex has trivial boundary maps. Similarly, ﬁnite cyclic summands of Hn(Y ) can be realized by wedge summands of the form S n−1 ∪ en in M(Hn(Y ), n − 1), contributing an n cell and an (n + 1) cell to X. This is Proposition 4C.1, but the present result is stronger because it tells us that a ﬁnite cyclic summand of Hn can be realized in one step by attaching the cone on a Moore space M(Zk, n − 1), rather than in two steps of attaching an n cell and then an (n + 1) cell. Exercises 1. Show that if A֓X is a coﬁbration of compact Hausdorﬀ spaces, then for any space Y, the map Y X→Y A obtained by restriction of functions is a ﬁbration. [If A ֓ X is a coﬁbration, so is A× Y ֓ X × Y for any space Y.] 2. Consider a pushout diagram as at the right, where B ⊔f X is B with X attached along A via f. Show that if A ֓ X is a coﬁbration, so is B ֓ B ⊔f X. 3. For ﬁbrations E1→B and E2→B, show that a ﬁber-preserving map E1→E2 that [This is dual to is a homotopy equivalence is in fact a ﬁber homotopy equivalence. Proposition 0.19.] 4. Deﬁne the dual of an iterated mapping cylinder precisely, in terms of maps from n, and use this to give a deﬁnition of ∇X, the dual of X, for X a complex of spaces. ∆ ∆ It sometimes happens that suspending a space has the eﬀect of simplifying its
homotopy type, as the suspension becomes homotopy equivalent to a wedge sum of Σ Stable Splittings of Spaces Section 4.I 467 smaller spaces. Much of the interest in such stable splittings comes from the fact that they provide a geometric explanation for algebraic splittings of homology and cohomology groups, as well as other algebraic invariants of spaces that are unaﬀected by suspension such as the cohomology operations studied in §4.L. (S 1 × S 1) is homotopy equivalent to S 2 ∨S 2 ∨S 3 since The simplest example of a stable splitting occurs for the torus S 1 × S 1. Here the (S 1 × S 1) reduced suspension is S 2 ∨ S 2 with a 3 cell attached by the suspension of the attaching map of the 2 cell of the torus, but the latter attaching map is the commutator of the two inclusions S 1 ֓ S 1 ∨ S 1, and the suspension of this commutator is trivial since it lies in the abelian group π2(S 2 ∨ S 2). Σ Σ By an easy geometric argument we will prove more generally: Proposition 4I.1. If X and Y are CW complexes, then (X × Y ) ≃ X∨ Y ∨ (X∧Y ). For example, (Sm × S n) ≃ Sm+1 ∨ S n+1 ∨ Sm+n+1. In view of the cup product structure on H ∗(Sm × S n) there can be no such splitting of Sm × S n before suspension. Σ Σ Σ Σ Proof: Consider the join X ∗ Y deﬁned in Chapter 0, consisting of all line segments joining points in X to points in Y. For our present purposes it is convenient to use the reduced version of the join, obtained by collapsing to a point the line segment joining the basepoints x0 ∈ X and y0 ∈ Y. We will still denote this reduced join by X ∗ Y. Consider the space obtained from X ∗ Y by attaching reduced cones CX and CY to the copies of X and Y at the two ends of X ∗ Y. If we col- lapse each of these cones to a point, we get the reduced suspension (X × Y ). Since each cone is
contractible, collapsing the cones gives a homotopy equivalence (X × Y ). Inside X ∗ Y there are also cones x0 ∗ Y and X ∗ y0 (X ∧ Y ) and X ∗ Y ∪ CX ∪ CY ≃ intersecting in a point. Collapsing these cones converts X ∗ Y into Σ Σ X ∗ Y ∪ CX ∪ CY into ( ⊔⊓ This result can be applied inductively to obtain splittings for suspensions of prod- Σ Σ Σ ucts of more than two spaces, using the fact that reduced suspension is smash product with S 1, and smash product is associative and commutative. For example, (X × Y × Z) ≃ X ∨ Y ∨ Z ∨ (X ∧ Y ) ∨ (X ∧ Z) ∨ (Y ∧ Z) ∨ (X ∧ Y ∧ Z) Σ Σ Σ Σ Σ Our next example involves the reduced product J(X) deﬁned in §3.2. An interesting case is J(S n), which has a CW structure of the form S n ∪ e2n ∪ e3n ∪ ···. All the cells ein for i > 1 are attached nontrivially since H ∗(J(S n); Q) is a polynomial ring Q[x] for n even and a tensor product Q[x] ⊗ Q[y] for n odd. However, after we J(S n), it is a rather surprising fact that all the attaching maps become Λ suspend to Σ Σ Σ trivial: Σ 468 Chapter 4 Homotopy Theory Proposition 4I.2. connected CW complex then J(S n) ≃ S n+1 ∨ S 2n+1 ∨ S 3n+1 ∨ ···. More generally, if X is a X ∧n where X ∧n denotes the smash product J(X) ≃ of n copies of X. Σ n W Σ Proof: The space J(X) is the union of an increasing sequence of subcomplexes Jk(X) with Jk(X) a quotient of the k fold product X × k. The quotient Jk(X)/Jk−1
(X) is X ∧k. Thus we have maps Σ X × k -→ Jk(X) -→ X ∧k = Jk(X)/Jk−1(X) By repeated application of the preceding proposition, X ∧k is a wedge summand of X × k, up to homotopy equivalence. The proof shows moreover that there is a map X ∧k→ X ∧k is homotopic to the Σ identity. This composition factors as Σ X × k such that the composition Σ X × k→ X ∧k→ Σ X ∧k -→ Σ X × k -→ Σ Jk(X) -→ Σ X ∧k so we obtain a map sk : topic to the identity. X ∧k→ Σ Jk(X) such that Σ Σ X ∧k sk-----→ Σ Jk(X)→ X ∧k is homo- Σ Σ Σ Σ Σ Jk(X), for the pair ( Jk−1(X)). Hence the map i ∨ sk : The map sk induces a splitting of the long exact sequence of homology groups Jk(X) induces an isomorphism on homology, where i denotes the inclusion map. It follows by induction that the map Jn(X) induces an isomorphism on homology for all ﬁnite n. This implies the corresponding statement for n = ∞ since X ∧n is (n − 1) connected if X is connected. Thus we have a map inducing an isomorphism on homology. By Whitehead’s theorem this map is a homo- Jk−1(X) ∨ X ∧k→ X ∧k→ X ∧k→ n k=1sk : J(X) n k= topy equivalence since the spaces are simply-connected CW complexes. Σ ⊔⊓ For our ﬁnal example the stable splitting will be constructed using the group structure on h X, Y i, the set of basepointed homotopy classes of maps X→Y. Σ Proposition 4I.3. For any prime power pn the suspension Σ K(Zpn, 1) is homotopy H∗(Xi; Z) equivalent to a wedge sum X1∨··
·∨Xp−1 where Xi is a CW complex having nonzero only in dimensions congruent to 2i mod 2p − 2. Σ e This result is best possible in a strong sense: No matter how many times any one of the spaces Xi is suspended, it never becomes homotopy equivalent to a nontrivial wedge sum. This will be shown in Example 4L.3 by studying cohomology operations in H ∗(K(Zpn, 1); Zp). There is also a somewhat simpler K–theoretic explanation for this; see [VBKT]. Proof: Let K = K(Zpn, 1). The multiplicative group of nonzero elements in the ﬁeld Zp is cyclic, so let the integer r represent a generator. By Proposition 1B.9 there is a map f : K→K inducing multiplication by r on π1(K). We will need to know that f induces multiplication by r i on H2i−1(K; Z) ≈ Zpn, and this can be seen as follows. Via Stable Splittings of Spaces Section 4.I 469 the natural isomorphism π1(K) ≈ H1(K; Z) we know that f induces multiplication by r on H1(K; Z). Via the universal coeﬃcient theorem, f also induces multiplication by r on H 1(K; Zpn ) and H 2(K; Zpn ). The cup product structure in H ∗(K; Zpn ) computed in Examples 3.41 and 3E.2 then implies that f induces multiplication by r i on H 2i−1(K; Zpn ), so the same is true for H2i−1(K; Z) by another application of the universal coeﬃcient theorem. K→ f − r j11, so hj inFor each integer j ≥ 0 let hj : K be the diﬀerence duces multiplication by r i − r j on H2i( K; Z) ≈ Zpn. By the choice of r we know Σ that p divides r i − r j iﬀ i ≡ j mod p − 1. This means that the map induced by hj Σ K; Z) has nontrivial kernel iﬀ i ≡ j mod p − 1. Therefore the
Some of these summands occur more than once, as we see in the case of Z2 × Z2. Σ K(Z2, 1) ∧ K(Z2, 1) Σ Σ Exercises 1. If a connected CW complex X retracts onto a subcomplex A, show that X ≃ (X/A). [One approach: Show the map A ∨ (X/A) induces an isomorphism on homology, where r : X→A is the retraction and q : X→X/A is the Σ quotient map.] → 470 Chapter 4 Homotopy Theory K(Zn, 1) 2. Using the K¨unneth formula, show that if m and n are relatively prime. Thus to determine stable splittings of K(Zn, 1) it suﬃces to do the case that n is a prime power, as in Proposition 4I.3. K(Zm × Zn, 1) ≃ K(Zm, 1) ∨ Σ Σ Σ 3. Extending Proposition 4I.3, show that the (2k + 1) skeleton of the suspension of a high-dimensional lens space with fundamental group of order pn is homotopy equivalent to the wedge sum of the (2k + 1) skeleta of the spaces Xi, if these Xi ’s are chosen to have the minimum number of cells in each dimension, as described in the remarks following the proof. Loopspaces appear at ﬁrst glance to be hopelessly complicated objects, but if one is only interested in homotopy type, there are many cases when great simpliﬁcations are possible. One of the nicest of these cases is the loopspace of a sphere. We show S n+1 has the weak homotopy type of the James reduced product X has the weak homotopy in this section that J(S n) introduced in §3.2. More generally, we show that type of J(X) for every connected CW complex X. If one wants, one can strengthen Ω ‘weak homotopy type’ to ‘homotopy type’ by quoting Milnor’s theorem, mentioned in §4.3, that the loopspace of a CW complex has the homotopy type of a CW complex. Part of the interest in X can
be attributed to its close connection with the sus- ΩΣ pension homomorphism πi(X)→πi+1( lence of ΩΣ X with J(X) to give another proof that the suspension homomorphism is X). We will use the weak homotopy equiva- ΩΣ an isomorphism in dimensions up to approximately double the connectivity of X. In addition, we will obtain an exact sequence that measures the failure of the suspension map to be an isomorphism in dimensions between double and triple the connectivity of X. An easy application of this, together with results proved elsewhere in the book, will be to compute πn+1(S n) and πn+2(S n) for all n. As a rough ﬁrst approximation to S n+1 obtained by regarding S n+1 as the reduced suspension S n+1 there is a natural inclusion of S n into S n, the quotient (S n × I)/(S n × ∂I∪{e}× I) where e is the basepoint of S n, then Ω associating to each point x ∈ S n the loop λ(x) in S n given by t ֏ (x, t). The ﬁgure shows what a few such loops look like. However, we cannot expect this inclusion S n ֓ S n+1 S n+1 is an H–space but to be a homotopy equivalence since S n is only an H–space when n = 1, 3, 7 by the theorem of Adams discussed in §4.B. The simplest way to correct this deﬁciency in S n would be to replace it by the free H–space that it generates, the reduced product J(S n). Re- Ω Ω Σ Σ Σ Ω The Loopspace of a Suspension Section 4.J 471 call from §3.2 that a point in J(S n) is a formal product x1 ··· xk of points xi ∈ S n, with the basepoint e acting as an identity element for the multiplication obtained by juxtaposition of formal products. We would like to deﬁne a map λ : J(S n)→ S n+1 by setting λ(x
1 ··· xk) = λ(x1) ··· λ(xk), the product of the loops λ(xi). The only diﬃculty is in the parametrization of this product, which needs to be adjusted so that λ is continuous. The problem is that when some xi approaches the basepoint e ∈ S n, one wants the loop λ(xi) to disappear gradually from the product λ(x1) ··· λ(xk), without disrupting the parametrization as simply deleting λ(e) would do. This can be achieved by ﬁrst making the time it takes to traverse each loop λ(xi) equal to the distance from xi to the basepoint of S n, then normalizing the resulting product of loops so that it takes unit time, giving a map I→ S n. Ω More generally, this same procedure deﬁnes a map λ : J(X)→ X for any con- nected CW complex X, where ‘distance to the basepoint’ is replaced by any map d : X→[0, 1] with d−1(0) = e, the basepoint of X. ΩΣ Σ Theorem 4J.1. The map λ : J(X)→ connected CW complex X. ΩΣ X is a weak homotopy equivalence for every Proof: The main step will be to compute the homology of it will be easy to deduce that λ induces an isomorphism on homology using the cal- X. After this is done, ΩΣ ΩΣ culation of the homology of J(X) in Proposition 3C.8, and from this conclude that λ is a weak homotopy equivalence. It will turn out to be suﬃcient to consider homology with coeﬃcients in a ﬁeld F. We know that H∗(J(X); F ) is the tensor algebra X; F ) has this same T H∗(X; F ) by Proposition 3C.8, so we want to show that H∗( structure, a result ﬁrst proved in [Bott & Samelson 1953]. e Let us write the reduced suspension Y =
X as the union of two reduced cones Σ Σ Ω Y+ = C+X and Y− = C−X intersecting in the equatorial X ⊂ X. Consider the path ﬁbration p : P Y →Y with ﬁber Y. Let P+Y = p−1(Y+) and P−Y = p−1(Y−), so P+Y consists of paths in Y starting at the basepoint and ending in Y+, and similarly for P−Y. Then P+Y ∩ P−Y is p−1(X), the paths from the basepoint to X. Since Y+ and Y− are deformation retracts of open neighborhoods U+ and U− in Y such that U+ ∩ U− deformation retracts onto Y+ ∩ Y− = X, the homotopy lifting property implies that P+Y, P−Y, and p+Y ∩ P−Y are deformation retracts, in the weak sense, of open neighborhoods p−1(U+), p−1(U−), and p−1(U+) ∩ p−1(U−), respectively. Therefore there is a Mayer–Vietoris sequence in homology for the decomposition of P Y as P+Y ∪P−Y. Taking reduced homology and using the fact that P Y is contractible, this gives an isomorphism (i) : H∗(P+Y ∩ P−Y ; F ) ≈------------→ H∗(P+Y ; F ) H∗(P−Y ; F ) The two coordinates of Φ e L are induced by the inclusions, with a minus sign in one case, e e but will still be an isomorphism if this minus sign is eliminated, so we may assume this has been done. Φ Φ 472 Chapter 4 Homotopy Theory We claim that the isomorphism can be rewritten as an isomorphism (ii) : H∗( Y × X; F ) ≈------------→ Φ H∗( Y ; F ) H∗( To see this, we observe that the ﬁbration P+Y →Y+ is ﬁber-homotopically trivial. This e is true since the cone Y+ is contractible, but we shall need an explicit ﬁ
ber homoY × Y+, and this is easily constructed as follows. Deﬁne topy equivalence P+Y ≃ f+ : P+Y → y is the obvious path in Y+ from y = (x, t) to the basepoint along the segment {x}× I. In the other y where the bar denotes the direction, deﬁne g+ : inverse path. Then f+g+ and g+f+ are ﬁber-homotopic to the respective identity maps since γ + Y × Y+ by f+(γ) = (γ γ+ Ω Y × Y+→P+Y by g+(γ, y) = γ γ + y, y) where y = γ(1) and γ+ y are homotopic to the constant paths. Ω y and γ+ y γ + y γ+ Ω In similar fashion the ﬁbration P−Y →Y− is ﬁber-homotopically trivial via maps f− and g−. By restricting a ﬁber-homotopy trivialization of either P+Y or P−Y to P+Y ∩ P−Y, we see that the ﬁbration P+Y ∩ P−Y is ﬁber-homotopy equivalent to the Y × X. Let us do this using the ﬁber-homotopy trivialization of P−Y. The product has coordinates induced groups in (i) can now be replaced by those in (ii). The map by inclusion, and it follows that the corresponding map Ω in (ii) has coordinates Y, (γ, x) ֏ γ λ(x) and (γ, x) ֏ γ. Namely, Θ Y, Y × X followed by projection to Φ is induced by f+g− \|\| the ﬁrst coordinate of Ω Ω and the second coordinate is the same but with f−g− in place of f+g−. Writing the two coordinates of as Θ Ω 1 and means that the restriction of K¨unneth formula we can write 1 to the kernel of Θ Y × X; F ) as Θ Θ where projection onto the latter summand is H∗(, the fact that is
an isomorphism 2 is an isomorphism. Via the gives an isomorphism L e Ω 1(γ, x) = Y ; F ). Since H∗(X; F ) H∗( Θ H∗( Θ 2. Hence Ω H∗( e from the ﬁrst summand H∗( (γ λ(x)), this means that the composed map H∗(X; F ) onto Θ Θ H∗( Y ; F ) ⊗ H∗(X; F ) e Ω 11 ⊗ λ∗ -----------------------------→ H∗( e Ω Θ Y ; F ) ⊗ H∗( Y ; F ) --------→ H∗( Y ; F ) induced by the two maps Y × X ) as the tensor algebra T Y ; F ), V = with the second map Pontryagin product, is an isomorphism. Now to ﬁnish the calcuΩ Ω H∗(X; F ), we apply the following algebraic lation of H∗( lemma, with A = H∗( H∗(X; F ), and i = λ∗. e Lemma 4J.2. Let A be a graded algebra over a ﬁeld F with A0 = F and let V be a graded vector space over F with V0 = 0. Suppose we have a linear map i : V→A preserving grading, such that the multiplication map µ : A ⊗ V→ A, µ(a ⊗ v) = ai(v), is an isomorphism. Then the canonical algebra homomorphism i : T V→A extending the previous i is an isomorphism. Ω e e For example, if V is a 1 dimensional vector space over F, as happens in the case X = S n, then this says that if the map A→ A given by right-multiplication by an element a = i(v) is an isomorphism, then A is the polynomial algebra F [a]. The e The Loopspace of a Suspension Section 4.J 473 general case can be viewed as the natural generalization of this to polynomials in any number of noncommuting variables. j aj
⊗ vj Proof: Since µ is an isomorphism, each element a ∈ An with n > 0 can be written j aji(vj ) for vj ∈ V and aj ∈ An(j), with uniquely in the form µ n(j) < n since V0 = 0. By induction on n, aj = i(αj) for a unique αj ∈ (T V )n(j). P Thus a = i so i is surjective. Since these representations are unique, i is also injective. The induction starts with the hypothesis that A0 = F, the scalars ⊔⊓ in T V. j αj ⊗ vj P P = Returning now to the proof of the theorem, we observe that λ is an H–map: The two maps J(X)× J(X)→ X, (x, y) ֏ λ(xy) and (x, y) ֏ λ(x)λ(y), are homotopic since the loops λ(xy) and λ(x)λ(y) diﬀer only in their parametrizations. Since λ is an H–map, the maps X ֓ J(X) duce the commutative diagram at the right. We X in- λ-----→ ΩΣ have shown that the downward map on the right ΩΣ is an isomorphism, and the same is true of the one on the left by the calculation of H∗(J(X); F ) in Proposition 3C.8. Hence λ∗ is an isomorphism. By Corollary 3A.7 this is also true for Z coeﬃcients. When X is simply-connected, so are J(X) and X, so after taking a CW approximation to X, Whitehead’s theorem implies that λ is a weak homotopy equivalence. In the general case that X is only connected, we obtain the same conclusion from the generalization ΩΣ ΩΣ of Whitehead’s theorem to abelian spaces, Proposition 4.74, since J(X) and H–spaces, with trivial action of π1 on all homotopy groups by Example 4A.3. X are
⊔⊓ ΩΣ Using the natural identiﬁcation πi( induces the suspension map πi(X)→πi+1( ΩΣ J(X), we can identify the relative groups πi( n connected then the pair (J(X), X) is (2n + 1) connected since we can replace X by a complex with n skeleton a point, and then the (2n + 1) skeleton of J(X) is X). Since this inclusion factors through X, X) with πi(J(X), X). ΩΣ If X is X) = πi+1( X), the inclusion X ֓ ΩΣ X Σ Σ contained in X. Thus we have: Corollary 4J.3. The suspension map πi(X)→πi+1( complex X is an isomorphism if i ≤ 2n and a surjection if i = 2n + 1. X) for an n connected CW ⊔⊓ Σ In the case of a sphere we can describe what happens in the ﬁrst dimension when suspension is not an isomorphism, namely the suspension π2n−1(S n)→π2n(S n+1) which the corollary guarantees only to be a surjection. The CW structure on J(S n) consists of a single cell in each dimension a multiple of n, so from exactness of -----→ π2n(S n+1) we see that the kernel of the suspension π2n(J(S n), S n) π2n−1(S n)→π2n(S n+1) is generated by the attaching map of the 2n cell of J(S n). This attaching map is the Whitehead product [11, 11], as we noted in §4.2 when we ∂-----→ π2n−1(S n) Σ 474 Chapter 4 Homotopy Theory deﬁned Whitehead products following Example 4.52. When n is even, the Hopf invariant homomorphism π2n−1(S n)→Z has the value ±2 on [11, 11], as we saw in §4.B. If
there is no map of Hopf invariant ±1, it follows that [11, 11] generates a Z summand of π2n−1(S n), and so the suspension homomorphism simply cancels this summand from π2n−1(S n). By Adams’ theorem, this is the situation for all even n except 2, 4, and 8. When n = 2 we have π3(S 2) ≈ Z generated by the Hopf map η with Hopf invari- ant 1, so 2η = ±[11, 11], generating the kernel of the suspension, hence: Corollary 4J.4. πn+1(S n) is Z2 for n ≥ 3, generated by the suspension or iterated ⊔⊓ suspension of the Hopf map. The situation for n = 4 and 8 is more subtle. We do not have the tools available here to do the actual calculation, but if we consult the table near the beginning of §4.1 we see that the suspension π7(S 4)→π8(S 5) is a map Z⊕ Z12→Z24. By our preceding remarks we know this map is surjective with kernel generated by the single element [11, 11]. Algebraically, what must be happening is that the coordinate of [11, 11] in the Z summand is twice a generator, while the coordinate in the Z12 summand is a generator. Thus a generator of the Z summand, which we may take to be the Hopf map S 7→S 4, suspends to a generator of the Z24. For n = 8 the situation is entirely similar, with the suspension π15(S 8)→π16(S 9) a homomorphism Z⊕ Z120→Z240. We can also obtain some information about suspension somewhat beyond the edge of the stable dimension range. Since S n is (n − 1) connected and (J(S n), S n) is (2n − 1) connected, we have isomorphisms πi(J(S n), S n) ≈ πi(J(S n)/S n) for i ≤ 3n − 2 by Proposition 4.28. The group πi(J(S n)/S n) is isomorphic to πi(S 2
n) in the same range i ≤ 3n − 2 since J(S n)/S n has S 2n as its (3n − 1) skeleton. Thus the terminal portion of the long exact sequence of the pair (J(S n), S n) starting with the term π3n−2(S n) can be written in the form π3n−2(S n) Σ-----→ π3n−1(S n+1)→π3n−2(S 2n)→π3n−3(S n) Σ-----→ π3n−2(S n+1)→ ··· This is known as the EHP sequence since its three maps were originally called E, H, and P. (The German word for ‘suspension’ begins with E, the H refers to a gen- eralization of the Hopf invariant, and the P denotes a connection with Whitehead products; see [Whitehead 1978] for more details.) Note that the terms πi(S 2n) in the EHP sequence are stable homotopy groups since i ≤ 3n − 2. Thus we have the curi- ous situation that stable homotopy groups are measuring the lack of stability of the groups πi(S n) in the range 2n − 1 ≤ i ≤ 3n − 2, the so-called metastable range. Specializing to the ﬁrst interesting case n = 2, the sequence becomes The Dold–Thom Theorem Section 4.K 475 From the Hopf bundle S 1→S 3→S 2 we have π4(S 2) ≈ π4(S 3) ≈ Z2, with π4(S 2) genη) where η is the Hopf map S 3→S 2. From exactness erated by the composition η( of the latter part of the sequence we deduce that the map π4(S 4)→π3(S 2) is injective, and hence that the suspension π4(S 2)→π5(S 3) is surjective, so π5(S 3) is either Z2 or 0. From the general suspension theorem, the suspension π5(S 3)→π6(S 4) is surjective as well, and the latter group is in the
stable range. We show in Proposition 4L.11 2 is nonzero, and so we conclude that πn+2(S n) ≈ Z2 for all that the stable group π s n ≥ 2, generated by the composition ( n−2η)( We will see in [SSAT] that the EHP sequence extends all the way to the left to form n−1η). Σ an inﬁnite exact sequence when n is odd, and when n is even a weaker statement holds: The sequence extends after factoring out all odd torsion. Replacing S n by any (n − 1) connected CW complex X, our derivation of the ﬁnite EHP sequence generalizes immediately to give an exact sequence π3n−2(X) Σ-----→ π3n−1( X)→π3n−2(X ∧ X)→π3n−3(X) Σ-----→ π3n−2( X)→ ··· using the fact that J2(X)/X = X ∧ X. Σ Σ The generalization of the results of this section to nX turns out to be of some importance in homotopy theory. In case we do not get to this topic in [SSAT], n the reader can begin to learn about it by looking at [Carlsson & Milgram 1995]. Ω Σ Σ Σ Exercise 1. Show that X for a nonconnected CW complex X reduces to the connected case α Xα X is homotopy equivalent to by showing that each path-component of where the Xα ’s are the components of X. ΩΣ ΩΣ W ΩΣ In the preceding section we studied the free monoid J(X) generated by a space X, and in this section we take up its commutative analog, the free abelian monoid generated by X. This is the inﬁnite symmetric product SP (X) introduced brieﬂy in §3.C. The main result will be a theorem of [Dold & Thom 1958] asserting that π∗SP (X) ≈ In particular this yields the surprising fact that SP (S n) is a K(Z, n), and more generally that the functor SP takes Moore spaces M(G
, n) to Eilenberg–MacLane spaces K(G, n). This leads to the H∗(X; Z) for all connected CW complexes X. e general result that for all connected CW complexes X, SP (X) has the homotopy type of a product of Eilenberg–MacLane spaces. In other words, the k invariants of SP (X) are all trivial. 476 Chapter 4 Homotopy Theory The main step in the proof of the Dold–Thom theorem will be to show that the homotopy groups π∗SP (X) deﬁne a homology theory. An easy computation of the coeﬃcient groups π∗SP (S n) will then show that this must be ordinary homology with Z coeﬃcients. A new idea needed for the proof of the main step is the notion of a quasiﬁbration, generalizing ﬁbrations and ﬁber bundles. In order to establish a few basic facts about quasiﬁbrations we ﬁrst make a small detour to prove an essentially elementary fact about relative homotopy groups. A Mayer–Vietoris Property of Homotopy Groups In this subsection we will be concerned largely with relative homotopy groups, and it will be impossible to avoid the awkward fact that there is no really good way to deﬁne the relative π0. What we will do as a compromise is to take π0(X, A, x0) to be the quotient set π0(X, x0)/π0(A, x0). This at least allows the long exact sequence of homotopy groups for (X, A) to end with the terms π0(A, x0)→π0(X, x0)→π0(X, A, x0)→0 An exercise for §4.1 shows that the ﬁve-lemma can be applied to the map of long exact sequences induced by a map (X, A)→(Y, B), provided the basepoint is allowed to vary. However, the long exact sequence of a triple cannot be extended through the π0 terms with this deﬁnition, so one must proceed with some caution. The excision theorem for homology involves a space X with sub
spaces A and B such that X is the union of the interiors of A and B. In this situation we call (X; A, B) an excisive triad. By a map f : (X; A, B)→(Y ; C, D) we mean f : X→Y with f (A) ⊂ C and f (B) ⊂ D. Proposition 4K.1. Let f : (X; A, B)→(Y ; C, D) be a map of excisive triads. If the induced maps πi(A, A ∩ B)→πi(C, C ∩ D) and πi(B, A ∩ B)→πi(D, C ∩ D) are bijections for i < n and surjections for i = n, for all choices of basepoints, then the same is true of the induced maps πi(X, A)→πi(Y, C). By symmetry the conclusion holds also for the maps πi(X, B)→πi(Y, D). The corresponding statement for homology is a trivial consequence of excision which says that Hi(X, A) ≈ Hi(B, A ∩ B) and Hi(Y, C) ≈ Hi(D, C ∩ D), so it is not necessary to assume anything about the map Hi(A, A ∩ B)→Hi(C, C ∩ D). With the failure of excision for homotopy groups, however, it is not surprising that the assumption on πi(A, A ∩ B)→πi(C, C ∩ D) cannot be dropped. An example is provided by the quotient map f : D2→S 2 collapsing ∂D2 to the north pole of S 2, with C and D the northern and southern hemispheres of S 2, and A and B their preimages under f. Proof: First we will establish a general fact about relative homotopy groups. Consider an inclusion (X, A) ֓ (Y, C). We will show the following three conditions are equivalent for each n ≥ 1 : The Dold–Thom Theorem Section 4.K 477 (i) For all choices of basepoints the map πi(X, A)
→πi(Y, C) induced by inclusion is surjective for i = n and has trivial kernel for i = n − 1. (ii) Let ∂Dn be written as the union of hemispheres ∂+Dn and ∂−Dn intersecting in S n−2. Then every map (Dn × {0} ∪ ∂+Dn × I, ∂−Dn × {0} ∪ S n−2 × I) -→ (Y, C) taking (∂+Dn × {1}, S n−2 × {1}) to (X, A) extends to a map (Dn × I, ∂−Dn × I)→(Y, C) taking (Dn × {1}, ∂−Dn × {1}) to (X, A). (iii) Condition (ii) with the added hypothesis that the restriction of the given map to ∂+Dn × I is independent of the I coordinate. It is obvious that (ii) and (iii) are equivalent since the stronger hypothesis in (iii) can always be achieved by composing with a homotopy of Dn × I that shrinks ∂+Dn × I to ∂+Dn × {1}. To see that (iii) implies (i), let f : (∂+Dn × {1}, S n−2× {1})→(X, A) represent an element of πn−1(X, A). If this is in the kernel of the map to πn−1(Y, C), then we get an extension of f over Dn × {0} ∪ ∂+Dn × I, with the constant homotopy on ∂+Dn × I and (Dn × {0}, ∂−Dn × {0}) mapping to (Y, C). Condition (iii) then gives an extension over Dn × I, whose restriction to Dn × {1} shows that f is zero in πn−1(X, A), so the kernel of πn−1(X, A)→πn−1(Y, C) is trivial. To check surjectivity of the map πn(X, A)→πn(Y, C), represent an element of πn
(Y, C) by a map f : Dn × {0}→Y taking ∂−Dn × {0} to C and ∂+Dn × {0} to a chosen basepoint. Extend f over ∂+Dn × I via the constant homotopy, then extend over Dn × I by applying (iii). The result is a homotopy of the given f to a map representing an element of the image of πn(X, A)→πn(Y, C). Now we show that (i) implies (ii). Given a map f as in the hypothesis of (ii), the injectivity part of (i) gives an extension of f over Dn × {1}. Choose a small disk En ⊂ ∂−Dn × I, shown shaded in the ﬁgure, intersecting ∂−Dn × {1} in a hemisphere ∂+En of its boundary. We may assume the extended f has a constant value x0 ∈ A on ∂+En. Viewing the extended f as representing an element of πn(Y, C, x0), the surjectivity part of (i) then gives an extension of f over Dn × I taking (En, ∂−En) to (X, A) and the rest of ∂−Dn × I to C. The argument is ﬁnished by composing this extended f with a deformation of Dn × I pushing En into Dn × {1}. Having shown the equivalence of (i)–(iii), let us prove the proposition. We may reduce to the case that the given f : (X; A, B)→(Y ; C, D) is an inclusion by using mapping cylinders. One’s ﬁrst guess would be to replace (Y ; C, D) by the triad of mapping cylinders (Mf ; Mf \|A, Mf \|B), where we view f \|\| A as a map A→C and f \|\| B as a map B→D. However, the triad (Mf ; Mf \|A, Mf \|B) need not be excisive, for example if X 478 Chapter 4 Homotopy Theory consists of two points A and B and Y is a single point. To remedy this problem, replace Mf \|A
by its union with f −1(C)× (1/2, 1) in Mf, and enlarge Mf \|B similarly. Now we prove the proposition for an inclusion (X; A, B) ֓ (Y ; C, D). The case n = 0 is trivial from the deﬁnitions, so let us assume n ≥ 1. In view of the equivalence of condition (i) with (ii) and (iii), it suﬃces to show that condition (ii) for the inclusions (A, A ∩ B) ֓ (C, C ∩ D) and (B, A ∩ B) ֓ (D, C ∩ D) implies (iii) for the inclusion (X, A) ֓ (Y, C). Let a map f : Dn × {0} ∪ ∂+Dn × I→Y as in the hypothesis of (iii) be given. The argument will involve subdivision of Dn into smaller disks, and for this it is more convenient to use the cube In instead of Dn, so let us identify In with Dn in such a way that ∂−Dn corresponds to the face In−1 × {1}, which we denote by ∂−In, and ∂+Dn corresponds to the remaining faces of In, which we denote by ∂+In. Thus we are given f on In × {0} taking ∂+In × {0} to X and ∂−In × {0} to C, and on ∂+In × I we have the constant homotopy. Since (Y ; C, D) is an excisive triad, we can subdivide each of the I factors of In × {0} into subintervals so that f takes each of the resulting n dimensional subcubes of In × {0} into either C or D. The extension of f we construct will have the following key property: If K is a one of the subcubes of In × {0}, or a lower-dimensional face of such a cube, then the extension of f takes (K × I, K × {1}) to (C, A) or (D, B) when- (∗) ever f takes K to C or D, respectively. Initially we have f deﬁned on ∂+In × I
with image in X, independent of the I coordinate, and we may assume the condition (∗) holds here since we may assume that A = X ∩ C and B = X ∩ D, these conditions holding for the mapping cylinder construction described above. Consider the problem of extending f over K × I for K one of the subcubes. We may assume that f has already been extended to ∂+K × I so that (∗) is satisﬁed, by induction on n and on the sequence of subintervals of the last coordinate of In × {0}. To extend f over K × I, let us ﬁrst deal with the cases that the given f takes (K, ∂−K) to (C, C ∩ D) or (D, C ∩ D). Then by (ii) for the inclusion (A, A ∩ B)֓ (C, C ∩ D) or (B, A ∩ B) ֓ (D, C ∩ D) we may extend f over K × I so that (∗) is still satisﬁed. If neither of these two cases applies, then the given f takes (K, ∂−K) just to (C, C) or (D, D), and we can apply (ii) trivially to ⊔⊓ construct the desired extension of f over K × I. Corollary 4K.2. Given a map f : X→Y and open covers {Ui} of X and {Vi} of Y with f (Ui) ⊂ Vi for all i, then if each restriction f : Ui→Vi and more generally each is a weak homotopy equivalence, so is f : X→Y. f : Ui1 ∩ ··· ∩ Uin→Vi1 ∩ ··· ∩ Vin Proof: First let us do the case of covers by two sets. By the ﬁve-lemma, the hypotheses imply that πn(Ui, U1 ∩ U2)→πn(Vi, V1 ∩ V2) is bijective for i = 1, 2, n ≥ 0, The Dold–Thom Theorem Section 4.K 479 and all choices of basepoints. The preceding proposition then implies that the maps πn(X
, U1)→πn(Y, V1) are isomorphisms. Hence by the ﬁve-lemma again, so are the maps πn(X)→πn(Y ). By induction, the case of ﬁnite covers by k > 2 sets reduces to the case of covers by two sets, by letting one of the two sets be the union of the ﬁrst k − 1 of the given sets and the other be the k th set. The case of inﬁnite covers reduces to the ﬁnite case since for surjectivity of πn(X)→πn(Y ), a map S n→Y has compact image covered ⊔⊓ by ﬁnitely many Vi ’s, and similarly for injectivity. Quasiﬁbrations A map p : E→B with B path-connected is a quasiﬁbration if the induced map p∗ : πi(E, p−1(b), x0)→πi(B, b) is an isomorphism for all b ∈ B, x0 ∈ p−1(b), and i ≥ 0. We have shown in Theorem 4.41 that ﬁber bundles and ﬁbrations have this property for i > 0, as a consequence of the homotopy lifting property, and the same reasoning applies for i = 0 since we assume B is path-connected. For example, consider the natural projection Mf →I of the mapping cylinder of a map f : X→Y. This projection will be a quasiﬁbration iﬀ f is a weak homotopy equivalence, since the lat= 0 = πi(I, b) ter condition is equivalent to having πi for all i and all b ∈ I. Note that if f is not surjective, there are paths in I that do not lift to paths in Mf with a prescribed starting point, so p will not be a ﬁbration in such cases. Mf, p−1(b) An alternative condition for a map p : E→B to be a quasiﬁbration is that the inclusion of each ﬁber p−1(b) into the homotopy ﬁber Fb of p over b is
a weak homotopy equivalence. Recall that Fb is the space of all pairs (x, γ) with x ∈ E and γ a path in B from p(x) to b. The actual ﬁber p−1(b) is included in Fb as the pairs with x ∈ p−1(b) and γ the constant path at x. To see the equivalence of the two deﬁnitions, consider the commutative triangle at the right, where Fb→Ep→B is the usual path-ﬁbration construction applied to p. The right-hand map in the diagram is an isomorphism for all i ≥ 0, and the upper map will be an isomorphism for all i ≥ 0 iﬀ the inclusion p−1(b) ֓ Fb is a weak equivalence since E ≃ Ep. Hence the two deﬁnitions are equivalent. Recall from Proposition 4.61 that all ﬁbers of a ﬁbration over a path-connected base are homotopy equivalent. Since we are only considering quasiﬁbrations over path-connected base spaces, this implies that all the ﬁbers of a quasiﬁbration have the same weak homotopy type. Quasiﬁbrations over a base that is not path-connected are considered in the exercises, but we will not need this generality in what follows. The following technical lemma gives various conditions for recognizing that a map is a quasiﬁbration, which will be needed in the proof of the Dold–Thom theorem. F 1(b) 480 Chapter 4 Homotopy Theory Lemma 4K.3. A map p : E→B is a quasiﬁbration if any one of the following conditions is satisﬁed : (a) B can be decomposed as the union of open sets V1 and V2 such that each of the restrictions p−1(V1)→V1, p−1(V2)→V2, and p−1(V1 ∩ V2)→V1 ∩ V2 is a quasiﬁbration. (b) B is the union of an increasing sequence of subspaces B1 ⊂ B2 ⊂ ··· with the property that each compact set in B lies
in some Bn, and such that each restriction p−1(Bn)→Bn is a quasiﬁbration. (c) There is a deformation Ft of E into a subspace E′, covering a deformation F t of B into a subspace B′, such that the restriction E′→B′ is a quasiﬁbration and F1 : p−1(b)→p−1 is a weak homotopy equivalence for each b ∈ B. By a ‘deformation’ in (c) we mean a deformation retraction in the weak sense as deﬁned in the exercises for Chapter 0, where the homotopy is not required to be the identity on the subspace. Proof: (a) To avoid some tedious details we will consider only the case that the ﬁbers of p are path-connected, which will suﬃce for our present purposes, leaving the general case as an exercise for the reader. This hypothesis on ﬁbers guarantees that all π0 ’s arising in the proof are trivial. In particular, by an exercise for §4.1 this allows us to terminate long exact sequences of homotopy groups of triples with zeros in the π0 positions. E, p−1(b) Uk, U1 ∩ U2, p−1(b) Let U1 = p−1(V1) and U2 = p−1(V2). The ﬁve-lemma for the long exact seand (Vk, V1 ∩ V2, b) quences of homotopy groups of the triples implies that the maps πi(Uk, U1 ∩ U2)→πi(Vk, V1 ∩ V2) are isomorphisms for k = 1, 2 and all i. Then Proposition 4K.1 implies that the maps πi(E, Uk)→πi(B, Vk) are iso→πi(Vk, b) are morphisms for all choices of basepoints. The maps πi isomorphisms by hypothesis, so from the ﬁve-lemma we can then deduce that the →πi(B, b) are isomorphisms for all b ∈ Vk, hence for all b ∈ B. maps π
i (b) Since each compact set in B lies in some Bn, each compact set in E lies in some subspace En = p−1(Bn), so πi En, p−1(b) just →πi(B, b) is an as πi(B, b) = lim--→πi(Bn, b). →πi(Bn, b) is an isomorphism isomorphism since each of the maps πi by assumption. We can take the point b to be an arbitrary point in B and then discard any initial spaces Bn in the sequence that do not contain b, so we can assume b lies in Bn for all n. is the direct limit lim--→πi E, p−1(b) En, p−1(b) Uk, p−1(b) It follows that the map πi E, p−1(b) (c) Consider the commutative diagram The Dold–Thom Theorem Section 4.K 481 where b is an arbitrary point in B. The upper map in the diagram is an isomorphism by the ﬁve-lemma since the hypotheses imply that F1 induces isomorphisms πi(E)→πi(E′) and πi(p−1(b))→πi p−1(F 1(b)) for all i. The hypotheses also imply that the lower map and the right-hand map are isomorphisms. Hence the left-hand map is an isomorphism. Symmetric Products ⊔⊓ Let us recall the deﬁnition from §3.C. For a space X the n fold symmetric product SPn(X) is the quotient space of the product of n copies of X obtained by factoring out the action of the symmetric group permuting the factors. A choice of basepoint e ∈ X gives inclusions SPn(X)֓SPn+1(X) induced by (x1, ···, xn)֏(x1, ···, xn, e), and SP (X) is deﬁned to be the union of this increasing sequence of spaces, with the direct limit topology. Note that SPn is a homotopy functor: A map f : X→Y induces f�
� : SPn(X)→SPn(Y ), and f ≃ g implies f∗ ≃ g∗. Hence X ≃ Y implies SPn(X) ≃ SPn(Y ). In similar fashion SP is a homotopy functor on the category of basepointed spaces and basepoint-preserving homotopy classes of maps. It follows that X ≃ Y implies SP (X) ≃ SP (Y ) for connected CW complexes X and Y since in this case requiring maps and homotopies to preserve basepoints does not aﬀect the relation of homotopy equivalence. Example 4K.4. An interesting special case is when X = S 2 because in this case SP (S 2) can be identiﬁed with CP∞ in the following way. We ﬁrst identify CPn with the nonzero polynomials of degree at most n with coeﬃcients in C, modulo scalar multiplication, by letting a0 + ··· + anzn correspond to the line containing (a0, ···, an). The sphere S 2 we view as C ∪ {∞}, and then we deﬁne f : (S 2)n→CPn by setting f (a1, ···, an) = (z + a1) ··· (z + an) with factors z + ∞ omitted, so in particular f (∞, ···, ∞) = 1. To check that f is continuous, suppose some ai approaches ∞, say an, and all the other aj ’s are ﬁnite. Then if we write (z + a1) ··· (z + an) = zn + (a1 + ··· + an)zn−1 + ··· + ai1 ··· aik zn−k + ··· + a1 ··· an Xi1<···<ik we see that, dividing through by an and letting an approach ∞, this polynomial approaches zn−1 + (a1 + ··· + an−1)zn−2 + ··· + a1 ··· an−1 = (z + a1) ··· (z + an−1). The same argument would apply if several ai ’s approach ∞ simultaneously. The
value f (a1, ···, an) is unchanged under permutation of the ai ’s, so there is an induced map SPn(S 2)→CPn which is a continuous bijection, hence a homeomorphism since both spaces are compact Hausdorﬀ. Letting n go to ∞, we then get a homeomorphism SP (S 2) ≈ CP∞. The same argument can be used to show that SPn(S 1) ≃ S 1 for all n, including n = ∞. Namely, the argument shows that SPn(C − {0}) can be identiﬁed with the 482 Chapter 4 Homotopy Theory polynomials zn + an−1zn−1 + ··· + a0 with a0 ≠ 0, or in other words, the n tuples (a0, ···, an−1) ∈ Cn with a0 ≠ 0, and this subspace of Cn deformation retracts onto a circle. The symmetric products of higher-dimensional spheres are more complicated, though things are not so bad for the 2 fold symmetric product: Example 4K.5. Let us show that SP2(S n) is homeomorphic to the mapping cone of a map S nRPn−1→S n where S nRPn−1 is the n fold unreduced suspension of RPn−1. H∗(S n+1RPn−1) from the long exact sequence of hoHence H∗(SP2(S n)) ≈ H∗(S n)⊕ mology groups for the pair (SP2(S n), S n), since SP2(S n)/S n is S n+1RPn−1 with no reduced homology below dimension n + 2. e If we view S n as Dn/∂Dn, then SP2(S n) becomes a certain quotient of Dn × Dn. Viewing Dn × Dn as the cone on its boundary Dn × ∂Dn ∪∂Dn × Dn, the identiﬁcations that produce SP2(S n) respect the various concentric copies of this boundary which ﬁll up the interior of Dn × Dn, so it suﬃces to analyze the ident
iﬁcations in all these copies of the boundary. The identiﬁcations on the boundary of Dn × Dn itself yield S n. This is clear since the identiﬁcation (x, y) ∼ (y, x) converts Dn × ∂Dn ∪∂Dn × Dn to Dn × ∂Dn, and all points of ∂Dn are identiﬁed in S n. It remains to see that the identiﬁcations (x, y) ∼ (y, x) on each concentric copy of the boundary in the interior of Dn × Dn produce S nRPn−1. Denote by Z the quotient of Dn × ∂Dn ∪ ∂Dn × Dn under these identiﬁcations. This is the same as the quotient of Dn × ∂Dn under the identiﬁcations (x, y) ∼ (y, x) for (x, y) ∈ ∂Dn × ∂Dn. Deﬁne f : Dn × RPn−1→Z by f (x, L) = (w, z) where x is equidistant from z ∈ ∂Dn and w ∈ Dn along the line through x parallel to L, as in the ﬁgure. If x is the midpoint of the segment zz′ then w = z′ and there is no way to distinguish between w and z, but since f takes values in the quotient space Z, this is not a problem. If x ∈ ∂Dn then w = z = x, independent of L. If x ∈ Dn − ∂Dn then w ≠ z, and conversely, given (w, z) ∈ Dn × ∂Dn with w ≠ z there is a unique (x, L) with f (x, L) = (w, z), namely x is the midpoint of the segment wz and L is the line parallel to this segment. In view of these remarks, we see that Z is the quotient space of Dn × RPn−1 under the identiﬁcations (x, L) ∼ (x, L′) if x ∈ ∂Dn.
This quotient is precisely S nRPn−1. This example illustrates that passing from a CW structure on X to a CW structure on SPn(X) or SP (X) is not at all straightforward. However, if X is a simplicial comcomplex structures on SPn(X) and SP (X), plex, there is a natural way of putting as follows. A simplicial complex structure on X gives a CW structure on the prod- uct of n copies of X, with cells n fold products of simplices. Such a product has ∆ a canonical barycentric subdivision as a simplicial complex, with vertices the points whose coordinates are barycenters of simplices of X. By induction over skeleta, this ∆ ∆ The Dold–Thom Theorem Section 4.K 483 just amounts to coning oﬀ a simplicial structure on the boundary of each product cell. This simplicial structure on the product of n copies of X is in fact a complex structure since the vertices of each of its simplices have a natural ordering given by the dimensions of the cells of which they are barycenters. The action of the symmet- ric group permuting coordinates respects this complex structure, taking simplices homeomorphically to simplices, preserving vertex-orderings, so there is an induced complex structure on the quotient SPn(X). With the basepoint of X chosen to complex be a vertex, SPn(X) is a subcomplex of SPn+1(X) so there is a natural ∆ structure on the inﬁnite symmetric product SP (X) as well. ∆ As usual with products, the CW topology on SPn(X) and SP (X) is in general diﬀerent from the topology arising from the original deﬁnition in terms of product topologies, but one can check that the two topologies have the same compact sets, so the distinction will not matter for our present purposes. For deﬁniteness, we will use the CW topology in what follows, which means restricting X to be a simplicial complex. Since every CW complex is homotopy equivalent to a simplicial complex by Theorem 2C.5, and SPn and SP are homotopy functors, there is no essential loss of generality in restricting from CW complexes to simplicial complexes. Here is the main
result of this section, the Dold–Thom theorem: Theorem 4K.6. The functor X ֏ πiSP (X) for i ≥ 1 coincides with the functor X ֏ Hi(X; Z) on the category of basepointed connected CW complexes. In particular this says that SP (S n) is a K(Z, n), and more generally that for a Moore space M(G, n), SP (M(G, n)) is a K(G, n). The fact that SP (X) is a commutative, associative H–space with a strict identity element limits its weak homotopy type considerably: Corollary 4K.7. A path-connected, commutative, associative H–space with a strict identity element has the weak homotopy type of a product of Eilenberg–MacLane spaces. In particular, if X is a connected CW complex then SP (X) is path-connected and n K(Hn(X), n). Thus the functor SP essentially has the weak homotopy type of reduces to Eilenberg–MacLane spaces. Q Proof: Let X be a path-connected, commutative, associative H–space with a strict identity element, and let Gn = πn(X). By Lemma 4.31 there is a map M(Gn, n)→X inducing an isomorphism on πn when n > 1 and an isomorphism on H1 when n = 1. We can take these maps to be basepoint-preserving, and then they comn M(Gn, n)→X. The very special H–space structure on X alIn general, W α SP (Xα) where this is the ‘weak’ inﬁnite product, the union of the ﬁnite products. This, together with the general fact that the map lows us to extend this to a homomorphism f : SP ( α Xα) can be identiﬁed with Gn, n) bine to give a map →X. n M SP ( W Q W 484 Chapter 4 Homotopy Theory πi(X)→πiSP (X) = Hi(X; Z) induced by the inclusion X = SP1(X) ֓ SP (X)
is the Hurewicz homomorphism, as we will see at the end of the proof of the Dold–Thom the- orem, implies that the map f induces an isomorphism on all homotopy groups. Thus →X, and as we noted above, n SP has only one we have a weak homotopy equivalence is a K(Gn, n). Finally, since each factor SP M(Gn, n) SP Q nontrivial homotopy group, the weak inﬁnite product has the same weak homotopy M(Gn, n) M(Gn, n) type as the ordinary inﬁnite product. ⊔⊓ The main step in the proof of the theorem will be to show that for a simplicial pair (X, A) with both X and A connected, there is a long exact sequence ··· →πiSP (A)→πiSP (X)→πiSP (X/A)→πi−1SP (A)→ ··· This would follow if the maps SP (A)→SP (X)→SP (X/A) formed a ﬁber bundle or ﬁbration. There is some reason to think this might be true, because all the ﬁbers of the projection SP (X)→SP (X/A) are homeomorphic to SP (A). In fact, in terms of the H–space structure on SP (X) as the free abelian monoid generated by X, the ﬁbers are exactly the cosets of the submonoid SP (A). The projection SP (X)→SP (X/A), however, fails to have the homotopy lifting property, even the special case of lifting paths. For if xt, t ∈ [0, 1), is a path in X − A approaching a point x1 = a ∈ A other than the basepoint, then regarding xt as a path in SP (X/A), any lift to SP (X) would have the form xtαt, αt ∈ SP (A), ending at x1α1 = aα1, a point of SP (A) which is a multiple of a, so in particular there would be no lift ending at the basepoint of SP (X). What
we will show is that the projection SP (X)→SP (X/A) has instead the weaker structure of a quasiﬁbration, which is still good enough to deduce a long exact se- quence of homotopy groups. Proof of 4K.6: As we have said, the main step will be to associate a long exact sequence of homotopy groups to each simplicial pair (X, A) with X and A connected. This will be the long exact sequence of homotopy groups coming from the quasiﬁbration SP (A)→SP (X)→SP (X/A), so the major work will be in verifying the quasiﬁbration property. Since SP is a homotopy functor, we are free to replace (X, A) by a homotopy equivalent pair, so let us replace (X, A) by (M, A) where M is the mapping cylinder of the inclusion A ֓ X. This new pair, which we still call (X, A), has some slight technical advantages, as we will see later in the proof. To begin the proof that the projection p : SP (X)→SP (X/A) is a quasiﬁbration, let Bn = SPn(X/A) and En = p−1(Bn). Thus En consists of those points in SP (X) having at most n coordinates in X − A. By Lemma 4K.3(b) it suﬃces to show that p : En→Bn is a quasiﬁbration. The proof of the latter fact will be by induction on n, starting with the trivial case n = 0 when B0 is a point. The induction step will consist of showing that p is a quasiﬁbration over a neighborhood of Bn−1 and over The Dold–Thom Theorem Section 4.K 485 Bn − Bn−1, then applying Lemma 4K.3(a). We ﬁrst tackle the problem of showing the quasiﬁbration property over a neighborhood of Bn−1. Let ft : X→X be a homotopy of the identity map deformation retracting a neighborhood N of A onto A. Since we have replaced the original X by the mapping cylinder of the inclusion A ֓ X, we can take ft
simply to slide points along the segments {a}× I in the mapping cylinder, with N = A× [0, 1/2). Let U ⊂ En consist of those points having at least one coordinate in N, or in other words, products with at least one factor in N. Thus U is a neighborhood of En−1 in En and p(U) is a neighborhood of Bn−1 in Bn. The homotopy ft induces a homotopy Ft : En→En whose restriction to U is a deformation of U into En−1, where by ‘deformation’ we mean deformation retraction in the weak sense. Since ft is the identity on A, Ft is the lift of a homotopy F t : Bn→Bn which restricts to a deformation of U = p(U) into Bn−1. We will deduce that the projection U→U is a quasiﬁbration by using Lemma 4K.3(c). To apply this to the case at hand we need to verify that F1 : p−1(b)→p−1 is a weak equivalence for all b. Each point w ∈ p−1(b) is a commuting product of points in X. Let be the subproduct whose factors are points in X − A, so we have w = wv for v a c product of points in A. Since f1 is the identity on A and F1 is a homomorphism, w)v ′v with v ′ also a product we have F1(w) = F1( w and let v vary over SP (A), we get all points of p−1(b) of points in A. If we ﬁx Æ c c exactly once, or in other words, we have p−1(b) expressed as the coset wSP (A). c w)v ′v, takes this coset to the coset The map F1, w)SP (A) by a map that would be a homeomorphism if the factor v ′ were not present. Since A is conÆ c t from v ′ to the basepoint, and so by replacing v ′ with v ′ nected, there is a path v ′ t in the product to a w)v ′v we obtain a homotopy from F1 : p
−1(b)→p−1 w)v, which can be written F 1(b) wv ֏ Æ c F1( F1( F1( F1( c c c w homeomorphism, so this map is a homotopy equivalence, as desired. F 1(b) It remains to see that p is a quasiﬁbration over Bn−Bn−1 and over the intersection of this set with U. The argument will be the same in both cases. c En − En−1, p−1(b) Identifying Bn − Bn−1 with SPn(X − A), the projection p : En − En−1→Bn − Bn−1 is the same as the operator w ֏ w. The inclusion SPn(X − A) ֓ En − En−1 gives a section for p : En − En−1→Bn − Bn−1, so p∗ : πi →πi(Bn − Bn−1, b) is surjective. To see that it is also injective, represent an element of its kernel by a map g : (Di, ∂Di)→ En − En−1, p−1(b). A nullhomotopy of pg gives a homotopy of g changing only its coordinates in X − A. This homotopy is through maps (Di, ∂Di)→, and ends with a map to p−1(b), so the kernel of En − En−1, p−1(b) p∗ is trivial. Thus the projection En − En−1→Bn − Bn−1 is a quasiﬁbration, at least if Bn − Bn−1 is path-connected. But by replacing the original X with the mapping cylinder of the inclusion A ֓ X, we guarantee that X − A is path-connected since it deformation retracts onto X. Hence the space Bn − Bn−1 = SPn(X − A) is also path-connected. Æ c 486 Chapter 4 Homotopy Theory This argument works equally well over any open subset of Bn − Bn−1 that is pathconnected, in particular over U ∩ (Bn − Bn−1), so via Lemma 4K.3(a
) this ﬁnishes the proof that SP (A)→SP (X)→SP (X/A) is a quasiﬁbration. Since the homotopy axiom is obvious, this gives us the ﬁrst two of the three axioms needed for the groups hi(X) = πiSP (X) to deﬁne a reduced homology theory. There remains only the wedge sum axiom, hi( α hi(Xα), but this is imα Xα) = mediate from the evident fact that SP ( α SP (Xα), where this is the ‘weak’ W α Xα) ≈ L product, the union of the products of ﬁnitely many factors. Q W The homology theory h∗(X) is deﬁned on the category of connected, basepointed simplicial complexes, with basepoint-preserving maps. The coeﬃcients of this homology theory, the groups hi(S n), are the same as for ordinary homology with Z coeﬃcients since we know this is true for n = 2 by the homeomorphism SP (S 2) ≈ CP∞, and there are isomorphisms hi(X) ≈ hi+1( X) in any reduced homology theory. If the homology theory h∗(X) were deﬁned on the category of all simplicial complexes, without basepoints, then Theorem 4.59 would give natural isomorphisms hi(X) ≈ Hi(X; Z) for all X, and the proof would be complete. However, it is easy to achieve this by deﬁning a new homology theory h′ X), since the suspension of an arbitrary complex is connected and the suspension of an arbitrary map i(X) = hi+1( Σ is basepoint-preserving, taking the basepoint to be one of the suspension points. Since h′ i(X) is naturally isomorphic to hi(X) if X is connected, we are done. ⊔⊓ Σ It is worth noting that the map πi(X)→πiSP (X) = Hi(X; Z) induced by the inclusion X = SP1(X) ֓ SP (X) is
the Hurewicz homomorphism. For by deﬁnition of the Hurewicz homomorphism and naturality this reduces to the case X = S i, where the map SP1(S i) ֓ SP (S i) induces on πi a homomorphism Z→Z, which one just has to check is an isomorphism, the Hurewicz homomorphism being determined only up to sign. The suspension isomorphism gives a further reduction to the case i = 1, where the inclusion SP1(S 1) ֓ SP (S 1) is a homotopy equivalence, hence induces an isomorphism on π1. Exercises 1. Show that Corollary 4K.2 remains valid when X and Y are CW complexes and the subspaces Ui and Vi are subcomplexes rather than open sets. 2. Show that a simplicial map f : K→L is a homotopy equivalence if f −1(x) is contractible for all x ∈ L. [Consider the cover of L by open stars of simplices and the cover of K by the preimages of these open stars.] n. 3. Show that SPn(I) = 4. Show that SP2(S 1) is a M¨obius band, and that this is consistent with the description of SP2(S n) as a mapping cone given in Example 4K.5. ∆ Steenrod Squares and Powers Section 4.L 487 5. A map p : E→B with B not necessarily path-connected is deﬁned to be a quasiﬁbration if the following equivalent conditions are satisﬁed: (i) For all b ∈ B and x0 ∈ p−1(b), the map p∗ : πi(E, p−1(b), x0)→πi(B, b) is an isomorphism for i > 0 and π0(p−1(b), x0)→π0(E, x0)→π0(B, b) is exact. (ii) The inclusion of the ﬁber p−1(b) into the homotopy ﬁber Fb of p over b is a weak homotopy equivalence for all b ∈ B. (iii
) The restriction of p over each path-component of B is a quasiﬁbration according to the deﬁnition in this section. Show these three conditions are equivalent, and prove Lemma 4K.3 for quasiﬁbrations over non-pathconnected base spaces. 6. Let X be a complex of spaces over a complex, as deﬁned in §4.G. Show that the natural projection is a quasiﬁbration if all the maps in X associated to X→ are weak homotopy equivalences. ∆ ∆ Γ Γ edges of Γ The main objects of study in this section are certain homomorphisms called Steen- rod squares and Steenrod powers: Sqi : H n(X; Z2)→H n+i(X; Z2) P i : H n(X; Zp)→H n+2i(p−1)(X; Zp) for odd primes p The terms ‘squares’ and ‘powers’ arise from the fact that Sqi and P i are related to the maps α֏ α2 and α֏ αp sending a cohomology class α to the 2 fold or p fold cup product with itself. Unlike cup products, however, the operations Sqi and P i are stable, that is, invariant under suspension. The operations Sqi generate an algebra A 2, called the Steenrod algebra, such that H ∗(X; Z2) is a module over A 2 for every space X, and maps between spaces induce 2 modules. Similarly, for odd primes p, H ∗(X; Zp) is a module homomorphisms of A over a corresponding Steenrod algebra A p generated by the P i ’s and Bockstein homomorphisms. Like the ring structure given by cup product, these module structures impose strong constraints on spaces and maps. For example, we will use them to show that there do not exist spaces X with H ∗(X; Z) a polynomial ring Z[α] unless α has dimension 2 or 4, where there are the familiar examples of CP∞ and HP∞. This rather lengthy section is divided into two main parts. The ﬁrst part describes the basic properties of Steenrod squares and powers and gives a number of examples and applications. The second part is devoted to
constructing the squares and powers and showing they satisfy the basic properties listed in the ﬁrst part. More extensive 488 Chapter 4 Homotopy Theory applications will be given in [SSAT] after spectral sequences have been introduced. Most applications of Steenrod squares and powers do not depend on how these op- erations are actually constructed, but only on their basic properties. This is similar to the situation for ordinary homology and cohomology, where the axioms generally suﬃce for most applications. The construction of Steenrod squares and powers and the veriﬁcation of their basic properties, or axioms, is rather interesting in its own way, but does involve a certain amount of work, particularly for the Steenrod powers, and this is why we delay the work until later in the section. We begin with a few generalities. A cohomology operation is a transformation X : Hm(X; G)→H n(X; H) deﬁned for all spaces X, with a ﬁxed choice of m, n, = Θ G, and H, and satisfying the naturality property that Θ for all maps f : X→Y there is a commuting diagram as shown at the right. For example, with coeﬃcients in a ring R the transformation Hm(X; R)→Hmp(X; R), α֏ αp, is a cohomology operation since f ∗(αp) = (f ∗(α))p. Taking R = Z, this example shows that cohomology operations need not be homomorphisms. On the other hand, when R = Zp with p prime, the operation α ֏ αp is a homomorphism. Other examples of cohomology operations we have already encountered are the Bockstein homomorphisms deﬁned in §3.E. As a more trivial example, a homomorphism G→H induces change-of-coeﬃcient homomorphisms Hm(X; G)→Hm(X; H) which can be viewed as cohomology operations. In spite of their rather general deﬁnition, cohomology operations can be described in somewhat more concrete terms: Proposition 4L.1. For ﬁxed m, n, G, and H there is a bijection between the
set of : Hm(X; G)→H n(X; H) and H n(K(G, m); H), deﬁned all cohomology operations by ֏ (ι) where ι ∈ Hm(K(G, m); G) is a fundamental class. Θ Θ Θ Proof: Via CW approximations to spaces, it suﬃces to restrict attention to CW complexes, so we can identify Hm(X; G) with hX, K(G, m)i when m > 0 by Theorem 4.57, and with [X, K(G, 0)] when m = 0. If an element α ∈ Hm(X; G) corresponds to a map ϕ : X→K(G, m), so ϕ∗(ι) = α, then is ֏ (ι) is injective. For surjectivity, given uniquely determined by an element α ∈ H n(K(G, m); H) corresponding to a map θ : K(G, m)→K(H, n), then composing with θ deﬁnes a transformation hX, K(G, m)i→hX, K(H, n)i, that amounts is, (ι) = α. The naturality property for (ϕ∗(ι)) = ϕ∗( (ι). Thus (ι)) and (α) = Θ Θ Θ Θ Θ Θ Θ : Hm(X; G)→H n(X; H), with to associativity of the compositions X Θ f-----→ Y ϕ-----→ K(G, m) θ-----→ K(H, n) and so Θ is a ⊔⊓ Θ Θ cohomology operation. A consequence of the proposition is that cohomology operations that decrease dimension are all rather trivial since K(G, m) is (m − 1) connected. Moreover, since Σ Steenrod Squares and Powers Section 4.L 489 Hm(K(G, m); H) ≈ Hom(G, H), it follows that the only cohomology operations that preserve dimension are given by coeﬃcient homomorphisms. The Steenrod squares Sqi
: H n(X; Z2)→H n+i(X; Z2), i ≥ 0, will satisfy the fol- lowing list of properties, beginning with naturality: (1) Sqi(f ∗(α)) = f ∗(Sqi(α)) for f : X→Y. (2) Sqi(α + β) = Sqi(α) + Sqi(β). (3) Sqi(α ` β) = (4) Sqi(σ (α)) = σ (Sqi(α)) where σ : H n(X; Z2)→H n+1( j Sqj (α) ` Sqi−j (β) (the Cartan formula). P X; Z2) is the suspension isomorphism given by reduced cross product with a generator of H 1(S 1; Z2). (5) Sqi(α) = α2 if i = \|α\|, and Sqi(α) = 0 if i > \|α\|. (6) Sq0 = 11, the identity. (7) Sq1 is the Z2 Bockstein homomorphism β associated with the coeﬃcient se- quence 0→Z2→Z4→Z2→0. The ﬁrst part of (5) says that the Steenrod squares extend the squaring operation α ֏ α2, which has the nice feature of being a homomorphism with Z2 coeﬃcients. Property (4) says that the Sqi ’s are stable operations, invariant under suspension. The actual squaring operation α ֏ α2 does not have this property since in a suspension X all cup products of positive-dimensional classes are zero, according to an exercise for §3.2. Σ The fact that Steenrod squares are stable operations extending the cup product square yields the following theorem, which implies that the stable homotopy groups of spheres π s 7 are nontrivial: 3, and π s 1, π s Theorem 4L.2. If f : S 2n−1→S n has Hopf invariant 1, then [f ] ∈ π s so the iterated suspensions kf : S 2n+k−1→S n+k are all homotopically nontriv
ial. n−1 is nonzero, Proof: Associated to a map f : S ℓ→Sm is the mapping cone Cf = Sm ∪f eℓ+1 with the cell eℓ+1 attached via f. Assuming f is basepoint-preserving, we have the relation C denotes reduced suspension. Σ f = Cf where Σ Σ If f : S 2n−1→S n has Hopf invariant 1, then by (5), Sqn : H n(Cf ; Z2)→H 2n(Cf ; Z2) kCf ; Z2) for all kCf →S n+k. The Σ is nontrivial. By (4) the same is true for Sqn : H n+k( k. If kf were homotopically trivial we would have a retraction r : kCf ; Z2)→H 2n+k( Σ Σ Σ diagram at the right would then commute by naturality of Sqn, but since the group in the lower left corner of the diagram is zero, this Σ gives a contradiction. ⊔⊓ The Steenrod power operations P i : H n(X; Zp)→H n+2i(p−1)(X; Zp) for p an odd prime will satisfy analogous properties: (1) P i(f ∗(α)) = f ∗(P i(α)) for f : X→Y. 490 Chapter 4 Homotopy Theory (2) P i(α + β) = P i(α) + P i(β). (3) P i(α ` β) = (4) P i(σ (α)) = σ (P i(α)) where σ : H n(X; Zp)→H n+1( j P j(α) ` P i−j(β) (the Cartan formula). P X; Zp) is the suspension iso- morphism given by reduced cross product with a generator of H 1(S 1; Zp). (5) P i(α) = αp if 2i = \|α\|, and P i(α) = 0 if 2i > \|α\|. (6) P 0 = 11, the identity. Σ The germinal property P
i(α) = αp in (5) can only be expected to hold for evendimensional classes α since for odd-dimensional α the commutativity property of cup product implies that α2 = 0 with Zp coeﬃcients if p is odd, and then αp = 0 since α2 = 0. Note that the formula P i(α) = αp for \|α\| = 2i implies that P i raises dimension by 2i(p − 1), explaining the appearance of this number. The Bockstein homomorphism β : H n(X; Zp)→H n+1(X; Zp) is not included as one of the P i ’s, but this is mainly a matter of notational convenience. As we shall see later when we discuss Adem relations, the operation Sq2i+1 is the same as the composition Sq1Sq2i = βSq2i, so the Sq2i ’s can be regarded as the P i ’s for p = 2. One might ask if there are elements of π s ∗ detectable by Steenrod powers in the same way that the Hopf maps are detected by Steenrod squares. The answer is yes for the operation P 1, as we show in Example 4L.6. It is a perhaps disappointing fact that no other squares or powers besides Sq1, Sq2, Sq4, Sq8, and P 1 detect elements of homotopy groups of spheres. ( Sq1 detects a map S n→S n of degree 2.) We will prove this for certain Sqi ’s and P i ’s later in this section. The general case for p = 2 is Adams’ theorem on the Hopf invariant discussed in §4.B, while the case of odd p is proved in [Adams & Atiyah 1966]; see also [VBKT]. The Cartan formulas can be expressed in a more concise form by deﬁning total Steenrod square and power operations by Sq = Sq0 + Sq1 + ··· and P = P 0 + P 1 + ···. These act on H ∗(X; Zp) since by property (5), only a ﬁnite number of Sqi ’s or P i ’s can be nonzero on a given cohomology class.
The Cartan formulas then say that Sq(α ` β) = Sq(α) ` Sq(β) and P (α ` β) = P (α) ` P (β), so Sq and P are ring homomorphisms. We can use Sq and P to compute the operations Sqi and P i for projective spaces and lens spaces via the following general formulas: (∗) Sqi(αn) = P i(αn) = n i αn+i for α ∈ H 1(X; Z2) αn+i(p−1) for α ∈ H 2(X; Zp) n i To derive the ﬁrst formula, properties (5) and (6) give Sq(α) = α + α2 = α(1 + α), αn+i. so Sq(αn) = Sq(α)n = αn(1 + α)n = i The second formula is obtained in similar fashion: P (α) = α + αp = α(1 + αp−1) so P αn+i(p−1). P (αn) = αn(1 + αp−1)n = αn+i and hence Sqi(αn Steenrod Squares and Powers Section 4.L 491 In Lemma 3C.6 we described how binomial coeﬃcients can be computed modulo a prime p : m n ≡ mi ni i Q mod p, where m = i mipi and n = i nipi are the p adic expansions of m and n. P P When p = 2 for example, the extreme cases of a dyadic expansion consisting of a single 1 or all 1 ’s give ) = α2k Sq(α2k Sq(α2k−1) = α2k−1 + α2k + α2k+1 + α2k+1 + ··· + α2k+1−2 for α ∈ H 1(X; Z2). More generally, the coeﬃcients of Sq(αn) can be read oﬀ from the (n + 1) st row of the mod 2 Pascal triangle, a portion of which is shown in the ﬁgure at the right, where dots denote zeros
. Example 4L.3: Stable Splittings. The formula (∗) tells us how to compute Steenrod squares for RP∞, hence also for any suspension of RP∞. The explicit formu) and Sq(α2k−1) above show that all the powers of the generator las for Sq(α2k α ∈ H 1(RP∞; Z2) are tied together by Steenrod squares since the ﬁrst formula connects α inductively to all the powers α2k and the second formula connects these kRP∞ has the hopowers to all the other powers. This shows that no suspension motopy type of a wedge sum X ∨ Y with both X and Y having nontrivial cohomology. In the case of RP∞ itself we could have deduced this from the ring structure of H ∗(RP∞; Z2) ≈ Z2[α], but cup products become trivial in a suspension. Σ The same reasoning shows that CP∞ and HP∞ have no nontrivial stable splittings. The Z2 cohomology in these cases is again Z2[α], though with α no longer 1 dimensional. However, we still have Sq(α) = α + α2 since these spaces have no nontrivial cohomology in the dimensions between α and α2, so we have Sq2i(αn) = αn+i for HP∞. Then the arguments from the n i n i real case carry over using the operations Sq2i and Sq4i in place of Sqi. αn+i for CP∞ and Sq4i(αn) = Suppose we consider the same question for K(Z3, 1) instead of RP∞. Taking cohomology with Z3 coeﬃcients, the Bockstein β is nonzero on odd-dimensional classes in H ∗(K(Z3, 1); Z3), thus tying them to the even-dimensional classes, so we only need to see which even-dimensional classes are connected by P i ’s. The evendimensional part of H ∗(K(Z3, 1); Z3) is a polynomial algebra Z3[α] with \|α\| = 2, so αn+2i by our earlier formula. Since P i raises
we have P i(αn) = dimension by 4i when p = 3, there is no chance that all the even-dimensional cohomology will be connected by the P i ’s. In fact, we showed in Proposition 4I.3 that K(Z3, 1) in dimensions congruent to 2 and 3 mod 4, while X2 has the remaining cohomology. Thus the best one Σ could hope would be that all the odd powers of α are connected by P i ’s and likewise all the even powers are connected, since this would imply that neither X1 nor X2 splits K(Z3, 1) ≃ X1 ∨ X2 where X1 has the cohomology of αn+i(p−1) = n i n i Σ 492 Chapter 4 Homotopy Theory n i nontrivially. This is indeed the case, as one sees by an examαn+2i. ination of the coeﬃcients in the formula P i(αn) = In the Pascal triangle mod 3, shown at the right, P (αn) is de- termined by the (n + 1) st row. For example the sixth row says that P (α5) = α5 + 2α7 + α9 + α11 + 2α13 + α15. A few moments’ thought shows that the rows that compute P (αn) for n = 3km − 1 have all nonzero entries, and these rows together with the rows right after them suﬃce to connect the powers of α in the desired way, so X1 and X2 have 2X1 and X2 are not homotopy equivalent, no stable splittings. One can also see that even stably, since the operations P i act diﬀerently in the two spaces. For example P 2 Σ is trivial on suspensions of α but not on suspensions of α2. The situation for K(Zp, 1) for larger primes p is entirely similar, with K(Zp, 1) splitting as a wedge sum of p − 1 spaces. The same arguments work more generally for K(Zpi, 1), though for i > 1 the usual Bockstein β is identically zero so one has to use instead a Bockstein involving Zpi coeﬃcients. We leave the details of these arguments as
exercises. Σ Example 4L.4: Maps of HP∞. We can use the operations P i together with a bit of number theory to demonstrate an interesting distinction between HP∞ and CP∞, namely, we will show that if a map f : HP∞→HP∞ has f ∗(γ) = dγ for γ a generator of H 4(HP∞; Z), then the integer d, which we call the degree of f, must be a square. By contrast, since CP∞ is a K(Z, 2), there are maps CP∞→CP∞ carrying a generator α ∈ H 2(CP∞; Z) onto any given multiple of itself. Explicitly, the map z ֏ zd, z ∈ C, induces a map f of CP∞ with f ∗(α) = dα, but commutativity of C is needed for this construction so it does not extend to the quaternionic case. n i We shall deduce the action of Steenrod powers on H ∗(HP∞; Zp) from their action on H ∗(CP∞; Zp), given by the earlier formula (∗) which says that P i(αn) = αn+i(p−1) for α a generator of H 2(CP∞; Zp). There is a natural quotient map CP∞→HP∞ arising from the deﬁnition of both spaces as quotients of S ∞. This map takes the 4 cell of CP∞ homeomorphically onto the 4 cell of HP∞, so the induced map on cohomology sends a generator γ ∈ H 4(HP∞; Zp) to α2, hence γn to α2n. γn+i(p−1)/2. 2n Thus the formula P i(α2n) = i For example, P 1(γ) = 2γ(p+1)/2. α2n+i(p−1) implies that P i(γn) = 2n i Now let f : HP∞→HP∞ be any map. Applying the formula P 1(γ) = 2γ(p+1)/2 in two ways, we get P 1f ∗(γ) = f ∗P 1
(γ) = f ∗(2γ(p+1)/2) = 2d(p+1)/2γ(p+1)/2 and P 1f ∗(γ) = P 1(dγ) = 2dγ(p+1)/2 Hence the degree d satisﬁes d(p+1)/2 ≡ d mod p for all odd primes p. Thus either d ≡ 0 mod p or d(p−1)/2 ≡ 1 mod p. In both cases d is a square mod p since the Steenrod Squares and Powers Section 4.L 493 congruence d(p−1)/2 ≡ 1 mod p is equivalent to d being a nonzero square mod p, the multiplicative group of nonzero elements of the ﬁeld Zp being cyclic of order p − 1. The argument is completed by appealing to the number theory fact that an integer which is a square mod p for all suﬃciently large primes p must be a square. This can be deduced from quadratic reciprocity and Dirichlet’s theorem on primes in arithmetic progressions as follows. Suppose on the contrary that the result is false for the integer d. Consider primes p not dividing d. Since the product of two squares in Zp is again a square, we may assume that d is a product of distinct primes q1, ···, qn, where one of these primes is allowed to be −1 if d is negative. In terms of the Legendre symbol d p which is deﬁned to be +1 if d is a square mod p and −1 otherwise, we have q1 p The left side is +1 for all large p by hypothesis, so it will suﬃce to see that p can be qn p d p ··· = chosen to give each term on the right an arbitrary preassigned value. The values of −1 p and 2 p depend only on p mod 8, and the four combinations of values are real- ized by the four residues 1, 3, 5, 7 mod 8. Having speciﬁed the value of p mod 8, the quadratic reciprocity law then says that for odd primes q, specifying is equiva- q p lent to specifying. Thus we need only choose p in the appropriate residue classes mod 8 and mod qi for each
odd qi. By the Chinese remainder theorem, this means specifying p modulo 8 times a product of odd primes. Dirichlet’s theorem guarantees p q that in fact inﬁnitely many primes p exist satisfying this congruence condition. It is known that the integers realizable as degrees of maps HP∞→HP∞ are exactly the odd squares and zero. The construction of maps of odd square degree will be given in [SSAT] using localization techniques, following [Sullivan 1974]. Ruling out nonzero even squares can be done using K–theory; see [Feder & Gitler 1978], which also treats maps HPn→HPn. The preceding calculations can also be used to show that every map HPn→HPn must have a ﬁxed point if n > 1. For, taking p = 3, the element P 1(γ) lies in H 8(HPn; Z3) which is nonzero if n > 1, so, when the earlier argument is specialized to the case p = 3, the congruence d(p+1)/2 ≡ d mod p becomes d2 = d in Z3, which is satisﬁed only by 0 and 1 in Z3. In particular, d is not equal to −1. The Lefschetz number λ(f ) = 1 + d + ··· + dn = (dn+1 − 1)/(d − 1) is therefore nonzero since the only integer roots of unity are ±1. The Lefschetz ﬁxed point theorem then gives the result. Example 4L.5: Vector Fields on Spheres. Let us now apply Steenrod squares to determine the maximum number of orthonormal tangent vector ﬁelds on a sphere in all cases except when the dimension of the sphere is congruent to −1 mod 16. The ﬁrst step is to rephrase the question in terms of Stiefel manifolds. Recall from the end of §3.D and Example 4.53 the space Vn,k of orthonormal k frames in Rn. Projection of a k frame onto its ﬁrst vector gives a map p : Vn,k→S n−1, and a section 494 Chapter 4 Homotopy Theory for this projection, that is, a map f : S n−1

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