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properties, we will use the transfer in the construction of a number of spaces whose Zp cohomology is a polynomial ring. Let π : X→X be an n sheeted covering space, for some ﬁnite n. to the induced map on singular chains π♯ : Ck( phism in the opposite direction τ : Ck(X)→Ck( k→X the sum of the n distinct lifts σ : σ : e In addition X)→Ck(X) there is also a homomorX) which assigns to a singular simplex e k→ e X. This is obviously a chain map, ∆ commuting with boundary homomorphisms, so it induces transfer homomorphisms e X; G)→H k(X; G) for any coeﬃcient group G. τ∗ : Hk(X; G)→Hk( We focus on cohomology in what follows, but similar statements hold for homology X; G) and τ ∗ : H k( ∆ e e e as well. The composition π♯τ is clearly multiplication by n, hence τ ∗π ∗ = n. This has the consequence that the kernel of π ∗ : H k(X; G)→H k( X; G) consists of torsion elements of order dividing n, since π ∗(α) = 0 implies τ ∗π ∗(α) = nα = 0. Thus the cohomology of X must be ‘larger’ than that of X except possibly for torsion of order dividing n. This can be a genuine exception as one sees from the examples of Sm covering RPm and lens spaces. More generally, if Sm→X is any n sheeted covering space, then the relation τ ∗π ∗ = n implies that H ∗(X; Z) consists entirely of torsion elements of order dividing n, apart from a possible Z in dimension m. (Since X is e e a closed manifold, its homology groups are ﬁnitely generated by Corollaries A.8 and e A.9 in the Appendix.) By studying the other composition π ∗τ ∗ we will prove: Proposition 3G.1. Let π : tion of a group e or a prime not dividing n, the map π ∗ : H
k(X; F )→H k( the subgroup H ∗( X→X be an n sheeted covering space deﬁned by an acX. Then with coeﬃcients in a ﬁeld F whose characteristic is 0 consisting of classes α such that γ∗(α) = α for all γ ∈ X; F ) is injective with image e X; F ) on Γ e Γ e. Γ 322 Chapter 3 Cohomology X; F ) Γ e Γ k→ X to the sum of all its images under the Proof: We have already seen that elements of the kernel of π ∗ have ﬁnite order dividing n, so π ∗ is injective for the coeﬃcient ﬁelds we are considering here. It remains to describe the image of π ∗. Note ﬁrst that τπ♯ sends a singular simplex γ∗(α) γ∗(α) for α ∈ H k( e ∆ equals nα, so if the coeﬃcient ﬁeld F has characteristic 0 or a prime not dividing n, we can write α = π ∗τ ∗(α/n) and thus α lies in the image of π ∗. Conversely, since, we have γ∗π ∗(α) = π ∗(α) for all α, and so the image of π ∗ π γ = π for all γ ∈ is contained in H ∗( ⊔⊓ action. Hence π ∗τ ∗(α) = X; F ), the sum X; F ). If α is ﬁxed under the action of on H k( P P γ∈ γ∈ e e Γ Γ. Γ Γ e Example 3G.2. Let, with X the n sheeted cover corresponding X is a circle with n S k ’s attached at equally to the index n subgroup of π1(X), so spaced points around the circle. The deck transformation group Zn acts by rotating the circle, permuting the S k ’s cyclically. Hence for any coeﬃcient group G, the inX; G
)Zn is all of H 0 and H 1, plus a copy of G in dimension variant cohomology H ∗( k, the cellular cohomology classes assigning the same element of G to each S k. Thus X; G)Zn is exactly the image of π ∗ for i = 0 and k, while the image of π ∗ in H i( X; G)Zn or not dedimension 1 is the subgroup nH 1( pends on G. For G = Q or Zp with p not dividing n, we have equality, but not for G = Z or Zp with p dividing n. In this last case the map π ∗ is not injective on H 1. X; G). Whether this equals H 1( e e e e e Spaces with Polynomial mod p Cohomology An interesting special case of the general problem of realizing graded commutative rings as cup product rings of spaces is the case of polynomial rings Zp[x1, ···, xn] over the coeﬃcient ﬁeld Zp, p prime. The basic question here is, which sets of numbers d1, ···, dn are realizable as the dimensions \|xi\| of the generators xi? From §3.2 we have the examples of products of CP∞ ’s and HP∞ ’s with di ’s equal to 2 or 4, for arbitrary p, and when p = 2 we can also take RP∞ ’s with di ’s equal to 1. As an application of transfer homomorphisms we will construct some examples with larger di ’s. examples realize everything that can be realized. But for two or more variables, more In the case of polynomials in one variable, it turns out that these sophisticated techniques are necessary to realize all the realizable cases; see the end of this section for further remarks on this. The construction can be outlined as follows. Start with a space Y already known to have polynomial cohomology H ∗(Y ; Zp) = Zp[y1, ···, yn], and suppose there is on Y. A simple trick called the Borel construction shows an action of a ﬁnite group that without loss of generality we may assume the action is free, deﬁning a covering space Y →
Y / H ∗(Y / that are invariant under the induced action of, ; Zp) is isomorphic to the subring of Zp[y1, ···, yn] consisting of polynomials Γ on H ∗(Y ; Zp). And in some cases. Then by Proposition 3G.1 above, if p does not divide the order of Γ Γ Γ this subring is itself a polynomial ring. Γ Transfer Homomorphisms Section 3.G 323 For example, if Y is the product of n copies of CP∞ then the symmetric group n acts on Y by permuting the factors, with the induced action on H ∗(Y ; Zp) ≈ Zp[y1, ···, yn] permuting the yi ’s. A standard theorem in algebra says that the Σ invariant polynomials form a polynomial ring Zp[σ1, ···, σn] where σi is the ith elementary symmetric polynomial, the sum of all products of i distinct yj ’s. Thus σi is a homogeneous polynomial of degree i. The order of n is n! so the condition that p not divide the order of amounts to p > n. Thus we realize the polynomial ring Zp[x1, ···, xn] with \|xi\| = 2i, provided that p > n. Σ This example is less than optimal since there happens to be another space, the Grassmann manifold of n dimensional linear subspaces of C∞, whose cohomology with any coeﬃcient ring R is R[x1, ···, xn] with \|xi\| = 2i, as we show in §4.D, so the restriction p > n is not really necessary. To get further examples the idea is to replace CP∞ by a space with the same Zp cohomology but with ‘more symmetry’, allowing for larger groups to act. The constructions will be made using K(π, 1) spaces, which were introduced in §1.B. For Γ complex Bπ with contractible universal cover a group π we constructed there a Eπ. The construction is functorial: A homomorphism ϕ : π→π ′ induces a map Bϕ
: Bπ→Bπ ′, Bϕ([g1\| ··· \|gn]) = [ϕ(g1)\| ··· \|ϕ(gn)], satisfying the functor properties B(ϕψ) = BϕBψ and B11 = 11. In particular, if is a group of automorphisms ∆ Γ of π, then acts on Bπ. Γ action of a group space Y ′. Namely, take Y ′ = Y × E where Γ acts on E The other ingredient we shall need is the Borel construction, which converts an on a space Y into a free action of on a homotopy equivalent with the diagonal action of, γ(y, z) = (γy, γz) as deck transformations. The diagonal action is free, in fact a covering space action, since this is true for the action in the second coordinate. The orbit space of this diagonal action is denoted Example 3G.3. Let π = Zp and let be the full automorphism group Aut(Zp). Automorphisms of Zp have the form x ֏ mx for (m, p) = 1, so is the multiplicative group of invertible elements in the ﬁeld Zp. By elementary ﬁeld theory this is a cyclic group, of order p − 1. The preceding constructions then give a covering space K(Zp, 1)→K(Zp, 1)/ assume we are in the nontrivial case p > 2. From the calculation of the cup product structure of lens spaces in Example 3.41 or Example 3E.2 we have H ∗(K(Zp, 1); Zp) ≈ acts on Zp [α] ⊗ Zp[β] with \|α\| = 1 and \|β\| = 2, and we need to ﬁgure out how ; Zp) ≈ H ∗(K(Zp, 1); Zp) with H ∗(K(Zp, 1)/. We may Γ Γ Γ Γ Γ this cohomology ring. Λ Let γ ∈ be a generator, say γ(x) = mx. The induced action of γ on π1K(Zp, 1) is also multiplication by m since
we have taken K(Zp, 1) = BZp × E and γ takes an edge loop [g] in BZp to [γ(g)] = [mg]. Hence γ acts on H1(K(Zp, 1); Z) by multiplication by m. It follows that γ(α) = mα and γ(β) = mβ since H 1(K(Zp, 1); Zp) ≈ Hom(H1(K(Zp, 1)), Zp) and H 2(K(Zp, 1); Zp) ≈ Ext(H1(K(Zp, 1)), Zp), and it is a gen- Γ Γ Γ 324 Chapter 3 Cohomology eral fact, following easily from the deﬁnitions, that multiplication by an integer m in an abelian group H induces multiplication by m in Hom(H, G) and Ext(H, G). Thus γ(βk) = mkβk and γ(αβk) = mk+1αβk. Since m was chosen to be a generator of the multiplicative group of invertible elements of Zp, it follows that the only elements of H ∗(K(Zp, 1); Zp) ﬁxed by γ, hence by, are the scalar multiples of [αβp−2] ⊗ Zp[βp−1], so we have βi(p−1) and αβi(p−1)−1. Thus H ∗(K(Zp, 1); Zp) = [x2p−3] ⊗ Zp[y2p−2], subscripts produced a space whose Zp cohomology ring is indicating dimension. Zp Λ Zp Γ Γ Λ Example 3G.4. As an easy generalization of the preceding example, replace the group there by a subgroup of Aut(Zp) of order d, where d is any divisor of p − 1. The is generated by the automorphism x ֏ m(p−1)/dx, and the same analysis new Γ [x2d−1] ⊗ Zp[y2d], subscripts shows that we obtain a space with Zp cohomology again
denoting dimension. For a given choice of d the condition that d divides p − 1 Zp Γ says p ≡ 1 mod d, which is satisﬁed by inﬁnitely many p ’s, according to a classical Λ theorem of Dirichlet. e Example 3G.5. The two preceding examples can be modiﬁed so as to eliminate the exterior algebra factors, by replacing Zp by Zp∞, the union of the increasing sequence Zp ⊂ Zp2 ⊂ Zp3 ⊂ ···. The ﬁrst step is to show that H ∗(K(Zp∞, 1); Zp) ≈ Zp[β] with H∗(K(Zpi, 1); Z) consists of Zpi ’s in odd dimensions. The in\|β\| = 2. We know that clusion Zpi ֓ Zpi+1 induces a map K(Zpi, 1)→K(Zpi+1, 1) that is unique up to homotopy. We can take this map to be a p sheeted covering space since the covering space of a K(Zpi+1, 1) corresponding to the unique index p subgroup of π1K(Zpi+1, 1) is a K(Zpi, 1). The homology transfer formula π∗τ∗ = p shows that the image of the induced map Hn(K(Zpi, 1); Z)→Hn(K(Zpi+1, 1); Z) for n odd contains the multiples of p, hence this map is the inclusion Zpi֓Zpi+1. We can use the universal coeﬃcient theorem to compute the induced map H ∗(K(Zpi+1, 1); Zp)→H ∗(K(Zpi, 1); Zp). Namely, the inclusion Zpi ֓ Zpi+1 induces the trivial map Hom(Zpi+1, Zp)→Hom(Zpi, Zp), so on odd-dimensional cohomology the induced map is trivial. On the other hand, the induced map on even-dimensional cohomology is an isomorphism since the map of free resolutions dualizes to Since Zp∞ is the union of the increasing
sequence of subgroups Zpi, the space BZp∞ is the union of the increasing sequence of subcomplexes BZpi. We can therefore apply Transfer Homomorphisms Section 3.G 325 Proposition 3F.5 to conclude that H ∗(K(Zp∞, 1); Zp) is zero in odd dimensions, while in even dimensions the map H ∗(K(Zp∞, 1); Zp)→H ∗(K(Zp, 1); Zp) induced by the inclusion Zp ֓ Zp∞ is an isomorphism. Thus H ∗(K(Zp∞, 1); Zp) ≈ Zp[β] as claimed. Next we show that the map Aut(Zp∞ )→Aut(Zp) obtained by restriction to the subgroup Zp ⊂ Zp∞ is a split surjection. Automorphisms of Zpi are the maps x֏mx for (m, p) = 1, so the restriction map Aut(Zpi+1 )→Aut(Zpi ) is surjective. Since ←-- Aut(Zpi ), the restriction map Aut(Zp∞ )→Aut(Zp) is also surjecAut(Zp∞ ) = lim tive. The order of Aut(Zpi ), the multiplicative group of invertible elements of Zpi, is pi − pi−1 = pi−1(p − 1) and p − 1 is relatively prime to pi−1, so the abelian group Aut(Zpi ) contains a subgroup of order p − 1. This subgroup maps onto the cyclic group Aut(Zp) of the same order, so Aut(Zpi )→Aut(Zp) is a split surjection, hence so is Aut(Zp∞ )→Aut(Zp). Thus we have an action of = Aut(Zp) on BZp∞ extending its natural action E ; Zp) onto the even-dimensional part of Γ ; Zp), a polynomial algebra Zp[y2p−2]. Similarly, if d is any divisor of to be the subgroup of Aut(Zp) of order d yields a space with Zp on BZp. The Borel construction then gives an inclusion BZp × �
� inducing an isomorphism of H ∗(BZp∞ × H ∗(BZp × p − 1, then taking cohomology the polynomial ring Zp[y2d]. ֓ BZp Example 3G.6. Now we enlarge the preceding example by taking products and bringing in the permutation group to produce a space with Zp cohomology the polynomial ring Zp[y2d, y4d, ···, y2nd] where d is any divisor of p − 1 and p > n. Let X be the product of n copies of BZp∞ and let be the group of homeomorphisms of X generated by permutations of the factors together with the actions of Zd in each factor constructed in the preceding example. We can view as a group of n× n matrices with entries in Zp, the matrices obtained by replacing some of the 1 ’s in a permutation matrix by elements of Zp of multiplicative order a divisor of n→0, and the order d. Thus there is a split short exact sequence 0→(Zd)n→ is dnn!. The product space X has H ∗(X; Zp) ≈ Zp[β1, ···, βn] with \|βi\| = 2, of so H ∗(X × provided that p does not divide the order of, which means p > n. For a polynomial to be invariant under the Zd action in each factor it must be a polynomial in the powers βd i, and to be invariant under Γ permutations of the variables it must be a symmetric polynomial in these powers. ; Zp) ≈ Zp[β1, ···, βn Since symmetric polynomials are exactly the polynomials in the elementary symmetric functions, the polynomials in the βi ’s invariant under Zp[y2d, y4d, ···, y2nd] with y2k the sum of all products of k distinct powers βd i. form a polynomial ring Γ Example 3G.7. As a further variant on the preceding example, choose a divisor q by its subgroup consisting of matrices for which the product of of d and replace the q th powers of the nonzero
entries is 1. This has the eﬀect of enlarging the ring of polynomials invariant under the action, and it can be shown that the invariant Γ 326 Chapter 3 Cohomology polynomials form a polynomial ring Zp[y2d, y4d, ···, y2(n−1)d, y2nq], with the last q i. For example, if n = 2 and q = 1 we obtain generator y2nd replaced by y2nq = i β Zp[y4, y2d] with y4 = β1β2 and y2d = βd 1 + βd in this case happens to be isomorphic to the dihedral group of order 2d. 2. The group Q Γ General Remarks The problem of realizing graded polynomial rings Zp[y] in one variable as cup product rings of spaces was discussed in §3.2, and Example 3G.5 provides the re- maining examples, showing that \|y\| can be any even divisor of 2(p − 1). In more variables the problem of realizing Zp[y1, ···, yn] with speciﬁed dimensions \|yi\| is more diﬃcult, but has been solved for odd primes p. Here is a sketch of the answer. Assuming that p is odd, the dimensions \|yi\| are even. Call the number di = \|yi\|/2 the degree of yi. In the examples above this was in fact the degree of yi as a polynomial in the 2 dimensional classes βj invariant under the action of. It was proved in [Dwyer, Miller, & Wilkerson 1992] that every realizable polynomial algebra Zp[y1, ···, yn] is the ring of invariant polynomials Zp[β1, ···, βn] Γ some ﬁnite group Γ for an action of on Zp[β1, ···, βn], where \|βi\| = 2. The basic examples, whose products yield all realizable polynomial algebras, can be divided into two categories. First there are classifying spaces of Lie groups, each of which realizes a polynomial algebra for all but ﬁnitely many primes p. These are
listed in the following table. Lie group S 1 SU(n) Sp(n) SO(2k) G2 F4 E6 E7 E8 degrees 1 2, 3, ···, n 2, 4, ···, 2n 2, 4, ···, 2k − 2, k 2, 6 2, 6, 8, 12 2, 5, 6, 8, 9, 12 2, 6, 8, 10, 12, 14 2, 8, 12, 14, 18, 20, 24, 30 primes all all all The remaining examples form two inﬁnite families plus 30 sporadic exceptions shown in the table on the next page. The ﬁrst row is the examples we have constructed, though our construction needed the extra condition that p not divide the order of. For all entries in both tables the order of the group Zp[y1, ···, yn] = Zp[β1, ···, βn] Γ When p does not divide this order, the method we used for the ﬁrst row can also be, turns out to equal the product of the degrees., the group such that Γ Γ applied to give examples for all the other rows. In some cases the congruence condip ) = GLn(Zp),. But when this is not the case automatically imply that p does not divide the order of tions on p, which are needed in order for to be a subgroup of Aut(Zn Γ a diﬀerent construction of a space with the desired cohomology is needed. To ﬁnd out more about this the reader can begin by consulting [Kane 1988] and [Notbohm 1999]. Γ Γ Local Coeﬃcients Section 3.H 327 degrees d, 2d, ···, (n − 1)d, nq with q \|d 2, d primes p ≡ 1 mod d p ≡ −1 mod d degrees 4, 6 6, 12 4, 12 12, 12 8, 12 8, 24 12, 24 24, 24 6, 8 8, 12 6, 24 12, 24 20, 30 20, 60 30, 60 primes p ≡ 1 mod 3 p ≡ 1 mod 3 p ≡ 1 mod 12 p ≡ 1 mod 12 p ≡ 1 mod 4 p ≡ 1 mod 8 p ≡ 1 mod 12 p ≡ 1 mod 24 p ≡
1, 3 mod 8 p ≡ 1 mod 8 p ≡ 1, 19 mod 24 p ≡ 1 mod 24 p ≡ 1 mod 5 p ≡ 1 mod 20 p ≡ 1 mod 15 For the prime 2 the realization problem has not yet been completely solved. Among the known examples are those in the table at the right. The construction for the last entry, which does not arise from a Lie group, is in [Dwyer & Wilkerson 1993]. (For p = 2 ‘degree’ means the actual co- homological dimension.) degrees 60, 60 12, 30 12, 60 12, 20 2, 6, 10 4, 6, 14 6, 9, 12 6, 12, 18 6, 12, 30 4, 8, 12, 20 2, 12, 20, 30 8, 12, 20, 24 12, 18, 24, 30 4, 6, 10, 12, 18 6, 12, 18, 24, 30, 42 primes p ≡ 1 mod 60 p ≡ 1, 4 mod 15 p ≡ 1, 49 mod 60 p ≡ 1, 9 mod 20 p ≡ 1, 4 mod 5 p ≡ 1, 2, 4 mod 7 p ≡ 1 mod 3 p ≡ 1 mod 3 p ≡ 1, 4 mod 15 p ≡ 1 mod 4 p ≡ 1, 4 mod 5 p ≡ 1 mod 4 p ≡ 1 mod 3 p ≡ 1 mod 3 p ≡ 1 mod 3 Lie group degrees O(1) SO(n) SU(n) Sp(n) P Sp(2n + 1) G2 Spin(7) Spin(8) Spin(9) F4 — 1 2, 3, ···, n 4, 6, ···, 2n 4, 8, ···, 4n 2, 3, 8, 12, ···, 8n + 4 4, 6, 7 4, 6, 7, 8 4, 6, 7, 8, 8 4, 6, 7, 8, 16 4, 6, 7, 16, 24 8, 12, 14, 15 Homology and cohomology with local coeﬃcients are fancier versions of ordi- nary homology and cohomology that can be deﬁned for nonsimply-connected spaces. In various situations these more reﬁned homology and cohomology theories arise naturally and inevitably. For example, the only way to extend Poincar´e duality with Z coe�
�cients to nonorientable manifolds is to use local coeﬃcients. In the overall scheme of algebraic topology, however, the role played by local coeﬃcients is fairly small. Local coeﬃcients bring an extra level of complication that one tries to avoid whenever possible. With this in mind, the goal of this section will not be to give a full exposition but rather just to sketch the main ideas, leaving the technical details for the interested reader to ﬁll in. 328 Chapter 3 Cohomology The plan for this section is ﬁrst to give the quick algebraic deﬁnition of homology and cohomology with local coeﬃcients, and then to reinterpret this deﬁnition more geometrically in a way that looks more like ordinary homology and cohomology. The reinterpretation also allows the familiar properties of homology and cohomology to be extended to the local coeﬃcient case with very little eﬀort. Local Coeﬃcients via Modules Let X be a path-connected space having a universal cover X and fundamental e group π, so that X is the quotient of X by the action of π by deck transforma- e e x ∈ x for γ ∈ π and X to the composition X induces an action of X. The action of π on X) of singular n chains in e e n σ-----→ x ֏ γ tions π on the group Cn( e γ-----→ n→ σ : X) makes X) a module over the group ring Z[π ], which consists of the ﬁnite formal sums Cn( ∆ ∆ i miγi with mi ∈ Z and γi ∈ π, with the natural addition i niγi = e i (mi + ni)γi and multiplication i,j minjγiγj. The boundX) are Z[π ] module homomorphisms since the action of P X)→Cn−1( ary maps ∂ : Cn( P π on these groups comes from an action on X, by sending a singular n simplex X. The action of π on Cn( i miγi + P X. i miγi j nj γj
If M is an arbitrary module over Z[π ], we would like to deﬁne Cn(X; M) to be X) ⊗ Z[π ]M, but for tensor products over a noncommutative ring one has to be a Cn( little careful with left and right module structures. In general, if R is a ring, possibly noncommutative, one deﬁnes the tensor product A ⊗R B of a right R module A and a left R module B to be the abelian group with generators a ⊗ b for a ∈ A and b ∈ B, e e subject to distributivity and associativity relations: (i) (a1 + a2) ⊗ b = a1 ⊗ b + a2 ⊗ b and a ⊗ (b1 + b2) = a ⊗ b1 + a ⊗ b2. (ii) ar ⊗ b = a ⊗ r b. In case R = Z[π ], a left Z[π ] module A can be regarded as a right Z[π ] module by setting aγ = γ−1a for γ ∈ π. So the tensor product of two left Z[π ] modules A and B is deﬁned, and the relation aγ ⊗ b = a ⊗ γb becomes γ−1a ⊗ b = a ⊗ γb, or equivalently a′ ⊗ b = γa′ ⊗ γb where a′ = γ−1a. Thus tensoring over Z[π ] has the eﬀect of factoring out the action of π. To simplify notation we shall write A ⊗Z[π ] B as A ⊗π B, emphasizing the fact that the essential part of a Z[π ] module structure is the action of π. X) ⊗π M is deﬁned if M is a left Z[π ] module. These chain In particular, Cn( X) ⊗π M form a chain complex with the boundary maps ∂ ⊗ 11. groups Cn(X; M) = Cn( e The homology groups Hn(X; M) of this chain complex are by de�
��nition homology groups with local coeﬃcients. e For cohomology one can set C n(X; M) = HomZ[π ](Cn( homomorphisms Cn( cohomology groups H n(X; M) are cohomology groups with local coeﬃcients. X), M), the Z[π ] module X)→M. These groups C n(X; M) form a cochain complex whose e e Local Coeﬃcients Section 3.H 329 → n→ σ ⊗ m = γ X). In Cn( ∆ e σ ⊗ m. Thus we can identify Cn( Example 3H.1. Let us check that when M is a trivial Z[π ] module, with γm = m for all γ ∈ π and m ∈ M, then Hn(X; M) is just ordinary homology with coeﬃcients in n→X, the various lifts the abelian group M. For a singular n simplex σ : X X) ⊗π M all these lifts are identiform an orbit of the action of π on Cn( e ∆ X) ⊗π M σ ⊗ γm = γ ﬁed via the relation e with Cn(X) ⊗ M, the chain group denoted Cn(X; M) in ordinary homology theory, so Hn(X; M) reduces to ordinary homology with coeﬃcients in M. The analogous statement for cohomology is also true since elements of HomZ[π ](Cn( X), M) are functions from singular n simplices a π orbit since the action of π on M is trivial, so HomZ[π ](Cn( ∆ with Hom(Cn(X), M), ordinary cochains with coeﬃcients in M. Example 3H.2. Suppose we take M = Z[π ], viewed as a module over itself via its ring structure. For a ring R with identity element, A ⊗R R is naturally isomorphic to A via the correspondence a ⊗ r ֏ ar. So we have a natural identiﬁcation of X). GenCn( eralizing
this, let X ′→X be the cover corresponding to a subgroup π ′ ⊂ π. Then the free abelian group Z[π /π ′] with basis the cosets γπ ′ is a Z[π ] module and X) ⊗Z[π ] Z[π /π ′] ≈ Cn(X ′), so Hn(X; Z[π /π ′]) ≈ Hn(X ′). More generally, if A is Cn( an abelian group then A[π /π ′] is a Z[π ] module and Hn(X; A[π /π ′]) ≈ Hn(X ′; A). So homology of covering spaces is a special case of homology with local coeﬃcients. X), and hence an isomorphism Hn(X; Z[π ]) ≈ Hn( X to M taking the same value on all elements of e X), M) is identiﬁable X) ⊗π Z[π ] with Cn( e e e e e e e The corresponding assertions for cohomology are not true, however, as we shall see later in the section. For a Z[π ] module M, let π ′ be the kernel of the homomorphism ρ : π→Aut(M) deﬁning the module structure, given by ρ(γ)(m) = γm, where Aut(M) is the group of automorphisms of the abelian group M. If X ′→X is the cover corresponding to X) ⊗π M ≈ Cn(X ′) ⊗π M ≈ Cn(X ′) ⊗Z[π /π ′] M. the normal subgroup π ′ of π, then Cn( This gives a more eﬃcient description of Hn(X; M). e Example 3H.3. As a special case, suppose that we take M = Z, so Aut(Z) ≈ Z2 = {±1}. For a nontrivial Z[π ] module structure on M, π ′ is a subgroup of index 2 and X ′→X is a 2 sheet
ed covering space. If τ is the nontrivial deck transformation of X ′, let n (X ′) = {α ∈ Cn(X ′) \| τ♯(α) = −α}. It C + n (X ′) = {α ∈ Cn(X ′) \| τ♯(α) = α} and C − n→X ′, and we have follows easily that C ± short exact sequences n (X ′) has basis the chains σ ± τσ for σ : 0 -→ C − 0 -→ C + n (X ′) -→ 0 n (X ′) ֓ Cn(X ′) n (X ′) -→ 0 n (X ′) ֓ Cn(X ′) (α) = α−τ♯(α). The homomorphism Cn(X)→C + -----→ C + -----→ C − ∆ Σ (α) = α+τ♯(α) and n (X ′) where sending a singular simplex in X to the sum of its two lifts to X ′ is an isomorphism. The quotient map Cn(X ′)→Cn(X ′) ⊗π Z has kernel C + n (X ′), so the second short exn (X ′) ≈ Cn(X ′) ⊗π Z. These isomorphisms are act sequence gives an isomorphism C − ∆ Σ ∆ 330 Chapter 3 Cohomology isomorphisms of chain complexes and the short exact sequences are short exact se- quence of chain complexes, so from the ﬁrst short exact sequence we get a long exact sequence of homology groups ··· -→ Hn(X; Z) -→ Hn(X ′) p∗---------→ Hn(X) -→ Hn−1(X; Z) -→ ··· where the symbol by the covering projection p : X ′→X. e Z indicates local coeﬃcients in the module Z and p∗ is induced e e Let us apply this exact sequence when X is a nonorientable n manifold M which is closed and connected. We shall use terminology and notation from §3.3. We can view Z as
a Z[π1M] module by letting a loop γ in M act on Z by multiplication by +1 or −1 according to whether γ preserves or reverses local orientations of M. The double cover X ′→X is then the 2 sheeted cover M orientable. The nonorientability of M implies that Hn(M) = 0. Since Hn+1(M) = 0, the exact seM) ≈ Z. This can be interpreted as saying quence above then gives Hn(M; that by taking homology with local coeﬃcients we obtain a fundamental class for a f M→M with Z) ≈ Hn( f f e nonorientable manifold. Local Coeﬃcients via Bundles of Groups Now we wish to reinterpret homology and cohomology with local coeﬃcients in more geometric terms, making it look more like ordinary homology and cohomology. Let us ﬁrst deﬁne a special kind of covering space with extra algebraic structure. A bundle of groups is a map p : E→X together with a group structure on each subset p−1(x), such that all these groups p−1(x) are isomorphic to a ﬁxed group G in the following special way: Each point of X has a neighborhood U for which there exists a homeomorphism hU : p−1(U)→U × G taking each p−1(x) to {x}× G by a group isomorphism. Since G is given the discrete topology, the projection p is a covering space. Borrowing terminology from the theory of ﬁber bundles, the subsets p−1(x) are called the ﬁbers of p : E→X, and one speaks of E as a bundle of groups with ﬁber G. It may be worth remarking that if we modify the deﬁnition by replacing the word ‘group’ with ‘vector space’ throughout, then we obtain the much more common notion of a vector bundle; see [VBKT]. Trivial examples are provided by products E = X × G. Nontrivial examples we have considered are the covering spaces MZ→M of nonorientable manifolds M deﬁned in §3.3. Here the group G is the
homology coeﬃcient group Z, though one could equally well deﬁne a bundle of groups MG→M for any abelian coeﬃcient group G. Homology groups of X with coeﬃcients in a bundle E of abelian groups may n→X is a sinbe deﬁned as follows. Consider ﬁnite sums n→E is a lifting of σi. The sum of two lifts ni gular n simplex in X and ni : and mi of the same σi is deﬁned by (ni + mi)(s) = ni(s) + mi(s), and is also a i niσi form an abelian group Cn(X; E), prolift of σi. In this way the ﬁnite sums vided we allow the deletion of terms niσi when ni is the zero-valued lift. A bound- i niσi where each σi : P P ∆ ∆ Local Coeﬃcients Section 3.H 331 \|\| [v0, ···, i,j (−1)jniσi ary homomorphism ∂ : Cn(X; E)→Cn−1(X; E) is deﬁned by the formula ∂ = vj, ···, vn] where ‘ ni ’ in the right side of the equation means the restricted lifting ni \|\| [v0, ···, vj, ···, vn]. The proof that the usual boundary hoP momorphism ∂ satisﬁes ∂2 = 0 still works in the present context, so the groups Cn(X; E) form a chain complex. We denote the homology groups of this chain complex by Hn(X; E). i niσi P b b Hn(X; E) = Hn(X; G), ordinary homology. uniquely determined by their value at one point s ∈ In case E is the product bundle X × G, lifts ni are simply elements of G, so n→E are n, and these values can be n is simply-connected, so the ni ’s can be thought of as speciﬁed arbitrarily since ∆ elements of p−1(σi(s)), a group
isomorphic to G. However if E is not a product, there is no canonical isomorphism between diﬀerent ﬁbers p−1(x), so one cannot identify Hn(X; E) with ordinary homology. In the general case, lifts ni : ∆ ∆ An alternative approach would be to take the coeﬃcients ni to be elements of n), say σi(v0). However, with such a the ﬁber group over a speciﬁc point of σi( deﬁnition the formula for the boundary operator ∂ becomes more complicated since there is no point of n that lies in all the faces. ∆ Our task now is to relate the homology groups Hn(X; E) to homology groups with coeﬃcients in a module, as deﬁned earlier. In §1.3 we described how covering spaces of X with a given ﬁber F can be classiﬁed in terms of actions of π1(X) on F, assuming X is path-connected and has the local properties guaranteeing the existence ∆ of a universal cover. It is easy to check that covering spaces that are bundles of groups with ﬁber a group G are equivalent to actions of π1(X) on G by automorphisms of G, that is, homomorphisms from π1(X) to Aut(G). For example, for the bundle MZ→M the action of a loop γ on the ﬁber Z is multiplication by ±1 according to whether γ preserves or reverses orientation in M→M M, that is, whether γ lifts to a closed loop in the orientable double cover or not. As another example, the action of π1(X) on itself by inner automorphisms corresponds to a bundle of groups p : E→X with ﬁbers p−1(x) = π1(X, x). This example is rather similar in spirit to the examples MZ→M. In both cases one has a functor associating a group to each point of a space, and all the groups at diﬀerent f points are isomorphic, but not canonically so. Diﬀerent choices of isomorphisms are obtained by choosing diﬀerent
paths between two points, and loops give rise to an action of π1 on the ﬁbers. In the case of bundles of groups p : E→X whose ﬁber G is abelian, an action of π1(X) on G by automorphisms is the same as a Z[π1X] module structure on G. Proposition 3H.4. If X is a path-connected space having a universal covering space, then the groups Hn(X; E) are naturally isomorphic to the homology groups Hn(X; G) with local coeﬃcients in the Z[π ] module G associated to E, where π = π1(X). 332 Chapter 3 Cohomology Proof: As noted earlier, a bundle of groups E→X with ﬁber G is equivalent to X is the universal an action of π on G. In more explicit terms this means that if e cover of X, then E is identiﬁable with the quotient of X × G by the diagonal action e e e e e P x, g) = (γ of π, γ( x, γg) where the action in the ﬁrst coordinate is by deck transi niσi ∈ Cn(X; E), the coeﬃcient ni gives a lift of formations of X. For a chain X × G. Thus we have natural surjecσi to E, and ni in turn has various lifts to X × G)→Cn(E)→Cn(X; E) expressing each of these groups as a quotient of tions Cn( e X) ⊗ Z[G] in the the preceding one. More precisely, identifying Cn( X) ⊗ Z[G] under the identiﬁcations obvious way, then Cn(E) is the quotient of Cn( X) ⊗π Z[G]. To pass to σ ⊗ g ∼ γ X) ⊗π Z[G] we need to take into account the the quotient Cn(X; E) of Cn(E) = Cn( e e n→E. This means that in sums sum operation in Cn(X; E), addition of lifts ni : σ ⊗ (g
1 + g2), the term g1 + g2 should be interpreted not in Z[G] σ ⊗ g1 + ∆ but in the natural quotient G of Z[G]. Hence Cn(X; E) is identiﬁed with the quoe e X) ⊗π Z[G]. This natural identiﬁcation commutes with the X) ⊗π G of Cn( tient Cn( ⊔⊓ boundary homomorphisms, so the homology groups are also identiﬁed. σ ⊗ γ g. This quotient is the tensor product Cn( X × G) with Cn( σ ⊗ g2 = e e e e e e e e More generally, if X has a number of path-components Xα with universal covers, so Hn(X; E) splits accordingly as a Xα, then Cn(X; E) = direct sum of the local coeﬃcient homology groups for the path-components Xα. e Xα) ⊗Z[π1(Xα)] G We turn now to the question of whether homology with local coeﬃcients satisﬁes α L Cn( e axioms similar to those for ordinary homology. The main novelty is with the behavior of induced homomorphisms. In order for a map f : X→X ′ to induce a map on homology with local coeﬃcients we must have bundles of groups E→X and E′→X ′ that are related in some way. The natural assumption to make is that there is a com- mutative diagram as at the right, such that f restricts to a homo- morphism in each ﬁber. With this hypothesis there is then a chain homomorphism f♯ : Cn(X; E)→Cn(X ′; E′) obtained by composing singular simplices with f and their lifts with f, hence there is an induced homomorphism f∗ : Hn(X; E)→Hn(X ′; E′). The ﬁbers of E and E′ need not be isomorphic groups, so change-of-coeﬃcient homomorphisms Hn(X; G
1)→Hn(X; G2) for ordinary homology are a special case. To avoid this extra complication we shall e e consider only the case that f restricts to an isomorphism on each ﬁber. With this condition, a commutative diagram as above will be called a bundle map. Here is a method for constructing bundle maps. Starting with a map f : X→X ′ and a bundle of groups p′ : E′→X ′, let E = (x, e′) ∈ X × E′ \|\|\|\| f (x) = p′(e′). f (x, e′) = This ﬁts into a commutative diagram as above if we deﬁne p(x, e′) = x and e′. In particular, the ﬁber p−1(x) consists of pairs (x, e′) with p′(e′) = f (x), so f is a bijection of this ﬁber with the ﬁber of E′→X ′ over f (x). We use this bijection e e e Local Coeﬃcients Section 3.H 333 to give p−1(x) a group structure. To check that p : E→X is a bundle of groups, let h′ : (p′)−1(U ′)→U ′ × G be an isomorphism as in the deﬁnition of a bundle of groups. Deﬁne h : p−1(U)→U × G over U = f −1(U ′) by h(x, e′) = (x, h′ 2 is the second coordinate of h′. An inverse for h is (x, g) ∈ (x, (h′)−1(f (x), g)), and h is clearly an isomorphism on each ﬁber. Thus p : E→X is a bundle of groups, called the pullback of E′→X ′ via f, or the induced bundle. The notation f ∗(E′) is often used for the pullback bundle. 2(e′)) where h′ Given any bundle map E→E′ as in the diagram above, it is routine to check f (e)), is an isomorphism of bundles over X,
that the map E→f ∗(E′), e ֏ (p(e), so the pullback construction produces all bundle maps. Thus we see one reason why homology with local coeﬃcients is somewhat complicated: Hn(X; E) is really a functor of two variables, covariant in X and contravariant in E. e Viewing bundles of abelian groups over X as Z[π1X] modules, the pullback construction corresponds to making a Z[π1X ′] module into a Z[π1X] module by deﬁning γg = f∗(γ)g for f∗ : π1(X)→π1(X ′). This follows easily from the deﬁnitions. In particular, this implies that homotopic maps f0, f1 : X→X ′ induce isomorphic pull1 (E′). Hence the map f∗ : Hn(X; E)→Hn(X ′; E′) induced by back bundles f ∗ a bundle map depends only on the homotopy class of f. 0 (E′), f ∗ Generalizing the deﬁnition of Hn(X; E) to pairs (X, A) is straightforward, starting with the deﬁnition of Hn(X, A; E) as the nth homology group of the chain complex of quotients Cn(X; E)/Cn(A; E) where p : E→X becomes a bundle of groups over A by restriction to p−1(A). Associated to the pair (X, A) there is then a long exact sequence of homology groups with local coeﬃcients in the bundle E. The excision property is proved just as for ordinary homology, via iterated barycentric subdivision. The ﬁnal axiom for homology, involving disjoint unions, extends trivially to homology with local coeﬃcients. Simplicial and cellular homology also extend without diﬃculty to the case of local coeﬃcients, as do the proofs that these forms of homology agree with singular homology for complexes and CW complexes, respectively. We leave the veriﬁcations of all these statements to the energetic
reader. ∆ Now we turn to cohomology. One might try deﬁning H n(X; E) by simply dualizing, taking Hom(Cn(X), E), but this makes no sense since E is not a group. Instead, the cochain group C n(X; E) is deﬁned to consist of all functions ϕ assigning In case E is the product to each singular simplex σ : n→X a lift ϕ(σ ) : n→E. X × G, this amounts to assigning an element of G to each σ, so this deﬁnition generalizes ordinary cohomology. Coboundary maps δ : C n(X; E)→C n+1(X; E) are deﬁned just as with ordinary cohomology, and satisfy δ2 = 0, so we have cohomology groups H n(X; E), and in the relative case, H n(X, A; E), deﬁned via relative cochains C n(X, A; E) = Ker ∆ ∆. For a path-connected space X with universal cover X and fundamental group π, we can identify H n(X; E) with H n(X; G), cohomology with local coeﬃcients in the C n(X; E)→C n(A; E) e 334 Chapter 3 Cohomology e X), G). e e Z[π ] module G corresponding to E, by identifying C n(X; E) with HomZ[π ](Cn( in the following way. An element ϕ ∈ C n(X; E) assigns to each σ : Regarding E as the quotient of X), G) n→X a lift to E. X × G under the diagonal action of π, a lift of σ to X × G. Such an orbit is a function f assigning to σ ) for all γ ∈ π, E is the same as an orbit of a lift to σ ) ∈ G such that f (γ σ ) = γf ( ∆ e e each lift X an element f ( that is, an element of HomZ[π ](Cn( e n→ σ : e ∆ The basic properties of ordinary cohomology
in §3.1 extend without great dif- e e e ∆ In order to deﬁne the map ﬁculty to cohomology groups with local coeﬃcients. f ∗ : H n(X ′; E′)→H n(X; E) induced by a bundle map as before, it suﬃces to observe σ ′ : σ = that a singular simplex σ : n→f ∗(E) of σ. To show that f ≃ g implies f ∗ = g∗ requires some mod(σ, e iﬁcation of the proof of the corresponding result for ordinary cohomology in §3.1, n→E′ of f σ deﬁne a lift n→X and a lift σ ′) : ∆ ∆ e ∆ which proceeded by dualizing the proof for homology. In the local coeﬃcient case one constructs a chain homotopy P ∗ satisfying g♯ − f ♯ = P ∗δ + δP ∗ directly from n × I used in the proof of the homology result. Similar remarks the subdivision of apply to proving excision and Mayer–Vietoris sequences for cohomology with local coeﬃcients. To prove the equivalence of simplicial and cellular cohomology with singular cohomology in the local coeﬃcient context, one should use the telescope argument from the proof of Lemma 2.34 to show that H n(X k; E) ≈ H n(X; E) for k > n. Once again details will be left to the reader. The diﬀerence between homology with local coeﬃcients and cohomology with lo- cal coeﬃcients is illuminated by comparing the following proposition with our earlier identiﬁcation of H∗(X; Z[π1X]) with the ordinary homology of the universal cover of X. Proposition 3H.5. If X is a ﬁnite CW complex with universal cover mental group π, then for all n, H n(X; Z[π ]) is isomorphic to H n c ( X with compact supports and ordinary integer coeﬃcients. of e X and fundaX; Z), cohom
ology e For example, consider the n dimensional torus T n, the product of n circles, with e fundamental group π = Zn and universal cover Rn. We have Hi(T n; Z[π ]) ≈ Hi(Rn), which is zero except for a Z in dimension 0, but H i(T n; Z[π ]) ≈ H i c(Rn) vanishes except for a Z in dimension n, as we saw in Example 3.34. To prove the proposition we shall use a few general facts about cohomology c (Y ; G)→C n with compact supports. One signiﬁcant diﬀerence between ordinary cohomology and cohomology with compact supports is in induced maps. A map f : X→Y induces f ♯ : C n c (X; G) provided that f is proper: The preimage f −1(K) of each compact set K in Y is compact in X. Thus if ϕ ∈ C n(Y ; G) vanishes on chains in Y − K then f ♯(ϕ) ∈ C n(X; G) vanishes on chains in X − f −1(K). Further, to guarantee that f ≃ g implies f ∗ = g∗ we should restrict attention to homotopies that are proper as maps X × I→Y. Relative groups c (X; G) and hence f ∗ : H n c (Y ; G)→H n Local Coeﬃcients Section 3.H 335 H n c (X, A; G) are deﬁned when A is a closed subset of X, which guarantees that the inclusion A ֓ X is a proper map. With these constraints the basic theory of §3.1 translates without diﬃculty to cohomology with compact supports. In particular, for a locally compact CW complex X one can compute H ∗ c (X; G) using ﬁnite cellular cochains, the cellular cochains vanishing on all but ﬁnitely many c (X n, X n−1; G) using excision one ﬁrst has to identify cells. Namely, to compute H n c (X n, N(X n−1); G) where N(X n−1) is a closed neighborhood of this group with
H n X n−1 in X n obtained by deleting an open n disk from the interior of each n cell. If X is locally compact, the obvious deformation retraction of N(X n−1) onto X n−1 is a proper homotopy equivalence. Hence via long exact sequences and the ﬁve-lemma c (X n, N(X n−1); G), and by excision the we obtain isomorphisms H n latter group can be identiﬁed with the ﬁnite cochains. c (X n, X n−1; G, X; Z) ϕ : Cn→Z[π ] by setting X; Z) using the groups C n f ( X n−1). Giving X the CW Proof of 3H.5: As noted above, we can compute H ∗ c ( of ﬁnite cellular cochains ϕ : Cn→Z, where Cn = Hn( e structure lifting the CW structure on X, then since X is compact, ﬁnite cellular cochains are exactly homomorphisms ϕ : Cn→Z such that for each cell en of X, ϕ(γen) is nonzero for only ﬁnitely many covering transformations γ ∈ π. Such a e γ ϕ(γ−1en)γ. The map ϕ determines a map ϕ is a Z[π ] homomorphism since if we replace the summation index γ in the right γ ϕ(γ−1en)ηγ. The function side of ϕ(ηen) = b X; Z)→HomZ[π ](Cn, Z[π ]) which is injective ϕ ֏ ϕ as the coeﬃcient of γ = 1. Furthermore, this hosince ϕ is recoverable from momorphism is surjective since a Z[π ] homomorphism ψ : M→Z[π ] has the form γ ψγ(x)γ with ψγ ∈ HomZ(M, Z) satisfying ψγ(x) = ψ1(γ−1x), so ψ1 deψ(x) = termines ψ. The isomorphisms C n X; Z) ≈ HomZ[π ](
Cn, Z[π ]) are isomorphisms of f ( X; Z) and H n(X; Z[π ]) cochain complexes, so the respective cohomology groups H n c ( ⊔⊓ are isomorphic. b γ ϕ(γ−1ηen)γ by ηγ, we get ϕ deﬁnes a homomorphism C n f ( ϕ(en Cup and cap product work easily with local coeﬃcients in a bundle of rings, the latter concept being deﬁned in the obvious way. The cap product can be used to give a version of Poincar´e duality for a closed n manifold M using coeﬃcients in a bundle of rings E under the same assumption as with ordinary coeﬃcients that there exists a fundamental class [M] ∈ Hn(M; E) restricting to a generator of Hn(M, M − {x}; E) for all x ∈ M. By excision the latter group is isomorphic to the ﬁber ring R of E. The same proof as for ordinary coeﬃcients then shows that [M]a : H k(M; E)→Hn−k(M; E) is an isomorphism for all k. Taking R to be one of the standard rings Z, Q, or Zp does not give anything new since the only ring automorphism these rings have is the identity, so the bundle of rings E must be the product M × R. To get something more interesting, suppose we take R to be the ring Z[i] of Gaussian integers, the complex numbers a + bi with 336 Chapter 3 Cohomology a, b ∈ Z. This has complex conjugation a + bi ֏ a − bi as a ring isomorphism. If M is nonorientable and connected we can use the homomorphism ω : π1(M)→{±1} that deﬁnes the bundle of groups MZ to build a bundle of rings E corresponding to the action of π1(M) on Z[i] given by γ(a + bi) = a + ω(γ)bi. The homology and cohomology groups of M with coeﬃcients in E depend only on the additive structure of Z[i
] so they split as the direct sum of their real and imaginary parts, which are just the homology or cohomology groups with ordinary coeﬃcients Z and Z) constructed in Z, respectively. The fundamental class in Hn(M; twisted coeﬃcients Example 3H.3 can be viewed as a pure imaginary fundamental class [M] ∈ Hn(M; E). Since cap product with [M] interchanges real and imaginary parts, we obtain: e e Theorem 3H.6. If M is a nonorientable closed connected n manifold then cap product with the pure imaginary fundamental class [M] gives isomorphisms H k(M; Z) ≈ ⊔⊓ Hn−k(M; Z) ≈ Hn−k(M; Z). Z) and H k(M; More generally this holds with Z replaced by other rings such as Q or Zp. There is also a version for noncompact manifolds using cohomology with compact supports. e e Exercises 1. Compute H∗(S 1; E) and H ∗(S 1; E) for E→S 1 the nontrivial bundle with ﬁber Z. 2. Compute the homology groups with local coeﬃcients Hn(M; MZ) for a closed nonorientable surface M. 3. Let B(X; G) be the set of isomorphism classes of bundles of groups E→X with ﬁber G, and let E0→BAut(G) be the bundle corresponding to the ‘identity’ action ρ : Aut(G)→Aut(G). Show that the map [X, BAut(G)]→B(X, G), [f ] ֏ f ∗(E0), is a bijection if X is a CW complex, where [X, Y ] denotes the set of homotopy classes of maps X→Y. 4. Show that if ﬁnite connected CW complexes X and Y are homotopy equivalent, then their universal covers X and Y are proper homotopy equivalent. e e X, 5. If X is a ﬁnite connected graph with π1(X) free on g > 0 generators, show that H n(X; Z[π1X]) is
zero unless n = 1, when it is Z when g = 1 and the direct sum of a countably inﬁnite number of Z ’s when g > 1. [Use Proposition 3H.5 and compute X) as lim--→H n( H n X − Ti) for a suitable sequence of ﬁnite subtrees T1 ⊂ T2 ⊂ ··· c ( i Ti = X with of e 6. Show that homology groups H ℓf n (X; G) can be deﬁned using locally ﬁnite chains, n→X with coeﬃcients σ gσ σ of singular simplices σ : which are formal sums gσ ∈ G, such that each x ∈ X has a neighborhood meeting the images of only ﬁnitely many σ ’s with gσ ≠ 0. Develop this version of homology far enough to show that for a ﬁnite-dimensional locally compact CW complex X, H ℓf n (X; G) can be computed using inﬁnite cellular chains X.] e P S ∆ e e e α gαen α. P Homotopy theory begins with the homotopy groups πn(X), which are the natural higher-dimensional analogs of the fundamental group. These higher homotopy groups have certain formal similarities with homology groups. For example, πn(X) turns out to be always abelian for n ≥ 2, and there are relative homotopy groups ﬁt- ting into a long exact sequence just like the long exact sequence of homology groups. However, the higher homotopy groups are much harder to compute than either ho- mology groups or the fundamental group, due to the fact that neither the excision property for homology nor van Kampen’s theorem for π1 holds for higher homotopy groups. In spite of these computational diﬃculties, homotopy groups are of great theo- retical signiﬁcance. One reason for this is Whitehead’s theorem that a map between CW complexes which induces isomorphisms on all homotopy groups is a homotopy equivalence. The stronger statement that two CW complexes with isomorphic homo- topy groups are homotopy equivalent is
usually false, however. One of the rare cases when a CW complex does have its homotopy type uniquely determined by its homo- topy groups is when it has just a single nontrivial homotopy group. Such spaces, known as Eilenberg–MacLane spaces, turn out to play a fundamental role in algebraic topology for a variety of reasons. Perhaps the most important is their close connec- tion with cohomology: Cohomology classes in a CW complex correspond bijectively with homotopy classes of maps from the complex into an Eilenberg–MacLane space. 338 Chapter 4 Homotopy Theory Thus cohomology has a strictly homotopy-theoretic interpretation, and there is an analogous but more subtle homotopy-theoretic interpretation of homology, explained in §4.F. A more elementary and direct connection between homotopy and homology is the Hurewicz theorem, asserting that the ﬁrst nonzero homotopy group πn(X) of a Hn(X). simply-connected space X is isomorphic to the ﬁrst nonzero homology group This result, along with its relative version, is one of the cornerstones of algebraic e topology. Though the excision property does not always hold for homotopy groups, in some important special cases there is a range of dimensions in which it does hold. This leads to the idea of stable homotopy groups, the beginning of stable homotopy theory. Perhaps the major unsolved problem in algebraic topology is the computation of the stable homotopy groups of spheres. Near the end of §4.2 we give some tables of known calculations that show quite clearly the complexity of the problem. Included in §4.2 is a brief introduction to ﬁber bundles, which generalize covering spaces and play a somewhat analogous role for higher homotopy groups. It would easily be possible to devote a whole book to the subject of ﬁber bundles, even the special case of vector bundles, but here we use ﬁber bundles only to provide a few basic examples and to motivate their more ﬂexible homotopy-theoretic generalization, ﬁbrations, which play a large role in §4.3. Among other things, ﬁbrations allow one to describe, in theory at least, how the homotopy type of
an arbitrary CW complex is built up from its homotopy groups by an inductive procedure of forming ‘twisted products’ of Eilenberg–MacLane spaces. This is the notion of a Postnikov tower. In favorable cases, including all simply-connected CW complexes, the additional data beyond homotopy groups needed to determine a homotopy type can also be described, in the form of a sequence of cohomology classes called the k invariants of a space. If these are all zero, the space is homotopy equivalent to a product of Eilenberg–MacLane spaces, and otherwise not. Unfortunately the k invariants are cohomology classes in rather complicated spaces in general, so this is not a practical way of classifying homotopy types, but it is useful for various more theoretical purposes. This chapter is arranged so that it begins with purely homotopy-theoretic notions, largely independent of homology and cohomology theory, whose roles gradually in- crease in later sections of the chapter. It should therefore be possible to read a good portion of this chapter immediately after reading Chapter 1, with just an occasional glimpse at Chapter 2 for algebraic deﬁnitions, particularly the notion of an exact se- quence which is just as important in homotopy theory as in homology and cohomology theory. Homotopy Groups Section 4.1 339 Perhaps the simplest noncontractible spaces are spheres, so to get a glimpse of the subtlety inherent in homotopy groups let us look at some of the calculations of the groups πi(S n) that have been made. A small sample is shown in the table below, extracted from [Toda 1962]. 5 0 6 0 7 i -→ Z2 Z2 Z12 Z2 0 0 Z Z2 Z2 Z12 Z2 Z2 Z2 Z2 πi(S n) 8 9 10 11 12 0 Z2 Z2 0 Z3 Z3 0 Z15 Z15 0 Z2 Z2 0 Z2 × Z2 Z2 × Z2 Z × Z12 Z2 × Z2 Z2 × Z2 Z24 × Z3 Z15 Z2 Z30 Z2 Z2 Z2 Z2 Z Z 0 0 Z24 0 0 Z2 Z24 Z2 Z2 Z2 0 Z24 Z2 Z24 Z2 Z2 Z This is an intriguing mixture of pattern and chaos. The most obvious feature
is the large region of zeros below the diagonal, and indeed πi(S n) = 0 for all i < n as we show in Corollary 4.9. There is also the sequence of zeros in the ﬁrst row, suggesting that πi(S 1) = 0 for all i > 1. This too is a fairly elementary fact, a special case of Proposition 4.1, following easily from covering space theory. The coincidences in the second and third rows can hardly be overlooked. These are the case n = 1 of isomorphisms πi(S 2n) ≈ πi−1(S 2n−1)× πi(S 4n−1) that hold for n = 1, 2, 4 and all i. The next case n = 2 says that each entry in the fourth row is the product of the entry diagonally above it to the left and the entry three units below it. Actually, these isomorphisms πi(S 2n) ≈ πi−1(S 2n−1)× πi(S 4n−1) hold for all n if one factors out 2 torsion, the elements of order a power of 2. This is a theorem of James that will be proved in [SSAT]. The next regular feature in the table is the sequence of Z ’s down the diagonal. This is an illustration of the Hurewicz theorem, which asserts that for a simply-connected space X, the ﬁrst nonzero homotopy group πn(X) is isomorphic to the ﬁrst nonzero homology group Hn(X). One may observe that all the groups above the diagonal are ﬁnite except for π3(S 2), π7(S 4), and π11(S 6). In §4.B we use cup products in cohomology to show that π4k−1(S 2k) contains a Z direct summand for all k ≥ 1. It is a theorem of Serre proved in [SSAT] that πi(S n) is ﬁnite for i > n except for π4k−1(S 2k), which is the direct sum of Z with a ﬁnite group. So all the complexity of the hom
otopy groups of spheres resides in ﬁnite abelian groups. The problem thus reduces to computing the p torsion in πi(S n) for each prime p. 340 Chapter 4 Homotopy Theory An especially interesting feature of the table is that along each diagonal the groups πn+k(S n) with k ﬁxed and varying n eventually become independent of n for large enough n. This stability property is the Freudenthal suspension theorem, proved in §4.2 where we give more extensive tables of these stable homotopy groups of spheres. Deﬁnitions and Basic Constructions Let In be the n dimensional unit cube, the product of n copies of the interval [0, 1]. The boundary ∂In of In is the subspace consisting of points with at least one coordinate equal to 0 or 1. For a space X with basepoint x0 ∈ X, deﬁne πn(X, x0) to be the set of homotopy classes of maps f : (In, ∂In)→(X, x0), where homotopies ft are required to satisfy ft(∂In) = x0 for all t. The deﬁnition extends to the case n = 0 by taking I0 to be a point and ∂I0 to be empty, so π0(X, x0) is just the set of path-components of X. When n ≥ 2, a sum operation in πn(X, x0), generalizing the composition opera- tion in π1, is deﬁned by (f + g)(s1, s2, ···, sn) = f (2s1, s2, ···, sn), g(2s1 − 1, s2, ···, sn), s1 ∈ [0, 1/2] s1 ∈ [1/2, 1] It is evident that this sum is well-deﬁned on homotopy classes. Since only the ﬁrst coordinate is involved in the sum operation, the same arguments as for π1 show that πn(X, x0) is a group, with identity element the constant map sending In to x0 and with inverses given by −f (s1, s2, ···, sn)
= f (1 − s1, s2, ···, sn). The additive notation for the group operation is used because πn(X, x0) is abelian for n ≥ 2. Namely, f + g ≃ g + f via the homotopy indicated in the following ﬁgures. The homotopy begins by shrinking the domains of f and g to smaller subcubes of In, with the region outside these subcubes mapping to the basepoint. After this has been done, there is room to slide the two subcubes around anywhere in In as long as they stay disjoint, so if n ≥ 2 they can be slid past each other, interchanging their positions. Then to ﬁnish the homotopy, the domains of f and g can be enlarged back to their original size. If one likes, the whole process can be done using just the coordinates s1 and s2, keeping the other coordinates ﬁxed. Maps (In, ∂In)→(X, x0) are the same as maps of the quotient In/∂In = S n to X taking the basepoint s0 = ∂In/∂In to x0. This means that we can also view πn(X, x0) as homotopy classes of maps (S n, s0)→(X, x0), where homotopies are through maps Homotopy Groups Section 4.1 341 of the same form (S n, s0)→(X, x0). In this interpretation of πn(X, x0), the sum f + g is the composition S n c-----→ S n ∨ S n f ∨g------------→ X where c collapses the equator S n−1 in S n to a point and we choose the basepoint s0 to lie in this S n−1. We will show next that if X is path-connected, diﬀerent choices of the basepoint x0 always produce isomorphic groups πn(X, x0), just as for π1, so one is justiﬁed in writing πn(X) for πn(X, x0) in these cases. Given a path γ : I→X from x0 = γ(0) to another bas
epoint x1 = γ(1), we may associate to each map f : (In, ∂In)→(X, x1) a new map γf : (In, ∂In)→(X, x0) by shrinking the domain of f to a smaller concentric cube in In, then inserting the path γ on each radial segment in the shell between this smaller cube and ∂In. When n = 1 the map γf is the composition of the three paths γ, f, and the inverse of γ, so the notation γf conﬂicts with the notation for composition of paths. Since we are mainly interested in the cases n > 1, we leave it to the reader to make the necessary notational adjustments when n = 1. A homotopy of γ or f through maps ﬁxing ∂I or ∂In, respectively, yields a homotopy of γf through maps (In, ∂In)→(X, x0). Here are three other basic properties: (1) γ(f + g) ≃ γf + γg. (2) (γη)f ≃ γ(ηf ). (3) 1f ≃ f, where 1 denotes the constant path. The homotopies in (2) and (3) are obvious. For (1), we ﬁrst deform f and g to be constant on the right and left halves of In, respectively, producing maps we may call f + 0 and 0 + g, then we excise a progressively wider symmetric middle slab of γ(f + 0) + γ(0 + g) until it becomes γ(f + g) : An explicit formula for this homotopy is ht(s1, s2, ···, sn) = γ(f + 0) γ(0 + g) (2 − t)s1, s2, ···, sn (2 − t)s1 + t − 1, s2, ···, sn, s1 ∈ [0, 1/2] s1 ∈ [1/2, 1], Thus we have γ(f + g) ≃ γ(f + 0) + γ(0 + g) ≃ γf + γg. If we
deﬁne a change-of-basepoint transformation βγ : πn(X, x1)→πn(X, x0) by βγ([f ]) = [γf ], then (1) shows that βγ is a homomorphism, while (2) and (3) imply that βγ is an isomorphism with inverse βγ where γ is the inverse path of γ, 342 Chapter 4 Homotopy Theory γ(s) = γ(1 − s). Thus if X is path-connected, diﬀerent choices of basepoint x0 yield isomorphic groups πn(X, x0), which may then be written simply as πn(X). Now let us restrict attention to loops γ at the basepoint x0. Since βγη = βγ βη, the association [γ] ֏ βγ deﬁnes a homomorphism from π1(X, x0) to Aut(πn(X, x0)), the group of automorphisms of πn(X, x0). This is called the action of π1 on πn, each element of π1 acting as an automorphism [f ] ֏ [γf ] of πn. When n = 1 this is the action of π1 on itself by inner automorphisms. When n > 1, the action makes the abelian group πn(X, x0) into a module over the group ring Z[π1(X, x0)]. i niγi with ni ∈ Z and γi ∈ π1, multiplication Elements of Z[π1] are ﬁnite sums being deﬁned by distributivity and the multiplication in π1. The module structure on i ni(γiα) for α ∈ πn. For brevity one sometimes says πn is given by πn is a π1 module rather than a Z[π1] module. i niγi α = P P P In the literature, a space with trivial π1 action on πn is called ‘ n simple’, and ‘simple’ means ‘ n simple for all n ’. In this book we will call a space ab
elian if it has trivial action of π1 on all homotopy groups πn, since when n = 1 this is the condition that π1 be abelian. This terminology is consistent with a long-established usage of the term ‘nilpotent’ to refer to spaces with nilpotent π1 and nilpotent action of π1 on all higher homotopy groups; see [Hilton, Mislin, & Roitberg 1975]. An important class of abelian spaces is H–spaces, as we show in Example 4A.3. We next observe that πn is a functor. Namely, a map ϕ : (X, x0)→(Y, y0) induces ϕ∗ : πn(X, x0)→πn(Y, y0) deﬁned by ϕ∗([f ]) = [ϕf ]. It is immediate from the deﬁnitions that ϕ∗ is well-deﬁned and a homomorphism for n ≥ 1. The functor properties (ϕψ)∗ = ϕ∗ψ∗ and 11∗ = 11 are also evident, as is the fact that if ϕt : (X, x0)→(Y, y0) is a homotopy then ϕ0∗ = ϕ1∗. In particular, a homotopy equivalence (X, x0) ≃ (Y, y0) in the basepointed sense induces isomorphisms on all homotopy groups πn. This is true even if basepoints are not required to be stationary during homotopies. We showed this for π1 in Proposition 1.18, and the generalization to higher n ’s is an exercise at the end of this section. Homotopy groups behave very nicely with respect to covering spaces: Proposition 4.1. A covering space projection p : ( phisms p∗ : πn( x0)→πn(X, x0) for all n ≥ 2. X, e e X, x0)→(X, x0) induces isomor- e e Proof: For surjectivity of p∗ we apply the lifting criterion in Proposition 1.33, which implies that every map (S
n, s0)→(X, x0) lifts to ( x0) provided that n ≥ 2 so that S n is simply-connected. Injectivity of p∗ is immediate from the covering homotopy ⊔⊓ property, just as in Proposition 1.31 which treated the case n = 1. X, e e In particular, πn(X, x0) = 0 for n ≥ 2 whenever X has a contractible universal cover. This applies for example to S 1, so we obtain the ﬁrst row of the table of homotopy groups of spheres shown earlier. More generally, the n dimensional torus T n, Homotopy Groups Section 4.1 343 the product of n circles, has universal cover Rn, so πi(T n) = 0 for i > 1. This is in marked contrast to the homology groups Hi(T n) which are nonzero for all i ≤ n. Spaces with πn = 0 for all n ≥ 2 are sometimes called aspherical. The behavior of homotopy groups with respect to products is very simple: Proposition 4.2. For a product spaces Xα there are isomorphisms πn Q α Xα of an arbitrary collection of path-connected α Xα ≈ α πn(Xα) for all n. Proof: A map f : Y → Taking Y to be S n and S n × I gives the result. Q α Xα is the same thing as a collection of maps fα : Y →Xα. ⊔⊓ Q Q Very useful generalizations of the homotopy groups πn(X, x0) are the relative homotopy groups πn(X, A, x0) for a pair (X, A) with a basepoint x0 ∈ A. To deﬁne these, regard In−1 as the face of In with the last coordinate sn = 0 and let J n−1 be the closure of ∂In − In−1, the union of the remaining faces of In. Then πn(X, A, x0) for n ≥ 1 is deﬁned to be the set of homotopy classes of maps (In, ∂In, J n−1)→(X, A, x0), with homotopies through maps of the same form
. There does not seem to be a completely satisfactory way of deﬁning π0(X, A, x0), so we shall leave this undeﬁned (but see the exercises for one possible deﬁnition). Note that πn(X, x0, x0) = πn(X, x0), so absolute homotopy groups are a special case of relative homotopy groups. A sum operation is deﬁned in πn(X, A, x0) by the same formulas as for πn(X, x0), except that the coordinate sn now plays a special role and is no longer available for the sum operation. Thus πn(X, A, x0) is a group for n ≥ 2, and this group is abelian for n ≥ 3. For n = 1 we have I1 = [0, 1], I0 = {0}, and J 0 = {1}, so π1(X, A, x0) is the set of homotopy classes of paths in X from a varying point in A to the ﬁxed basepoint x0 ∈ A. In general this is not a group in any natural way. Just as elements of πn(X, x0) can be regarded as homotopy classes of maps (S n, s0)→(X, x0), there is an alternative deﬁnition of πn(X, A, x0) as the set of homotopy classes of maps (Dn, S n−1, s0)→(X, A, x0), since collapsing J n−1 to a point converts (In, ∂In, J n−1) into (Dn, S n−1, s0). From this viewpoint, addition is done via the map c : Dn→Dn ∨ Dn collapsing Dn−1 ⊂ Dn to a point. A useful and conceptually enlightening reformulation of what it means for an element of πn(X, A, x0) to be trivial is given by the following compression criterion: A map f : (Dn, S n−1, s0)→(X, A, x0) represents zero in πn(X, A,
x0) iﬀ it is homotopic rel S n−1 to a map with image contained in A. For if we have such a homotopy to a map g, then [f ] = [g] in πn(X, A, x0), and [g] = 0 via the homotopy obtained by composing g with a deformation retraction of Dn onto s0. Conversely, if [f ] = 0 via a homotopy F : Dn × I→X, then by restricting F to a family of n disks in Dn × I starting with Dn × {0} and ending with the disk Dn × {1} ∪ S n−1 × I, all the disks in the family having the same boundary, then we get a homotopy from f to a map into A, stationary on S n−1. 344 Chapter 4 Homotopy Theory A map ϕ : (X, A, x0)→(Y, B, y0) induces maps ϕ∗ : πn(X, A, x0)→πn(Y, B, y0) which are homomorphisms for n ≥ 2 and have properties analogous to those in the absolute case: (ϕψ)∗ = ϕ∗ψ∗, 11∗ = 11, and ϕ∗ = ψ∗ if ϕ ≃ ψ through maps (X, A, x0)→(Y, B, y0). Probably the most useful feature of the relative groups πn(X, A, x0) is that they ﬁt into a long exact sequence i∗-----→ πn(X, x0) j∗-----→ πn(X, A, x0) ∂-----→ πn−1(A, x0) -→ ··· -→ π0(X, x0) ··· -→ πn(A, x0) Here i and j are the inclusions (A, x0) ֓ (X, x0) and (X, x0, x0) ֓ (X, A, x0). The map ∂ comes from restricting maps (In, ∂In, J n−1)→(X, A, x0) to In−1, or
by restricting maps (Dn, S n−1, s0)→(X, A, x0) to S n−1. The map ∂, called the boundary map, is a homomorphism when n > 1. Theorem 4.3. This sequence is exact. Near the end of the sequence, where group structures are not deﬁned, exactness still makes sense: The image of one map is the kernel of the next, those elements mapping to the homotopy class of the constant map. Proof: With only a little more eﬀort we can derive the long exact sequence of a triple (X, A, B, x0) with x0 ∈ B ⊂ A ⊂ X : ··· -→ πn(A, B, x0) i∗-----→ πn(X, B, x0) j∗-----→ πn(X, A, x0) ∂-----→ πn−1(A, B, x0) -→ ··· -→ π1(X, A, x0) When B = x0 this reduces to the exact sequence for the pair (X, A, x0), though the latter sequence continues on two more steps to π0(X, x0). The veriﬁcation of exactness at these last two steps is left as a simple exercise. Exactness at πn(X, B, x0) : First note that the composition j∗i∗ is zero since every map (In, ∂In, J n−1)→(A, B, x0) represents zero in πn(X, A, x0) by the compression criterion. To see that Ker j∗ ⊂ Im i∗, let f : (In, ∂In, J n−1)→(X, B, x0) represent zero in πn(X, A, x0). Then by the compression criterion again, f is homotopic rel ∂In to a map with image in A, hence the class [f ] ∈ πn(X, B, x0) is in the image of i∗. Exactness at πn(X, A, x0) : The composition ∂j∗ is zero since the restriction of
a map (In, ∂In, J n−1)→(X, B, x0) to In−1 has image lying in B, and hence represents zero in πn−1(A, B, x0). Conversely, suppose the restriction of f : (In, ∂In, J n−1)→(X, A, x0) to In−1 represents zero in πn−1(A, B, x0). Then f \|\| In−1 is homotopic to a map with image in B via a homotopy F : In−1 × I→A rel ∂In−1. We can tack F onto f to get a new map (In, ∂In, J n−1)→(X, B, x0) which, as a map (In, ∂In, J n−1)→(X, A, x0), is homotopic to f by the homotopy that tacks on increasingly longer initial segments of F. So [f ] ∈ Im j∗. Homotopy Groups Section 4.1 345 Exactness at πn(A, B, x0) : The composition i∗∂ is zero since the restriction of a map f : (In+1, ∂In+1, J n)→(X, A, x0) to In is homotopic rel ∂In to a constant map via f itself. The converse is easy if B is a point, since a nullhomotopy ft : (In, ∂In)→(X, x0) of f0 : (In, ∂In)→(A, x0) gives a map F : (In+1, ∂In+1, J n)→(X, A, x0) with ∂([F ]) = [f0]. Thus the proof is ﬁnished in this case. For a general B, let F be a nullhomotopy of f : (In, ∂In, J n−1)→(A, B, x0) through maps (In, ∂In, J n−1)→(X, B, x0), and let g be the restriction of F to In−1 × I, as in the ﬁrst of the two pictures below. Reparam
etrizing the nth and (n + 1) st coordinates as shown in the second picture, we see that f with g tacked on is in the image of ∂. But as we noted in the preceding paragraph, tacking g onto f gives the same element of πn(A, B, x0). ⊔⊓ Example 4.4. Let CX be the cone on a path-connected space X, the quotient space of X × I obtained by collapsing X × {0} to a point. We can view X as the subspace X × {1} ⊂ CX. Since CX is contractible, the long exact sequence of homotopy groups for the pair (CX, X) gives isomorphisms πn(CX, X, x0) ≈ πn−1(X, x0) for all n ≥ 1. Taking n = 2, we can thus realize any group G, abelian or not, as a relative π2 by choosing X to have π1(X) ≈ G. The long exact sequence of homotopy groups is clearly natural: A map of basepointed triples (X, A, B, x0)→(Y, C, D, y0) induces a map between the associated long exact sequences, with commuting squares. There are change-of-basepoint isomorphisms βγ for relative homotopy groups analogous to those in the absolute case. One starts with a path γ in A ⊂ X from x0 to x1, and this induces βγ : πn(X, A, x1)→πn(X, A, x0) by setting βγ([f ]) = [γf ] where γf is deﬁned as in the picture, by placing a copy of f in a smaller cube with its face In−1 centered in the corresponding face of the larger cube. This construction satisﬁes the same basic properties as in the absolute case, with very similar proofs that we leave to the exercises. Separate proofs must be given in the two cases since the deﬁnition of γf in the relative case does not specialize to the deﬁnition of γf in the absolute case. The isomorphisms βγ show that πn(X, A, x0
) is independent of x0 when A is path-connected. In this case πn(X, A, x0) is often written simply as πn(X, A). Restricting to loops at the basepoint, the association γ ֏ βγ deﬁnes an action of π1(A, x0) on πn(X, A, x0) analogous to the action of π1(X, x0) on πn(X, x0) in the absolute case. In fact, it is not hard to see that π1(A, x0) acts on the whole long exact sequence of homotopy groups for (X, A, x0), the action commuting with the various maps in the sequence. 346 Chapter 4 Homotopy Theory A space X with basepoint x0 is said to be n connected if πi(X, x0) = 0 for i ≤ n. Thus 0 connected means path-connected and 1 connected means simplyconnected. Since n connected implies 0 connected, the choice of the basepoint x0 is not signiﬁcant. The condition of being n connected can be expressed without mention of a basepoint since it is an easy exercise to check that the following three conditions are equivalent. (1) Every map S i→X is homotopic to a constant map. (2) Every map S i→X extends to a map Di+1→X. (3) πi(X, x0) = 0 for all x0 ∈ X. Thus X is n connected if any one of these three conditions holds for all i ≤ n. Similarly, in the relative case it is not hard to see that the following four conditions are equivalent, for i > 0 : (1) Every map (Di, ∂Di)→(X, A) is homotopic rel ∂Di to a map Di→A. (2) Every map (Di, ∂Di)→(X, A) is homotopic through such maps to a map Di→A. (3) Every map (Di, ∂Di)→(X, A) is homotopic through such maps to a constant map Di→A. (4) πi(X, A, x0) = 0 for all x0 ∈ A. When i = 0 we
did not deﬁne the relative π0, and (1)–(3) are each equivalent to saying that each path-component of X contains points in A since D0 is a point and ∂D0 is empty. The pair (X, A) is called n connected if (1)–(4) hold for all i ≤ n, i > 0, and (1)–(3) hold for i = 0. Note that X is n connected iﬀ (X, x0) is n connected for some x0 and hence for all x0. Whitehead’s Theorem Since CW complexes are built using attaching maps whose domains are spheres, it is perhaps not too surprising that homotopy groups of CW complexes carry a lot of information. Whitehead’s theorem makes this explicit: Theorem 4.5. If a map f : X→Y between connected CW complexes induces isomorphisms f∗ : πn(X)→πn(Y ) for all n, then f is a homotopy equivalence. In case f is the inclusion of a subcomplex X ֓ Y, the conclusion is stronger: X is a deformation retract of Y. The proof will follow rather easily from a more technical result that turns out to be very useful in quite a number of arguments. For convenient reference we call this the compression lemma. Lemma 4.6. Let (X, A) be a CW pair and let (Y, B) be any pair with B ≠ ∅. For each n such that X − A has cells of dimension n, assume that πn(Y, B, y0) = 0 for all y0 ∈ B. Then every map f : (X, A)→(Y, B) is homotopic rel A to a map X→B. Homotopy Groups Section 4.1 347 When n = 0 the condition that πn(Y, B, y0) = 0 for all y0 ∈ B is to be regarded as saying that (Y, B) is 0 connected. Φ If Proof: Assume inductively that f has already been homotoped to take the skeleton X k−1 to B. is the characteristic map of a cell ek of X − A, the composition : (Dk, ∂Dk)→(Y, B) can be homotoped into B
rel ∂Dk in view of the hypothesis f that πk(Y, B, y0) = 0 if k > 0, or that (Y, B) is 0 connected if k = 0. This homotopy Φ induces a homotopy of f on the quotient space X k−1 ∪ ek of X k−1 ∐ Dk, a of f homotopy rel X k−1. Doing this for all k cells of X − A simultaneously, and taking the constant homotopy on A, we obtain a homotopy of f \|\| X k ∪ A to a map into B. By the homotopy extension property in Proposition 0.16, this homotopy extends to a Φ homotopy deﬁned on all of X, and the induction step is completed. Finitely many applications of the induction step ﬁnish the proof if the cells of X − A are of bounded dimension. In the general case we perform the homotopy of the induction step during the t interval [1 − 1/2k, 1 − 1/2k+1]. Any ﬁnite skeleton X k is eventually stationary under these homotopies, hence we have a well-deﬁned ⊔⊓ homotopy ft, t ∈ [0, 1], with f1(X) ⊂ B. Proof of Whitehead’s Theorem: In the special case that f is the inclusion of a subcomplex, consider the long exact sequence of homotopy groups for the pair (Y, X). Since f induces isomorphisms on all homotopy groups, the relative groups πn(Y, X) are zero. Applying the lemma to the identity map (Y, X)→(Y, X) then yields a deformation retraction of Y onto X. The general case can be proved using mapping cylinders. Recall that the mapping cylinder Mf of a map f : X→Y is the quotient space of the disjoint union of X × I and Y under the identiﬁcations (x, 1) ∼ f (x). Thus Mf contains both X = X × {0} and Y as subspaces, and Mf deformation retracts onto Y. The map f becomes the composition of the inclusion X֓Mf with the ret
raction Mf →Y. Since this retraction is a homotopy equivalence, it suﬃces to show that Mf deformation retracts onto X if f induces isomorphisms on homotopy groups, or equivalently, if the relative groups πn(Mf, X) are all zero. If the map f happens to be cellular, taking the n skeleton of X to the n skeleton of Y for all n, then (Mf, X) is a CW pair and so we are done by the ﬁrst paragraph of the proof. If f is not cellular, we can either appeal to Theorem 4.8 which says that f is homotopic to a cellular map, or we can use the following argument. First apply the preceding lemma to obtain a homotopy rel X of the inclusion (X ∪Y, X)֓(Mf, X) to a map into X. Since the pair (Mf, X ∪ Y ) obviously satisﬁes the homotopy extension property, this homotopy extends to a homotopy from the identity map of Mf to a map g : Mf →Mf taking X ∪ Y into X. Then apply the lemma again to the composition g-----→ (Mf, X) to ﬁnish the construction of a (X × I ∐ Y, X × ∂I ∐ Y ) -→ (Mf, X ∪ Y ) ⊔⊓ deformation retraction of Mf onto X. 348 Chapter 4 Homotopy Theory Whitehead’s theorem does not say that two CW complexes X and Y with isomor- phic homotopy groups are homotopy equivalent, since there is a big diﬀerence be- tween saying that X and Y have isomorphic homotopy groups and saying that there is a map X→Y inducing isomorphisms on homotopy groups. For example, consider X = RP2 and Y = S 2 × RP∞. These both have fundamental group Z2, and Proposition 4.1 implies that their higher homotopy groups are isomorphic since their universal covers S 2 and S 2 × S ∞ are homotopy equivalent, S ∞ being contractible. But RP2 and S 2 × RP∞ are not homotopy equivalent since their homology groups are vastly different, S 2
× RP∞ having nonvanishing homology in inﬁnitely many dimensions since it retracts onto RP∞. Another pair of CW complexes that are not homotopy equivalent but have isomorphic homotopy groups is S 2 and S 3 × CP∞, as we shall see in Example 4.51. In fact it turns out to be quite rare that the homotopy type of a CW complex is determined by its homotopy groups. One very special case when the homotopy type of a CW complex is determined by its homotopy groups is when all the homotopy groups are trivial, for then the inclusion map of a 0 cell into the complex induces an isomorphism on homotopy groups, so the complex deformation retracts to the 0 cell. Somewhat similar in spirit to the compression lemma is the following rather basic extension lemma: Lemma 4.7. Given a CW pair (X, A) and a map f : A→Y with Y path-connected, then f can be extended to a map X→Y if πn−1(Y ) = 0 for all n such that X − A has cells of dimension n. Proof: Assume inductively that f has been extended over the (n − 1) skeleton. Then an extension over an n cell exists iﬀ the composition of the cell’s attaching map S n−1→X n−1 with f : X n−1→Y is nullhomotopic. ⊔⊓ Cellular Approximation An intuitively appealing strategy for proving that πn(S k) = 0 for n < k would be to show ﬁrst that every map S n→S k can be deformed to make its image miss at least one point of S k, and then use the fact that the complement of a point in S k is contractible to ﬁnish the proof. One might think that the ﬁrst step was unnecessary, that no continuous map S n→S k could be surjective when n < k, but it is not hard to use space-ﬁlling curves from point-set topology to produce such maps. Thus to make this strategy into a valid proof some work must be done to construct homotopies eliminating the strange behavior of these dimension-raising maps. For maps between CW complexes it turns out to be suﬃcient for
this and many other purposes in homotopy theory to require just that cells map to cells of the same or lower dimension. Such a map f : X→Y, satisfying f (X n) ⊂ Y n for all n, is called Homotopy Groups Section 4.1 349 a cellular map. It is a fundamental fact that arbitrary maps can always be deformed to be cellular. This is the cellular approximation theorem: Theorem 4.8. Every map f : X→Y of CW complexes is homotopic to a cellular map. If f is already cellular on a subcomplex A ⊂ X, the homotopy may be taken to be stationary on A. Corollary 4.9. πn(S k) = 0 for n < k. Proof: If S n and S k are given their usual CW structures, with the 0 cells as basepoints, then every basepoint-preserving map S n→S k can be homotoped, ﬁxing the basepoint, ⊔⊓ to be cellular, and hence constant if n < k. Linear maps cannot increase dimension, so one might try to prove the theorem by showing that arbitrary maps between CW complexes can be homotoped to have some sort of linearity properties. For simplicial complexes the simplicial approximation theorem, Theorem 2C.1, achieves this, and cellular approximation can be regarded as a CW analog of simplicial approximation since simplicial maps are cellular. However, simplicial maps are much more rigid than cellular maps, which perhaps explains why subdivision of the domain is required for simplicial approximation but not for cellular approximation. The core of the proof of cellular approximation will be a weak form of simplicial approximation that can be proved by a rather elementary direct argument. Proof of 4.8: Suppose inductively that f : X→Y is already cellular on the skeleton X n−1, and let en be an n cell of X. The closure of en in X is compact, being the image of a characteristic map for en, so f takes the closure of en to a compact set in Y. Since a compact set in a CW complex can meet only ﬁnitely many cells by Proposition A.1 in the Appendix, it follows that f (en) meets only ﬁnitely many cells of Y. Let ek ⊂ Y be a cell of highest dimension meeting f (en). We may assume k
> n, otherwise f is already cellular on en. We will show below that it is possible to deform f \|\| X n−1 ∪ en, staying ﬁxed on X n−1, so that f (en) misses some point p ∈ ek. Then we can deform f \|\| X n−1 ∪ en rel X n−1 so that f (en) misses the whole cell ek by composing with a deformation retraction of Y k − {p} onto Y k − ek. By ﬁnitely many iterations of this process we eventually make f (en) miss all cells of dimension greater than n. Doing this for all n cells simultaneously, staying ﬁxed on n cells in A where f is already cellular, we obtain a homotopy of f \|\| X n rel X n−1 ∪ An to a cellular map. The induction step is then completed by appealing to the homotopy extension property in Proposition 0.16 to extend this homotopy, together with the constant homotopy on A, to a homotopy deﬁned on all of X. Letting n go to ∞, the resulting possibly inﬁnite string of homotopies can be realized as a single homotopy by performing the nth homotopy during the t interval [1 − 1/2n, 1 − 1/2n+1]. This makes sense since each point of X lies in some X n, which is eventually stationary in the inﬁnite chain of homotopies. 350 Chapter 4 Homotopy Theory To ﬁll in the missing step in this argument we will need a technical lemma about deforming maps to create some linearity. Deﬁne a polyhedron in Rn to be a subspace that is the union of ﬁnitely many convex polyhedra, each of which is a compact set form obtained by intersecting ﬁnitely many half-spaces deﬁned by linear inequalities of the i aixi ≤ b. By a PL (piecewise linear) map from a polyhedron to Rk we shall mean a map which is linear when restricted to each convex polyhedron in some such P decomposition of the polyhedron into convex polyhedra. Lemma 4.10. Let f : In→Z be a map
, where Z is obtained from a subspace W by rel f −1(W ) attaching a cell ek. Then there is a homotopy ft : from f = f0 to a map f1 for which there is a polyhedron K ⊂ In such that : (a) f1(K) ⊂ ek and f1 \|\| K is PL with respect to some identiﬁcation of ek with Rk. (b) K ⊃ f −1 1 (U) for some nonempty open set U in ek. In, f −1(ek) → Z, ek Before proving the lemma, let us see how it ﬁnishes the proof of the cellular approximation theorem. Composing the given map f : X n−1 ∪ en→Y k with a characteristic map In→X for en, we obtain a map f as in the lemma, with Z = Y k and W = Y k − ek. The homotopy given by the lemma is ﬁxed on ∂In, hence induces a homotopy ft of f \|\| X n−1 ∪ en ﬁxed on X n−1. The image of the resulting map f1 intersects the open set U in ek in a set contained in the union of ﬁnitely many hyperplanes of dimension at most n, so if n < k there will be points p ∈ U not in the ⊔⊓ image of f1. Proof of 4.10: Identifying ek with Rk, let B1, B2 ⊂ ek be the closed balls of radius 1 and 2 centered at the origin. Since f −1(B2) is closed and therefore compact in In, it follows that f is uniformly continuous on f −1(B2). Thus there exists ε > 0 such that \|x − y\| < ε implies \|f (x) − f (y)\| < 1/2 for all x, y ∈ f −1(B2). Subdivide the interval I so that the induced subdivision of In into cubes has each cube lying in a ball of diameter less than ε. Let K1 be the union of all the cubes meeting f −1(B1), and let K2 be the union of all the cubes meeting K1. We may assume ε is
chosen smaller than half the distance between the compact sets f −1(B1) and In − f −1(int(B2)), and then we will have K2 ⊂ f −1(B2). Homotopy Groups Section 4.1 351 Now we subdivide all the cubes of K2 into simplices. This can be done inductively. The boundary of each cube is a union of cubes of one lower dimension, so assuming these lower-dimensional cubes have already been subdivided into simplices, we obtain a subdivision of the cube itself by taking its center point as a new vertex and joining this by a cone to each simplex in the boundary of the cube. Let g : K2→ek = Rk be the map that equals f on all vertices of simplices of the subdivision and is linear on each simplex. Let ϕ : K2→[0, 1] be the map that is linear on simplices and has the value 1 on vertices in K1 and 0 on vertices in K2 − K1. Thus ϕ(K1) = 1. Deﬁne a homotopy ft : K2→ek by the formula (1 − tϕ)f + (tϕ)g, so f0 = f and f1 \|\| K1 = g \|\|K1. Since ft is the constant homotopy on simplices in K2 disjoint from K1, and in particular on simplices in the closure of In − K2, we may extend ft to be the constant homotopy of f on In − K2. The map f1 takes the closure of In − K1 to a compact set C which, we claim, is disjoint from the centerpoint 0 of B1 and hence from a neighborhood U of 0. This will prove the lemma, with K = K1, since we will then have f −1 1 (U) ⊂ K1 with f1 PL on K1 where it is equal to g. The veriﬁcation of the claim has two steps: (1) On In − K2 we have f1 = f, and f takes In − K2 outside B1 since f −1(B1) ⊂ K2 by construction. (2) For a simplex σ of K2 not in K1 we have f (σ ) contained in some ball Bσ of radius 1/
2 by the choice of ε and the fact that K2 ⊂ f −1(B2). Since f (σ ) ⊂ Bσ and Bσ is convex, we must have g(σ ) ⊂ Bσ, hence also ft(σ ) ⊂ Bσ for all t, and in particular f1(σ ) ⊂ Bσ. We know that Bσ is not contained in B1 since σ contains points outside K1 hence outside f −1(B1). The radius of Bσ is half that ⊔⊓ of B1, so it follows that 0 is not in Bσ, and hence 0 is not in f1(σ ). Example 4.11: Cellular Approximation for Pairs. Every map f : (X, A)→(Y, B) of CW pairs can be deformed through maps (X, A)→(Y, B) to a cellular map. This follows from the theorem by ﬁrst deforming the restriction f : A→B to be cellular, then extending this to a homotopy of f on all of X, then deforming the resulting map to be cellular staying ﬁxed on A. As a further reﬁnement, the homotopy of f can be taken to be stationary on any subcomplex of X where f is already cellular. An easy consequence of this is: Corollary 4.12. A CW pair (X, A) is n connected if all the cells in X − A have dimension greater than n. In particular the pair (X, X n) is n connected, hence the inclusion X n ֓ X induces isomorphisms on πi for i < n and a surjection on πn. Proof: Applying cellular approximation to maps (Di, ∂Di)→(X, A) with i ≤ n gives the ﬁrst statement. The last statement comes from the long exact sequence of the pair (X, X n). ⊔⊓ 352 Chapter 4 Homotopy Theory CW Approximation A map f : X→Y is called a weak homotopy equivalence if it induces isomorphisms πn(X, x0)→πn for all n ≥ 0 and all choices of basepoint x0. Whitehead’s theorem can be restated as saying
that a weak homotopy equivalence Y, f (x0) between CW complexes is a homotopy equivalence. It follows easily that this holds also for spaces homotopy equivalent to CW complexes. In general, however, weak homotopy equivalence is strictly weaker than homotopy equivalence. For example, there exist noncontractible spaces whose homotopy groups are all trivial, such as the ‘quasi-circle’ according to an exercise at the end of this section, and for such spaces a map to a point is a weak homotopy equivalence that is not a homotopy equivalence. We will show that for every space X there is a CW complex Z and a weak homotopy equivalence f : Z→X. Such a map f : Z→X is called a CW approximation to X. A weak homotopy equivalence induces isomorphisms on all homology and cohomol- ogy groups, as we will see, so CW approximations allow many general statements in algebraic topology to be proved using cell-by-cell arguments for CW complexes. CW approximations to a given space X are unique up to homotopy equivalence since if f : Z→X and f ′ : Z ′→X are two CW approximations, then the composition Z→X ֓ Mf ′ can be deformed into Z ′ by the compression lemma, giving a map Z→Z ′ which is a weak homotopy equivalence and hence a homotopy equivalence. The construction of a CW approximation f : Z→X for a space X is inductive, so let us describe the induction step. Suppose given a CW complex A with a map f : A→X and suppose we have chosen a basepoint 0 cell aγ in each component of A. Then for an integer k ≥ 0 we will attach k cells to A to form a CW complex B with a map f : B→X extending the given f, such that: (∗) For each basepoint aγ the induced map f∗ : πi(B, aγ )→πi(X, f (aγ )) is injective for i = k − 1 (when k > 0 ) and surjective for i = k. There are two steps in the construction, the ﬁrst of which is omitted when k = 0 : (1)
Choose maps ϕα : (S k−1, s0)→(A, aγ) representing all nontrivial elements of the kernels of the maps f∗ : πk−1(A, aγ)→πk−1(X, f (aγ )) for all the basepoints aγ. We may assume the maps ϕα are cellular, where S k−1 has its standard CW structure with s0 as 0 cell. Attaching cells ek α to A via the maps ϕα then produces a CW complex, and the map f extends over these cells using nullhomotopies of the compositions f ϕα, which exist by the choice of the ϕα ’s. (2) Choose maps fβ : S k→X representing all nontrivial elements of πk(X, f (aγ )) for β to A via the constant maps at the appropriate all the aγ ’s, then attach cells ek basepoints aγ and extend f over the resulting spheres S k β via the maps fβ. The surjectivity condition in (∗) then holds by construction. For the injectivity condition, a nontrivial element of the kernel of f∗ : πk−1(B, aγ)→πk−1(X, f (aγ)) can be Homotopy Groups Section 4.1 353 represented by a cellular map h : S k−1→B. This has image in A, so is in the kernel of f∗ : πk−1(A, aγ)→πk−1(X, f (aγ )) and hence is homotopic to some ϕα and is therefore nullhomotopic in B. More eﬃciently, in step (1) it suﬃces to attach cells just for generators of the kernels when k > 1, and just for generators of πk(X, f (aγ)) in step (2) when k > 0. Note that if the given map f : A→X happened to be injective or surjective on πi for some i < k − 1 or i < k, respectively, then this remains true after attaching the k cells. This is because attaching k cells does not aﬀect πi if i < k − 1, by cellular approximation, nor
does it destroy surjectivity on πk−1 or indeed any πi, obviously. Now to construct a CW approximation f : Z→X one can start with A consisting of one point for each path-component of X, with f : A→X mapping each of these points to the corresponding path-component. Having now a bijection on π0, attach 1 cells to A to create a surjection on π1 for each path-component, then 2 cells to improve this to an isomorphism on π1 and a surjection on π2, and so on for each successive πi in turn. After all cells have been attached one has a CW complex Z with a weak homotopy equivalence f : Z→X. This proves: Proposition 4.13. Every space X has a CW approximation f : Z→X. If X is pathconnected, Z can be chosen to have a single 0 cell, with all other cells attached by basepoint-preserving maps. Thus every connected CW complex is homotopy equiva- lent to a CW complex with these additional properties. ⊔⊓ Example 4.14. One can also apply this technique to produce a CW approximation to a pair (X, X0). First construct a CW approximation f0 : Z0→X0, then starting with the composition Z0→X0 ֓ X, attach cells to Z0 to create a weak homotopy equivalence f : Z→X extending f0. By the ﬁve-lemma, the map f : (Z, Z0)→(X, X0) induces isomorphisms on relative as well as absolute homotopy groups. Here is another application of the technique, giving a more geometric interpreta- tion to the homotopy-theoretic notion of n connectedness: Proposition 4.15. If (X, A) is an n connected CW pair, then there exists a CW pair (Z, A) ≃ (X, A) rel A such that all cells of Z − A have dimension greater than n. Proof: Starting with the inclusion A ֓ X, attach cells to A of dimension n + 1 and higher to produce a CW complex Z and a map f : Z→X that is the identity on A and induces an injection on πn and isomorphisms on all higher homotopy groups. The induced map on π
n is also surjective since this is true for the composition f-----→ X by the hypothesis that (X, A) is n connected. In dimensions below n, f A֓Z induces isomorphisms on homotopy groups since both inclusions A֓ Z and A֓ X do. Thus f is a weak homotopy equivalence, and hence a homotopy equivalence. To see that f is a homotopy equivalence rel A we could apply Proposition 0.19, but here is an alternative argument. Let W be the quotient space of the mapping cylin- 354 Chapter 4 Homotopy Theory der Mf obtained by collapsing each segment {a}× I to a point, for a ∈ A. Assuming f has been made cellular, W is a CW complex containing X and Z as subcomplexes, and W deformation retracts to X just as Mf does. Also, πi(W, Z) = 0 for all i since f induces isomorphisms on all homotopy groups, so W deformation retracts onto Z by Theorem 4.5. These two deformation retractions of W onto X and Z are stationary on A, hence give a homotopy equivalence X ≃ Z rel A. ⊔⊓ Example 4.16: Postnikov Towers. We can also apply the technique to construct, for each connected CW complex X and each integer n ≥ 1, a CW complex Xn containing X as a subcomplex such that: (a) πi(Xn) = 0 for i > n. (b) The inclusion X ֓ Xn induces an isomorphism on πi for i ≤ n. To do this, all we have to do is apply the general construction to the constant map of X to a point, starting at the stage of attaching cells of dimension n + 2. Thus we attach (n + 2) cells to X using cellular maps S n+1→X that generate πn+1(X) to form a space with πn+1 trivial, then for this space we attach (n + 3) cells to make πn+2 trivial, and so on. The result is a CW complex Xn with the desired properties. The inclusion X ֓ Xn extends to a map Xn+1→Xn since Xn+1 is obtained from X by attaching cells of dimension n + 3
and greater, and πi(Xn) = 0 for i > n so we can apply Lemma 4.7, the extension lemma. Thus we have a commu- tative diagram as at the right. This is a called a Postnikov tower for X. One can regard the spaces Xn as truncations of X which provide successively better approximations to X as n increases. Postnikov towers turn out to be quite powerful tools for proving general theorems, and we will study them further in §4.3. Now that we have seen several varied applications of the technique of attaching cells to make a map f : A→X more nearly a weak homotopy equivalence, it might be useful to give a name to the properties that the construction can achieve. To simplify the description, we may assume without loss of generality that the given f is an inclusion A ֓ X by replacing X by the mapping cylinder of f. Thus, starting with a pair (X, A) where the subspace A ⊂ X is a nonempty CW complex, we deﬁne an n connected CW model for (X, A) to be an n connected CW pair (Z, A) and a map f : Z→X with f \|\| A the identity, such that f∗ : πi(Z)→πi(X) is an isomorphism for i > n and an injection for i = n, for all choices of basepoint. Since (Z, A) is n connected, the map πi(A)→πi(Z) is an isomorphism for i < n and a surjection f-----→ X induce a composition for i = n. In the critical dimension n, the maps A ֓ Z πn(A)→πn(Z)→πn(X) factoring the map πn(A)→πn(X) as a surjection followed by an injection, just as any homomorphism ϕ : G→H can be factored (uniquely) as a surjection ϕ : G→ Im ϕ followed by an injection Im ϕ ֓ H. One can think of Z as Homotopy Groups Section 4.1 355 a sort of homotopy-theoretic hybrid of A and X. As n increases, the hybrid looks more and more like A, and less and less like X
. Our earlier construction shows: Proposition 4.17. For every pair (X, A) with A a nonempty CW complex there exist n connected CW models f : (Z, A)→(X, A) for all n ≥ 0, and these models can be chosen to have the additional property that Z is obtained from A by attaching cells of dimension greater than n. ⊔⊓ The construction of n connected CW models involves many arbitrary choices, so it may be somewhat surprising that they turn out to be unique up to homotopy equivalence. This will follow easily from the next proposition. Another application of the proposition will be to build a tower like the Postnikov tower from the various n -connected CW models for a given pair (X, A). Proposition 4.18. Suppose we are given : (i) an n connected CW model f : (Z, A)→(X, A), (ii) an n′ connected CW model f ′ : (Z ′, A′)→(X ′, A′), (iii) a map g : (X, A)→(X ′, A′). Then if n ≥ n′, there is a map h : Z→Z ′ such that h \|\| A = g and gf ≃ f ′h rel A, so the diagram above is commutative up to homotopy rel A. Furthermore, such a map h is unique up to homotopy rel A. Proof: By Proposition 4.15 we may assume all cells of Z − A have dimension greater than n. Let W be the quotient space of the mapping cylinder of f ′ obtained by collapsing each line segment {a′}× I to a point, for a′ ∈ A′. We can think of W as a relative mapping cylinder, and like the ordinary mapping cylinder, W contains copies of Z ′ and X ′, the latter as a deformation retract. The assumption that (Z ′, A′) is an n′ connected CW model for (X ′, A′) implies that the relative groups πi(W, Z ′) are zero for i > n′. Via the inclusion X ′ ֓ W we can view gf as a map Z→W. As a map of pairs (Z, A)→(W, Z ′), gf is homotopic rel A
to a map h with image in Z ′, by the compression lemma and the hypothesis n ≥ n′. This proves the ﬁrst assertion. For the second, suppose h0 and h1 are two maps Z→Z ′ whose compositions with f ′ are homotopic to gf rel A. Thus if we regard h0 and h1 as maps to W, they are homotopic rel A. Such a homotopy gives a map (Z × I, Z × ∂I ∪ A× I)→(W, Z ′), and by the compression lemma again this map can be deformed rel Z × ∂I ∪ A× I to a map with image in Z ′, which gives the desired homotopy h0 ≃ h1 rel A. ⊔⊓ Corollary 4.19. An n connected CW model for (X, A) is unique up to homotopy equivalence rel A. Proof: Given two n connected CW models (Z, A) and (Z ′, A) for (X, A), we apply the proposition twice with g the identity map to obtain maps h : Z→Z ′ and h′ : Z ′→Z. The uniqueness statement gives homotopies hh′ ≃ 11 and h′h ≃ 11 rel A. ⊔⊓ 356 Chapter 4 Homotopy Theory Taking n = n′ in the proposition, we obtain also a functoriality property for n connected CW models. For example, a map X→X ′ induces a map of CW approximations Z→Z ′, which is unique up to homotopy. The proposition allows us to relate n connected CW models (Zn, A) for (X, A) for varying n, by means of maps Zn→Zn−1 that form a tower as shown in the diagram, with commutative triangles on the left and homotopy-commutative triangles on the right. We can make the triangles on the right strictly commutative by replacing the maps Zn→X by the compositions through Z0. Example 4.20: Whitehead Towers. If we take X to be an arbitrary CW complex with the subspace A a point, then the resulting tower of n connected CW models amounts to a sequence of maps ··· →Z2→Z1→Z0→X
with Zn n connected and the map Zn→X inducing an isomorphism on all homotopy groups πi with i > n. The space Z0 is path-connected and homotopy equivalent to the component of X containing A, so one may as well assume Z0 equals this component. The next space Z1 is simply-connected, and the map Z1→X has the homotopy properties of the universal cover of the component Z0 of X. For larger values of n one can by analogy view the map Zn→X as an ‘ n connected cover’ of X. For n > 1 these do not seem to arise so frequently in nature as in the case n = 1. A rare exception is the Hopf map S 3→S 2 deﬁned in Example 4.45, which is a 2 connected cover. Now let us show that CW approximations behave well with respect to homology and cohomology: Proposition 4.21. A weak homotopy equivalence f : X→Y induces isomorphisms f∗ : Hn(X; G)→Hn(Y ; G) and f ∗ : H n(Y ; G)→H n(X; G) for all n and all coeﬃcient groups G. Proof: Replacing Y by the mapping cylinder Mf and looking at the long exact sequences of homotopy, homology, and cohomology groups for (Mf, X), we see that it suﬃces to show: If (Z, X) is an n connected pair of path-connected spaces, then Hi(Z, X; G) = 0 and H i(Z, X; G) = 0 for all i ≤ n and all G. P Let α = j njσj be a relative cycle representing an element of Hk(Z, X; G), for singular k simplices σj : complex K from a disjoint union of k simplices, one for each σj, by identifying all (k − 1) dimensional faces of these k simplices for which the corresponding restrictions of the σj ’s are equal. Thus the σj ’s induce a map σ : K→Z. Since α is a relative cycle, ∂α is a chain in X. Let k→Z. Build a ﬁnite ∆ �
� Homotopy Groups Section 4.1 357 α in K, with ∂ L ⊂ K be the subcomplex consisting of (k − 1) simplices corresponding to the singular (k − 1) simplices in ∂α, so σ (L) ⊂ X. The chain α is the image under the chain map σ♯ of a chain α a chain in L. In relative homology we then α] = [α]. If we assume πi(Z, X) = 0 for i ≤ k, then σ : (K, L)→(Z, X) is have σ∗[ e homotopic rel L to a map with image in X, by the compression lemma. Hence σ∗[ α] is in the image of the map Hk(X, X; G)→Hk(Z, X; G), and since Hk(X, X; G) = 0 we e conclude that [α] = σ∗[ α] = 0. This proves the result for homology, and the result for cohomology then follows by the universal coeﬃcient theorem. ⊔⊓ e e e CW approximations can be used to reduce many statements about general spaces to the special case of CW complexes. For example, the cup product version of the K¨unneth formula in Theorem 3.15, asserting that H ∗(X × Y ; R) ≈ H ∗(X; R) ⊗ H ∗(Y ; R) under certain conditions, can now be extended to non-CW spaces since if X and Y are CW approximations to spaces Z and W, respectively, then X × Y is a CW approximation to Z × W. Here we are giving X × Y the CW topology rather than the product topology, but this has no eﬀect on homotopy groups since the two topologies have the same compact sets, as explained in the Appendix. Similarly, the general K¨unneth formula for homology in §3.B holds for arbitrary products X × Y. The condition for a map Y →Z to be a weak homotopy equivalence involves only maps of spheres into Y and Z, but in fact weak homotopy equivalences Y →Z behave nicely with respect to maps of arbitrary CW complexes into Y and Z, not
just spheres. The following proposition gives a precise statement, using the notations [X, Y ] for the set of homotopy classes of maps X→Y and hX, Y i for the set of basepointpreserving-homotopy classes of basepoint-preserving maps X→Y. (The notation hX, Y i is not standard, but is intended to suggest ‘pointed homotopy classes’.) Proposition 4.22. A weak homotopy equivalence f : Y -→ Z induces bijections [X, Y ]→[X, Z] and hX, Y i→hX, Zi for all CW complexes X. Proof: Consider ﬁrst [X, Y ]→[X, Z]. We may assume f is an inclusion by replacing Z by the mapping cylinder Mf as usual. The groups πn(Z, Y, y0) are then zero for all n and all basepoints y0 ∈ Y, so the compression lemma implies that any map X→Z can be homotoped to have image in Y. This gives surjectivity of [X, Y ]→[X, Z]. A relative version of this argument shows injectivity since we can deform a homotopy (X × I, X × ∂I)→(Z, Y ) to have image in Y. In the case of hX, Y i→hX, Zi the same argument applies if Mf is replaced by the reduced mapping cylinder, the quotient of Mf obtained by collapsing the segment {y0}× I to a point, for y0 the basepoint of Y. This collapsed segment then serves as the common basepoint of Y, Z, and the reduced mapping cylinder. The reduced mapping cylinder deformation retracts to Z just as the unreduced one does, but with the advantage that the basepoint does not move. ⊔⊓ 358 Chapter 4 Homotopy Theory Exercises 1. Suppose a sum f +′g of maps f, g : (In, ∂In)→(X, x0) is deﬁned using a coordinate of In other than the ﬁrst coordinate as in the usual sum f + g. Verify the formula (f + g) +′ (h + k) = (f +′ h) + (g +′ k)
, and deduce that f +′ k ≃ f + k so the two sums agree on πn(X, x0), and also that g +′ h ≃ h + g so the addition is abelian. 2. Show that if ϕ : X→Y is a homotopy equivalence, then the induced homomorphisms ϕ∗ : πn(X, x0)→πn(Y, ϕ(x0)) are isomorphisms for all n. [The case n = 1 is Proposition 1.18.] 3. For an H–space (X, x0) with multiplication µ : X × X→X, show that the group operation in πn(X, x0) can also be deﬁned by the rule (f + g)(x) = µ. X→X be the universal cover of a path-connected space X. Show that X), which holds for n ≥ 2, the action of 4. Let p : under the isomorphism πn(X) ≈ πn( π1(X) on πn(X) corresponds to the action of π1(X) on πn( tion of π1(X) on γp∗(α) = p∗ homomorphism induced by the action of γ on where γ ∈ π1(X, x0), α ∈ πn( X as deck transformations. More precisely, prove a formula like x0), and γ∗ denotes the X) induced by the ac- γ(γ∗(α)) e f (x), g(x) X. X, β e e e e e e 5. For a pair (X, A) of path-connected spaces, show that π1(X, A, x0) can be identiﬁed in a natural way with the set of cosets Hα of the subgroup H ⊂ π1(X, x0) represented by loops in A at x0. e X, x0)→(X, A, x0) is a covering space with 6. If p : ( A, map p∗ : πn( A, X, e e x0)→πn(X, A, x0) is an isomorphism
for all n > 1. e e A = p−1(A), show that the e e e 7. Extend the results proved near the beginning of this section for the change-ofbasepoint maps βγ to the case of relative homotopy groups. ∂-----→ π0(A, x0) -→ π0(X, x0) is exact. 8. Show the sequence π1(X, x0) -→ π1(X, A, x0), 9. Suppose we deﬁne π0(X, A, x0) to be the quotient set π0(X, x0)/i∗ π0(A, x0) so that the long exact sequence of homotopy groups for the pair (X, A) extends to ··· -→ π0(A, x0) (a) Show that with this extension, the ﬁve-lemma holds for the map of long exact sequences induced by a map (X, A, x0)→(Y, B, y0), in the following form: One of the maps between the two sequences is a bijection if the four surrounding maps are bijections for all choices of x0. i∗-----→ π0(X, x0) -→ π0(X, A, x0) -→ 0. (b) Show that the long exact sequence of a triple (X, A, B, x0) can be extended only to the term π0(A, B, x0) in general, and that the ﬁve-lemma holds for this extension. 10. Show the ‘quasi-circle’ described in Exercise 7 in §1.3 has trivial homotopy groups but is not contractible, hence does not have the homotopy type of a CW complex. 11. Show that a CW complex is contractible if it is the union of an increasing sequence of subcomplexes X1 ⊂ X2 ⊂ ··· such that each inclusion Xi ֓ Xi+1 is nullhomotopic, a condition sometimes expressed by saying Xi is contractible in Xi+1. An example is Homotopy Groups Section 4.1 359 S ∞, or more generally the inﬁnite suspension S ∞X of any CW complex X
, the union of the iterated suspensions S nX. 12. Show that an n connected, n dimensional CW complex is contractible. 13. Use the extension lemma to show that a CW complex retracts onto any contractible subcomplex. 14. Use cellular approximation to show that the n skeletons of homotopy equivalent CW complexes without cells of dimension n + 1 are also homotopy equivalent. 15. Show that every map f : S n→S n is homotopic to a multiple of the identity map by the following steps. (a) Use Lemma 4.10 (or simplicial approximation, Theorem 2C.1) to reduce to the case that there exists a point q ∈ S n with f −1(q) = {p1, ···, pk} and f is an invertible linear map near each pi. (b) For f as in (a), consider the composition gf where g : S n→S n collapses the complement of a small ball about q to the basepoint. Use this to reduce (a) further to the case k = 1. (c) Finish the argument by showing that an invertible n× n matrix can be joined by a path of such matrices to either the identity matrix or the matrix of a reﬂection. (Use Gaussian elimination, for example.) 16. Show that a map f : X→Y between connected CW complexes factors as a composition X→Zn→Y where the ﬁrst map induces isomorphisms on πi for i ≤ n and the second map induces isomorphisms on πi for i ≥ n + 1. 17. Show that if X and Y are CW complexes with X m connected and Y n connected, then (X × Y, X ∨ Y ) is (m + n + 1) connected, as is the smash product X ∧ Y. 18. Give an example of a weak homotopy equivalence X→Y for which there does not exist a weak homotopy equivalence Y →X. 19. Consider the equivalence relation ≃w generated by weak homotopy equivalence: X ≃w Y if there are spaces X = X1, X2, ···, Xn = Y with weak homotopy equivalences Xi→Xi+1 or Xi← Xi+1 for each i. Show that X ≃w Y iﬀ X and
Y have a common CW approximation. 20. Show that [X, Y ] is ﬁnite if X is a ﬁnite connected CW complex and πi(Y ) is ﬁnite for i ≤ dim X. 21. For this problem it is convenient to use the notations X n for the nth stage in a Postnikov tower for X and Xm for an (m − 1) connected covering of X, where X is a connected CW complex. Show that (X n)m ≃ (Xm)n, so the notation X n m) ≈ πi(X) for m ≤ i ≤ n and all other homotopy groups of X n m is unambiguous. Thus πi(X n m are zero. 22. Show that a path-connected space X has a CW approximation with countably many cells iﬀ πn(X) is countable for all n. [Use the results on simplicial approximations to maps and spaces in §2.C.] 360 Chapter 4 Homotopy Theory 23. If f : X→Y is a map with X and Y homotopy equivalent to CW complexes, show that the pair (Mf, X) is homotopy equivalent to a CW pair, where Mf is the mapping cylinder. Deduce that the mapping cone Cf has the homotopy type of a CW complex. We have not yet computed any nonzero homotopy groups πn(X) with n ≥ 2. In Chapter 1 the two main tools we used for computing fundamental groups were van Kampen’s theorem and covering spaces. In the present section we will study the higher-dimensional analogs of these: the excision theorem for homotopy groups, and ﬁber bundles. Both of these are quite a bit weaker than their fundamental group analogs, in that they do not directly compute homotopy groups but only give relations between the homotopy groups of diﬀerent spaces. Their applicability is thus more limited, but suﬃces for a number of interesting calculations, such as πn(S n) and more generally the Hurewicz theorem relating the ﬁrst nonzero homotopy and homology groups of a space. Another noteworthy application is the Freudenthal suspension theorem, which leads to stable homotopy groups and in fact the whole subject of stable homotopy theory
. Excision for Homotopy Groups What makes homotopy groups so much harder to compute than homology groups is the failure of the excision property. However, there is a certain dimension range, depending on connectivities, in which excision does hold for homotopy groups: Theorem 4.23. Let X be a CW complex decomposed as the union of subcomplexes A and B with nonempty connected intersection C = A ∩ B. If (A, C) is m connected and (B, C) is n connected, m, n ≥ 0, then the map πi(A, C)→πi(X, B) induced by inclusion is an isomorphism for i < m + n and a surjection for i = m + n. This yields the Freudenthal suspension theorem: Corollary 4.24. The suspension map πi(S n)→πi+1(S n+1) is an isomorphism for i < 2n − 1 and a surjection for i = 2n − 1. More generally this holds for the suspension πi(X)→πi+1(SX) whenever X is an (n − 1) connected CW complex. Proof: Decompose the suspension SX as the union of two cones C+X and C−X intersecting in a copy of X. The suspension map is the same as the map πi(X) ≈ πi+1(C+X, X) -→ πi+1(SX, C−X) ≈ πi+1(SX) Elementary Methods of Calculation Section 4.2 361 where the two isomorphisms come from long exact sequences of pairs and the middle map is induced by inclusion. From the long exact sequence of the pair (C±X, X) we see that this pair is n connected if X is (n − 1) connected. The preceding theorem then says that the middle map is an isomorphism for i + 1 < 2n and surjective for ⊔⊓ i + 1 = 2n. In Corollary 4.25. πn(S n) ≈ Z, generated by the identity map, for all n ≥ 1. particular, the degree map πn(S n)→Z is an isomorphism. Proof: From the preceding corollary we know that in the
sequence of suspension maps π1(S 1)→π2(S 2)→π3(S 3)→ ··· the ﬁrst map is surjective and all the subsequent maps are isomorphisms. Since π1(S 1) is Z generated by the identity map, it follows that πn(S n) for n ≥ 2 is a ﬁnite or inﬁnite cyclic group independent of n, generated by the identity map. The fact that this cyclic group is inﬁnite can be deduced from homology theory since there exist basepoint-preserving maps S n→S n of arbitrary degree, and degree is a homotopy invariant. Alternatively, if one wants to avoid appealing to homology theory one can use the Hopf bundle S 1→S 3→S 2 described in Example 4.45, whose long exact sequence of homotopy groups gives an isomorphism π1(S 1) ≈ π2(S 2). The degree map πn(S n)→Z is an isomorphism since the map z ֏ zk of S 1 has ⊔⊓ degree k, as do its iterated suspensions by Proposition 2.33. Proof of 4.23: We proceed by proving successively more general cases. The ﬁrst case contains the heart of the argument, and suﬃces for the calculation of πn(S n). Case 1: A is obtained from C by attaching cells em+1 and B is obtained from C by attaching a cell en+1. To show surjectivity of πi(A, C)→πi(X, B) we start with a map f : (Ii, ∂Ii, J i−1)→(X, B, x0). This has compact image, meeting only ﬁnitely and en+1. By Lemma 4.10 we may homotope f through maps many of the cells em+1 (Ii, ∂Ii, J i−1)→(X, B, x0) so that there are simplices n+1 ⊂ en+1 for which f −1( n+1) are ﬁnite unions of convex polyhedra, on each ) and f −1( of
which f is the restriction of a linear map from Ri to Rm+1 or Rn+1. We may assume these linear maps are surjections by rechoosing smaller simplices and n+1 in the complement of the images of the nonsurjective linear maps. ⊂ em+1 α m+1 α m+1 α m+1 α and ∆ ∆ ∆ ∆ α α ∆ Claim: If i ≤ m+n, then there exist points pα ∈ ∆ n+1, and a map ϕ : Ii−1→[0, 1) such that: q ∈ (a) f −1(q) lies below the graph of ϕ in Ii−1 × I = Ii. (b) f −1(pα) lies above the graph of ϕ for each α. (c) ϕ = 0 on ∂Ii−1. ∆ ∆ m+1 α, Granting this, let ft be a homotopy of f excising the region under the graph of ϕ by restricting f to the region above the graph of tϕ for 0 ≤ t ≤ 1. By (b), ft(Ii−1) is α{pα} for all t, and by (a), f1(Ii) is disjoint from Q = {q}. This disjoint from P = S 362 Chapter 4 Homotopy Theory means that in the commutative diagram at the right the given element [f ] in the upper-right group, when regarded as an element of the lower-right group, is equal to the element [f1] in the image of the lower horizontal map. Since the vertical maps are isomorphisms, this proves the surjectivity statement. ∆ ∆ ∆ m+1 α π (f −1(q)) Now we prove the Claim. For any q ∈ n+1, f −1(q) is a ﬁnite union of convex n+1) is a ﬁnite union of convex polyhedra of dimension ≤ i − n − 1 since f −1( polyhedra on each of which f is the restriction of a linear surjection Ri→Rn+1. so that not only is f −1(q) disjoint from We wish to choose the points pα ∈ f
−1(pα) for each α, but also so that f −1(q) and f −1(pα) have disjoint images under the projection π : Ii→Ii−1. This is equivalent to saying that f −1(pα) is disjoint from, the union of all segments {x}× I meeting f −1(q). This set T is T = π −1 a ﬁnite union of convex polyhedra of dimension ≤ i − n since f −1(q) is a ﬁnite union of convex polyhedra of dimension ≤ i − n − 1. Since linear maps cannot increase dimension, f (T ) ∩ ≤ i − n. Thus if m + 1 > i − n, there is a point pα ∈ not in f (T ). This gives f −1(q) f −1(pα) ∩ T = ∅ if i ≤ m + n. Hence we can choose a neighborhood U of π for all α. Then there exists ϕ : Ii−1→[0, 1) having in Ii−1 disjoint from π f −1(pα) support in U, with f −1(q) lying under the graph of ϕ. This veriﬁes the Claim, and so ﬁnishes the proof of surjectivity in Case 1. is also a ﬁnite union of convex polyhedra of dimension m+1 α m+1 α ∆ ∆ For injectivity in Case 1 the argument is very similar. Suppose we have two maps f0, f1 : (Ii, ∂Ii, J i−1)→(A, C, x0) representing elements of πi(A, C, x0) having the same image in πi(X, B, x0). Thus there is a homotopy from f0 to f1 in the form of a map F : (Ii, ∂Ii, J i−1)× [0, 1]→(X, B, x0). After a preliminary deformation of F via Lemma 4.10, we construct a function ϕ : Ii−1 × I→[0, 1) separating F −1(q) from the sets F −
1(pα) as before. This allows us to excise F −1(q) from the domain of F, from which it follows that f0 and f1 represent the same element of πi(A, C, x0). Since Ii × I now plays the role of Ii, the dimension i is replaced by i + 1 and the dimension restriction i ≤ m + n becomes i + 1 ≤ m + n, or i < m + n. Case 2: A is obtained from C by attaching (m + 1) cells as in Case 1 and B is obtained from C by attaching cells of dimension ≥ n + 1. To show surjectivity of πi(A, C)→πi(X, B), consider a map f : (Ii, ∂Ii, J i−1)→(X, B, x0) representing an element of πi(X, B). The image of f is compact, meeting only ﬁnitely many cells, and by repeated applications of Case 1 we can push f oﬀ the cells of B − C one at a time, in order of decreasing dimension. Injectivity is quite similar, starting with a homotopy F : (Ii, ∂Ii, J i−1)× [0, 1]→(X, B, x0) and pushing this oﬀ cells of B − C. Case 3: A is obtained from C by attaching cells of dimension ≥ m + 1 and B is as in Case 2. We may assume all cells of A − C have dimension ≤ m + n + 1 since higherdimensional cells have no eﬀect on πi for i ≤ m + n, by cellular approximation. Let Elementary Methods of Calculation Section 4.2 363 Ak ⊂ A be the union of C with the cells of A of dimension ≤ k and let Xk = Ak ∪ B. We prove the result for πi(Ak, C)→πi(Xk, B) by induction on k. The induction starts with k = m + 1, which is Case 2. For the induction step consider the following commutative diagram formed by the exact sequences of the triples (Ak, Ak−1, C) and (Xk, Xk−1, B) : When i < m + n the ﬁrst and fourth
vertical maps are isomorphisms by Case 2, while by induction the second and ﬁfth maps are isomorphisms, so the middle map is an isomorphism by the ﬁve-lemma. Similarly, when i = m + n the second and fourth maps are surjective and the ﬁfth map is injective, which is enough to imply the middle map is surjective by one half of the ﬁve-lemma. When i = 2 the diagram may contain nonabelian groups and the two terms on the right may not be groups, but the ﬁve- lemma remains valid in this generality, with trivial modiﬁcations to the proof in §2.1. When i = 1 the assertion about π1(A, C)→π1(X, B) follows by a direct argument: If m ≥ 1 then both terms are trivial, while if m = 0 then n ≥ 1 and the result follows by cellular approximation. After these special cases we can now easily deal with the general case. The con- nectivity assumptions on the pairs (A, C) and (B, C) imply by Proposition 4.15 that they are homotopy equivalent to pairs (A′, C) and (B′, C) as in Case 3, via homotopy equivalences ﬁxed on C, so these homotopy equivalences ﬁt together to give a homotopy equivalence A ∪ B ≃ A′ ∪ B′. Thus the general case reduces to Case 3. ⊔⊓ Q and, sions a multiple of n, the pair ( α S n α ֓ W α. We can regard α, where S n α has the product CW structure. Since Example 4.26. The calculation of πn(S n) can be extended to show that πn( n ≥ 2 is free abelian with basis the homotopy classes of the inclusions S n Suppose ﬁrst that there are only ﬁnitely many summands S n as the n skeleton of the product α ) for is given its usual CW structure W α has cells only in dimenα S n α ) is (2n − 1) connected. Hence from the long exact sequence of homotopy groups for this pair we see that the inclusion W α S n α
induces an isomorphism on πn if n ≥ 2. By Proposition 4.2 we have α ), a free abelian group with basis the inclusions S n α πn(S n α S n α ) ≈ α, πn( Q W so the same is true for α ֓ α. This takes care of the case of ﬁnitely many S n α ’s. Q Q To reduce the case of inﬁnitely many summands S n α S n W α πn(S n is surjective since any map f : S n→ α to the ﬁnite case, consider the α S n α. α has compact image contained in W α ’s, so by the ﬁnite case already proved, [f ] is in. Similarly, a nullhomotopy of f has compact image contained in a α ) induced by the inclusions S n Then Φ the wedge sum of ﬁnitely many S n Φ the image of ﬁnite wedge sum of S n α ’s, so the ﬁnite case also implies that α ֓ α S n homomorphism α S n W α S n W α )→πn is injective. Φ 364 Chapter 4 Homotopy Theory Example 4.27. Let us show that πn(S 1 ∨ S n) for n ≥ 2 is free abelian on a countably inﬁnite number of generators. By Proposition 4.1 we may compute πi(S 1 ∨ S n) for i ≥ 2 by passing to the universal cover. This consists of a copy of R with a sphere S n k attached at each integer point k ∈ R, so it is homotopy equivalent to k. The k S n preceding Example 4.26 says that πn( k ) is free abelian with basis represented by the inclusions of the wedge summands. So a basis for πn of the universal cover of S 1 ∨ S n is represented by maps that lift the maps obtained from the inclusion S n ֓ S 1 ∨ S n by the action of the various elements of π1(S 1 ∨ S n) ≈ Z. This means that πn(S 1 ∨ S n) is a free Z[π1
(S 1 ∨ S n)] module on a single basis element, the homotopy class of the inclusion S n ֓ S 1 ∨ S n. Writing a generator of π1(S 1 ∨ S n) as t, the group ring Z[π1(S 1 ∨ S n)] becomes Z[t, t−1], the Laurent polynomials in t and t−1 with Z coeﬃcients, and we have πn(S 1 ∨ S n) ≈ Z[t, t−1]. k S n W W This example shows that the homotopy groups of a ﬁnite CW complex need not be ﬁnitely generated, in contrast to the homology groups. However, if we restrict attention to spaces with trivial action of π1 on all πn ’s, then a theorem of Serre, proved in [SSAT], says that the homotopy groups of such a space are ﬁnitely generated iﬀ the homology groups are ﬁnitely generated. In this example, πn(S 1 ∨ S n) is ﬁnitely generated as a Z[π1] module, but there are ﬁnite CW complexes where even this fails. This happens in fact for π3(S 1 ∨ S 2), according to Exercise 38 at the end of this section. In §4.A we construct more complicated examples for each πn with n > 1, in particular for π2. A useful tool for more complicated calculations is the following general result: Proposition 4.28. If a CW pair (X, A) is r connected and A is s connected, with r, s ≥ 0, then the map πi(X, A)→πi(X/A) induced by the quotient map X→X/A is an isomorphism for i ≤ r + s and a surjection for i = r + s + 1. Proof: Consider X ∪ CA, the complex obtained from X by attaching a cone CA along A ⊂ X. Since CA is a contractible subcomplex of X ∪ CA, the quotient map X ∪ CA→(X ∪ CA)/CA = X/A is a homotopy equivalence by Proposition 0.17. So we have a comm
utative diagram where the vertical isomorphism comes from a long exact sequence. Now apply the excision theorem to the ﬁrst map in the diagram, using the fact that (CA, A) is (s + 1) connected if A is s connected, which comes from the exact sequence for the pair (CA, A). ⊔⊓ Example 4.29. Suppose X is obtained from a wedge of spheres via basepoint-preserving maps ϕβ : S n→ cells en+1 α by attaching α, with n ≥ 2. By cellular α S n W β α S n W Elementary Methods of Calculation Section 4.2 365 approximation we know that πi(X) = 0 for i < n, and we shall show that πn(X) is α Z by the subgroup generated the quotient of the free abelian group πn by the classes [ϕβ]. Any subgroup can be realized in this way, by choosing maps W ϕβ to represent a set of generators for the subgroup, so it follows that every abelian group can be realized as πn(X) for such a space X =. This is the higher-dimensional analog of the construction in Corollary 1.28 of a 2 dimensional β en+ CW complex with prescribed fundamental group. To see that πn(X) is as claimed, consider the following portion of the long exact sequence of the pair (X, α ) : α S n W πn+1(X, ∂-------------→ πn( α S n W α is a wedge of spheres S n+1 α S n α ) W α ) --------→ πn(X) ----→ 0 α S n The quotient X/ Example 4.26 imply that πn+1(X, W the cells en+1 follows., so the preceding proposition and α ) is free with basis the characteristic maps of. The boundary map ∂ takes these to the classes [ϕβ], and the result α S n W β β Eilenberg–MacLane Spaces A space X having just one nontrivial homotopy group πn(X) ≈ G is called an Eilenberg–MacLane space K(G, n). The case n = 1 was considered in §1.B, where the condition that πi(X)
= 0 for i > 1 was replaced by the condition that X have a contractible universal cover, which is equivalent for spaces that have a universal cover of the homotopy type of a CW complex. We can build a CW complex K(G, n) for arbitrary G and n, assuming G is abelian if n > 1, in the following way. To begin, let X be an (n − 1) connected CW complex of dimension n + 1 such that πn(X) ≈ G, as was constructed in Example 4.29 above when n > 1 and in Corollary 1.28 when n = 1. Then we showed in Example 4.16 how to attach higher-dimensional cells to X to make πi trivial for i > n without aﬀecting πn or the lower homotopy groups. By taking products of K(G, n) ’s for varying n we can then realize any sequence of groups Gn, abelian for n > 1, as the homotopy groups πn of a space. A fair number of K(G, 1) ’s arise naturally in a variety of contexts, and a few of these are mentioned in §1.B. By contrast, naturally occurring K(G, n) ’s for n ≥ 2 are rare. It seems the only real example is CP∞, which is a K(Z, 2) as we shall see in Example 4.50. One could of course trivially generalize this example by taking a product of CP∞ ’s to get a K(G, 2) with G a product of Z ’s. Actually there is a fairly natural construction of a K(Z, n) for arbitrary n, the inﬁnite symmetric product SP (S n) deﬁned in §3.C. In §4.K we prove that the functor SP has the surprising property of converting homology groups into homotopy groups, ≈ Hi(X; Z) for all i > 0 and all connected CW complexes X. Taking namely πi X to be a sphere, we deduce that SP (S n) is a K(Z, n). More generally, SP M(G, n) is a K(G, n) for each Moore space M(G, n). SP (X) 366 Chapter 4 Homotopy Theory
Having shown the existence of K(G, n) ’s, we now consider the uniqueness ques- tion, which has the nicest possible answer: Proposition 4.30. The homotopy type of a CW complex K(G, n) is uniquely determined by G and n. The proof will be based on a more technical statement: Lemma 4.31. Let X be a CW complex of the form for some n ≥ 1. Then for every homomorphism ψ : πn(X)→πn(Y ) with Y path-connected there exists a map f : X→Y with f∗ = ψ. β en+ Proof: To begin, let f send the natural basepoint of α to a chosen basepoint y0 ∈ Y. Extend f over each sphere S n α via a map representing ψ([iα]) where iα α ֓ X. Thus for the map f : X n→Y constructed so far we have is the inclusion S n f∗([iα]) = ψ([iα]) for all α, hence f∗([ϕ]) = ψ([ϕ]) for all basepoint-preserving maps ϕ : S n→X n since the iα ’s generate πn(X n). To extend f over a cell en+1 all we need is that the composition of the attaching map ϕβ : S n→X n for this cell with f be nullhomotopic in Y. But this composition f ϕβ represents f∗([ϕβ]) = ψ([ϕβ]), and ψ([ϕβ]) = 0 because [ϕβ] is zero in πn(X) since ϕβ is nullhomotopic in X. Thus we obtain an extension f : X→Y. This has via the characteristic map of en+1 f∗ = ψ since the elements [iα] generate πn(X n) and hence also πn(X) by cellular ⊔⊓ approximation. β β Proof of 4.30: Suppose K and K′ are K(G, n) CW complexes. Since homotopy equivalence is an equivalence relation, there is no loss of generality if we assume K is a particular K(G, n), namely one constructed from a space X as in the le
mma by attaching cells of dimension n + 2 and greater. By the lemma there is a map f : X→K′ inducing an isomorphism on πn. To extend this f over K we proceed inductively. For each cell en+2, the composition of its attaching map with f is nullhomotopic in K′ since πn+1(K′) = 0, so f extends over this cell. The same argument applies for all the higher-dimensional cells in turn. The resulting f : K→K′ is a homotopy ⊔⊓ equivalence since it induces isomorphisms on all homotopy groups. The Hurewicz Theorem Using the calculations of homotopy groups done above we can easily prove the simplest and most often used cases of the Hurewicz theorem: Theorem 4.32. If a space X is (n − 1) connected, n ≥ 2, then Hi(X) = 0 for i < n and πn(X) ≈ Hn(X). If a pair (X, A) is (n − 1) connected, n ≥ 2, with A simplyconnected and nonempty, then Hi(X, A) = 0 for i < n and πn(X, A) ≈ Hn(X, A). e Thus the ﬁrst nonzero homotopy and homology groups of a simply-connected space occur in the same dimension and are isomorphic. One cannot expect any nice Elementary Methods of Calculation Section 4.2 367 relationship between πi(X) and Hi(X) beyond this. For example, S n has trivial homology groups above dimension n but many nontrivial homotopy groups in this range when n ≥ 2. In the other direction, Eilenberg–MacLane spaces such as CP∞ have trivial higher homotopy groups but many nontrivial homology groups. The theorem can sometimes be used to compute π2(X) if X is a path-connected X is the universal cover, then space that is nice enough to have a universal cover. For if π2(X) ≈ π2( X) and the latter group is isomorphic to H2( X well enough to compute H2( So if one can understand X) by the Hurewicz theorem. e X), one can compute π2(X
). e In the part of the theorem dealing with relative groups, notice that X must be e e e simply-connected as well as A since (X, A) is 1 connected by hypothesis. There is a more general version of the relative Hurewicz theorem given later in Theorem 4.37 that allows A and X to be nonsimply-connected, but this requires πn(X, A) to be replaced by a certain quotient group. Proof: We may assume X is a CW complex and (X, A) is a CW pair by taking CW approximations to X and (X, A). For CW pairs the relative case then reduces to the absolute case since πi(X, A) ≈ πi(X/A) for i ≤ n by Proposition 4.28, while Hi(X, A) ≈ Hi(X/A) for all i by Proposition 2.22. α e α S n β en+1 In the absolute case we can apply Proposition 4.15 to replace X by a homoHi(X) = 0 for topy equivalent CW complex with (n − 1) skeleton a point, hence i < n. To show πn(X) ≈ Hn(X), we can further simplify by throwing away cells of dimension greater than n + 1 since these have no eﬀect on πn or Hn. Thus X has the form. We may assume the attaching maps ϕβ of the cells en+1 are basepoint-preserving since this is what the proof of Proposition 4.15 gives. β Example 4.29 then applies to compute πn(X) as the cokernel of the boundary map πn+1(X, X n)→πn(X n), a map α Z. This is the same as the cellular boundary map d : Hn+1(X n+1, X n)→Hn(X n, X n−1) since for a cell en+1, the coeﬃcients of L den+1 are the degrees of the compositions qαϕβ where qα collapses all n cells exβ α to a point, and the isomorphism πn(S n) ≈ Z in Corollary 4.25 is given by cept en degree. Since there are no (n − 1) cells, we
have Hn(X) ≈ Coker d. ⊔⊓ β Z→ S W L e β β Since homology groups are usually more computable than homotopy groups, the following version of Whitehead’s theorem is often easier to apply: Corollary 4.33. A map f : X→Y between simply-connected CW complexes is a homotopy equivalence if f∗ : Hn(X)→Hn(Y ) is an isomorphism for each n. Proof: After replacing Y by the mapping cylinder Mf we may take f to be an inclusion X ֓ Y. Since X and Y are simply-connected, we have π1(Y, X) = 0. The relative Hurewicz theorem then says that the ﬁrst nonzero πn(Y, X) is isomorphic to the ﬁrst nonzero Hn(Y, X). All the groups Hn(Y, X) are zero from the long exact sequence of homology, so all the groups πn(Y, X) also vanish. This means that the inclusion 368 Chapter 4 Homotopy Theory X ֓ Y induces isomorphisms on all homotopy groups, and therefore this inclusion ⊔⊓ is a homotopy equivalence. Example 4.34: Uniqueness of Moore Spaces. Let us show that the homotopy type of a CW complex Moore space M(G, n) is uniquely determined by G and n if n > 1, so M(G, n) is simply-connected. Let X be an M(G, n) as constructed in Example 2.40 by attaching (n + 1) cells to a wedge sum of n spheres, and let Y be any other M(G, n) CW complex. By Lemma 4.31 there is a map f : X→Y inducing an isomorphism on πn. If we can show that f also induces an isomorphism on Hn, then the preceding corollary will imply the result. One way to show that f induces an isomorphism on Hn would be to use a more reﬁned version of the Hurewicz theorem giving an isomorphism between πn and Hn that is natural with respect to maps between spaces, as in Theorem 4.37 below. However, here is a
direct argument which avoids naturality questions. For the mapping cylinder Mf we know that πi(Mf, X) = 0 for i ≤ n. If this held also for i = n + 1 then the relative Hurewicz theorem would say that Hi(Mf, X) = 0 for i ≤ n + 1 and hence that f∗ would be an isomorphism on Hn. To make this argument work, let us temporarily enlarge Y by attaching (n + 2) cells to make πn+1 zero. The new mapping cylinder Mf then has πn+1(Mf, X) = 0 from the long exact sequence of the pair. So for the enlarged Y the map f induces an isomorphism on Hn. But attaching (n + 2) cells has no eﬀect on Hn, so the original f : X→Y had to be an isomorphism on Hn. It is certainly possible for a map of nonsimply-connected spaces to induce isomor- phisms on all homology groups but not on homotopy groups. Nonsimply-connected acyclic spaces, for which the inclusion of a point induces an isomorphism on ho- mology, exhibit this phenomenon in its purest form. Perhaps the simplest nontrivial acyclic space is the 2 dimensional complex constructed in Example 2.38 with funda- mental group a, b \|\|\|\| a5 = b3 = (ab)2 of order 120. It is also possible for a map between spaces with abelian fundamental groups to induce isomorphisms on homology but not on higher homotopy groups, as the next example shows. Example 4.35. We construct a space X = (S 1 ∨ S n) ∪ en+1, for arbitrary n > 1, such that the inclusion S 1 ֓ X induces an isomorphism on all homology groups and on πi for i < n, but not on πn. From Example 4.27 we have πn(S 1 ∨ S n) ≈ Z[t, t−1]. Let X be obtained from S 1 ∨ S n by attaching a cell en+1 via a map S n→S 1 ∨ S n corresponding to 2t − 1 ∈ Z[t, t−1]. By looking in the universal cover we see that
πn(X) ≈ Z[t, t−1]/(2t − 1), where (2t − 1) denotes the ideal in Z[t, t−1] generated by 2t − 1. Note that setting t = 1/2 embeds Z[t, t−1]/(2t − 1) in Q as the subring Z[1/2] consisting of rationals with denominator a power of 2. From the long exact sequence of homotopy groups for the (n − 1) connected pair (X, S 1) we see that the inclusion Elementary Methods of Calculation Section 4.2 369 S 1 ֓ X induces an isomorphism on πi for i < n. The fact that this inclusion also induces isomorphisms on all homology groups can be deduced from cellular homology. The key point is that the cellular boundary map Hn+1(X n+1, X n)→Hn(X n, X n−1) is an isomorphism since the degree of the composition of the attaching map S n→S 1 ∨ S n of en+1 with the collapse S 1 ∨ S n→S n is 2 − 1 = 1. This example relies heavily on the nontriviality of the action of π1(X) on πn(X), so one might ask whether the simple-connectivity assumption in Corollary 4.33 can be weakened to trivial action of π1 on all πn ’s. This is indeed the case, as we will show in Proposition 4.74. The form of the Hurewicz theorem given above asserts merely the existence of an isomorphism between homotopy and homology groups, but one might want a more precise statement which says that a particular map is an isomorphism. In fact, there are always natural maps from homotopy groups to homology groups, deﬁned in the following way. Thinking of πn(X, A, x0) for n > 0 as homotopy classes of maps f : (Dn, ∂Dn, s0)→(X, A, x0), the Hurewicz map h : πn(X, A, x0)→Hn(X, A) is deﬁned by h([f ]) = f∗(
α) where α is a ﬁxed generator of Hn(Dn, ∂Dn) ≈ Z and f∗ : Hn(Dn, ∂Dn)→Hn(X, A) is induced by f. If we have a homotopy f ≃ g through maps (Dn, ∂Dn, s0)→(X, A, x0), or even through maps (Dn, ∂Dn)→(X, A) not preserving the basepoint, then f∗ = g∗, so h is well-deﬁned. Proposition 4.36. The Hurewicz map h : πn(X, A, x0)→Hn(X, A) is a homomorphism, assuming n > 1 so that πn(X, A, x0) is a group. Proof: It suﬃces to show that for maps f, g : (Dn, ∂Dn)→(X, A), the induced maps on homology satisfy (f + g)∗ = f∗ + g∗, for if this is the case then h([f + g]) = (f + g)∗(α) = f∗(α) + g∗(α) = h([f ]) + h([g]). Our proof that (f + g)∗ = f∗ + g∗ will in fact work for any homology theory. Let c : Dn→Dn ∨ Dn be the map collapsing the equatorial Dn−1 to a point, and let q1, q2 : Dn ∨ Dn→Dn be the quotient maps onto the two summands, collapsing the other summand to a point. We then have a diagram The map q1∗ ⊕ q2∗ is an isomorphism with inverse i1∗ + i2∗ where i1 and i2 are the inclusions of the two summands Dn ֓ Dn ∨ Dn. Since q1c and q2c are homotopic to the identity through maps (Dn, ∂Dn)→(Dn, ∂Dn), the composition (q1∗ ⊕ q2∗)c∗ is the
diagonal map x ֏ (x, x). From the equalities (f ∨ g)i1 = f and (f ∨ g)i2 = g we deduce that (f ∨ g)∗(i1∗ + i2∗) sends (x, 0) to f∗(x) and (0, x) to g∗(x), hence (x, x) to f∗(x) + g∗(x). Thus the composition across the top of the diagram is 370 Chapter 4 Homotopy Theory x ֏ f∗(x) + g∗(x). On the other hand, f + g = (f ∨ g)c, so this composition is also ⊔⊓ (f + g)∗. There is also an absolute Hurewicz map h : πn(X, x0)→Hn(X) deﬁned in a similar way by setting h([f ]) = f∗(α) for f : (S n, s0)→(X, x0) and α a chosen generator of Hn(S n). For example, if X = S n then f∗(α) is (deg f )α by the deﬁnition of degree, so we can view h in this case as the degree map πn(S n)→Z, which we know is an isomorphism by Corollary 4.25. The proof of the preceding proposition is readily modiﬁed to show that the absolute h is a homomorphism for n ≥ 1. The absolute and relative Hurewicz maps can be combined in a diagram of long exact sequences An easy deﬁnition check which we leave to the reader shows that this diagram com- mutes up to sign at least. With more care in the choice of the generators α it can be made to commute exactly. Another elementary property of Hurewicz maps is that they are natural: A map f : (X, x0)→(Y, y0) induces a commutative diagram as at the right, and similarly in the relative case. It is easy to construct nontrivial elements of the kernel of the Hurewicz homomorphism h : πn(X, x0)→Hn(X
) if π1(X, x0) acts nontrivially on πn(X, x0), namely elements of the form [γ][f ]−[f ]. This is because γf and f, viewed as maps S n→X, are homotopic if we do not require the basepoint to be ﬁxed during the homotopy, so (γf )∗(α) = f∗(α) for α a generator of Hn(S n). Similarly in the relative case the kernel of h : πn(X, A, x0)→Hn(X, A) contains the elements of the form [γ][f ] − [f ] for [γ] ∈ π1(A, x0). For example the Hurewicz map πn(S 1 ∨ S n, S 1)→Hn(S 1 ∨ S n, S 1) is the homomorphism Z[t, t−1]→Z sending all powers of t to 1. Since the pair (S 1 ∨ S n, S 1) is (n − 1) connected, this example shows that the hypothesis π1(A, x0) = 0 in the relative form of the Hurewicz theorem proved earlier cannot be dropped. If we deﬁne π ′ n(X, A, x0) to be the quotient group of πn(X, A, x0) obtained by factoring out the subgroup generated by all elements of the form [γ][f ] − [f ], or the normal subgroup generated by such elements in the case n = 2 when π2(X, A, x0) n(X, A, x0)→Hn(X, A). may not be abelian, then h induces a homomorphism h′ : π ′ The general form of the Hurewicz theorem deals with this homomorphism: Elementary Methods of Calculation Section 4.2 371 Theorem 4.37. If (X, A) is an (n − 1) connected pair of path-connected spaces n(X, A, x0)→Hn(X, A) is an isomorphism and with n ≥ 2 and A ≠ ∅, then h′ : π �
� Hi(X, A) = 0 for i < n. Note that this statement includes the absolute form of the theorem by taking A to be the basepoint. Before starting the proof of this general Hurewicz theorem we have a preliminary step: α are attached for α, then πn(W, X) is a free α of the α, provided that the map π1(X)→π1(W ) induced by inclusion is an isomorIn the general n = 2 case, α to- Lemma 4.38. If X is a connected CW complex to which cells en a ﬁxed n ≥ 2, forming a CW complex W = X π1(X) module with basis the homotopy classes of the characteristic maps cells en phism. In particular, this is always the case if n ≥ 3. π2(W, X) is generated by the classes of the characteristic maps of the cells e2 gether with their images under the action of π1(X). α en S Φ If the characteristic maps α : (Dn, ∂Dn)→(W, X) do not take a basepoint s0 in ∂Dn to the basepoint x0 in X, then they will deﬁne elements of πn(W, X, x0) only Φ α(s0) to x0. Diﬀerafter we choose change-of-basepoint paths from the points ent choices of such paths yield elements of πn(W, X, x0) related by the action of π1(X, x0), so the basis for πn(W, X, x0) is well-deﬁned up to multiplication by invertible elements of Z[π1(X)]. Φ The situation when n = 2 and the map π1(X)→π1(W ) is not an isomorphism is more complicated because the relative π2 can be nonabelian in this case. Whitehead analyzed what happens here and showed that π2(W, X) has the structure of a ‘free crossed π1(X) module’. See [Whitehead 1949] or [Sieradski 1993]. Proof: Since W /X = α, we have πn(W, X)
. The inclusion (W, X)֓(W, Z) is a homotopy equivalence of pairs. Homotopy excision W gives a surjection π2(Y, Y 1)→π2(W, Z). The universal cover Y of Y is obtained from αβ of the disks D2 α. the universal cover α. Let Y = X α int(D2 X of X by taking the wedge sum with lifts e α, and let Z = W − α D2 D2 S Hence we have isomorphisms e π2(Y, Y 1) ≈ π2( ≈ π2( Y, Y 1) D2 αβ, e e αβ D2 W αβ ∂ e W αβ) where Y 1 is the 1 skeleton of Y e αβ ∂ D2 αβ) since D2 e since e αβ X is contractible e αβ is contractible e ≈ π1( W W αβ D2 D2 αβ, e D2 D2 This last group is free with basis the loops ∂ αβ αβ). This implies that π2(Y, Y 1) is generated by form a basis for π2( α ֓ Y and their images under the action of loops in X. The same is the inclusions D2 true for π2(W, Z) via the surjection π2(Y, Y 1)→π2(W, Z). Using the isomorphism π2(W, Z) ≈ π2(W, X), we conclude that π2(W, X) is generated by the characteristic maps of the cells e2 ⊔⊓ α and their images under the action of π1(X). αβ, so the inclusions αβ ֓ αβ ∂ D2 W W W αβ e e e e e e Proof of the general Hurewicz Theorem: Since the pair (X, A) is (n − 1) connected we may assume it is a CW pair such that the cells of X − A have dimension ≥ n. We may assume also that X = X n+1 since higher-dimensional cells have no eﬀect on πn or Hn. Consider the commutative diagram where the maps q are the quotient maps. The
ﬁrst and third rows are exact sequences for the triple (X, X n ∪ A, A). To construct the map ∂′ would take a small extra argument but we will not actually need this map so it can be ignored for the proof. If we did have this map and we knew the middle row was exact, the theorem would follow from the ﬁve-lemma once we show that the middle h′ is an isomorphism and the left-hand h′ is surjective. Instead we will use exactness of the upper row and a diagram chase similar to one in the proof of the ﬁve-lemma. Elementary Methods of Calculation Section 4.2 373 The map d in the lower row is just the cellular boundary map for computing Hn(X, A) as Coker(d) in the cellular chain complex for (X, A) since X = X n+1. In particular Hn+1(X, X n∪A) and Hn(X n ∪A, A) are free with bases the (n+1) cells and n cells of X −A, respectively, so the left-hand and middle maps h′ are surjective. The preceding lemma implies that the middle h′ is in fact an isomorphism when n ≥ 3. When n = 2 the lemma implies that π ′ 2(X 2 ∪ A, A) is generated by the characteristic maps of the 2 cells of X − A. The images of these generators under h′ form a basis 2(X 2 ∪ A, A) is abelian by Lemma 4.39 below, and h′ for H2(X 2 ∪ A, A). The group π ′ is a homomorphism from this group to a free abelian group taking a set of generators to a basis, hence h′ is an isomorphism. n(X, A) be an element with h′(x) = 0. The map i′ Now for the diagram chase to show that the right-hand h′ is an isomorphism. Surjectivity is immediate since the lower i∗ and the middle h′ are surjective. For injectivity, let x ∈ π ′ ∗ is surjective since the maps q and the upper map i∗ are surjective, so x = i′ ∗
(y) for some y ∈ π ′ n(X n∪A, A). We have i∗h′(y) = 0 in the lower row so h′(y) comes from an element of Hn+1(X, X n ∪ A) hence also from some z ∈ πn+1(X, X n ∪ A) since the maps in the left column are surjective. We have h′q∂(z) = h′(y) by commutativity, hence q∂(z) = y since the middle h′ is injective. Then x = i′ ∗q∂(z) = qi∗∂(z) = 0 ⊔⊓ since i∗∂ = 0. ∗(y) = i′ Lemma 4.39. For any (X, A, x0), the formula a + b − a = (∂a)b holds for all a, b ∈ π2(X, A, x0), where ∂ : π2(X, A, x0)→π1(A, x0) is the usual boundary map and (∂a)b denotes the action of ∂a on b. Hence π ′ 2(X, A, x0) is abelian. Here the ‘ + ’ and ‘ − ’ in a + b − a refer to the group operation in the nonabelian group π2(X, A, x0). Proof: The formula is obtained by constructing a homotopy from a + b − a to (∂a)b ⊔⊓ as indicated in the pictures below. The Plus Construction There are quite a few situations in algebraic topology where having a nontriv- ial fundamental group complicates matters considerably. We shall next describe a construction which in certain circumstances allows one to modify a space so as to eliminate its fundamental group, or at least simplify it, without aﬀecting homology or cohomology. Here is the simplest case: 374 Chapter 4 Homotopy Theory Proposition 4.40. Let X be a connected CW complex with H1(X) = 0. Then there is a simply-connected CW complex X + and a map X→X + inducing isomorphisms on all hom
ology groups. Proof: Choose loops ϕα : S 1→X 1 generating π1(X) and use these to attach cells e2 to X to form a simply-connected CW complex X ′. The homology exact sequence α 0 -→ H2(X) -→ H2(X ′) -→ H2(X ′, X) -→ 0 = H1(X) splits since H2(X ′, X) is free with basis the cells e2 α. Thus we have an isomorphism H2(X ′) ≈ H2(X)⊕ H2(X ′, X). Since X ′ is simply-connected, the Hurewicz theorem gives an isomorphism H2(X ′) ≈ π2(X ′), and so we may represent a basis for the free summand H2(X ′, X) by maps ψα : S 2→X ′. We may assume these are cellular maps, α to X ′ forming a simply-connected CW complex and then use them to attach cells e3 X +, with the inclusion X ֓ X + an isomorphism on all homology groups. ⊔⊓ e In the preceding proposition, the condition H1(X) = 0 means that π1(X) is equal to its commutator subgroup, that is, π1(X) is a perfect group. Suppose more generally that X is a connected CW complex and H ⊂ π1(X) is a perfect subgroup. X) ≈ H is perLet p : X ֓ X +. fect and H1( X + and the mapping cylinder Mp Let X + be obtained from the disjoint union of e by identifying the copies of X→X be the covering space corresponding to H, so π1( X) = 0. From the previous proposition we get an inclusion X in these two spaces. Thus e e e e e e we have the commutative diagram of inclusion maps shown e at the right. From the van Kampen theorem, the induced map π1(X)→π1(X +) is surjective with kernel the normal subgroup generated by H. Further, since X +/Mp is homeomorphic to have H∗(X +, Mp
) = H∗( on homology. X we X) = 0, so the map X→X + induces an isomorphism X +/ X +, e e This construction X→X +, killing a perfect subgroup of π1(X) while preserving homology, is known as the Quillen plus construction. In some of the main applications X is a K(G, 1) where G has perfect commutator subgroup, so the map X→X + abelianizes π1 while preserving homology. The space X + need no longer be a K(π, 1), and in fact its homotopy groups can be quite interesting. The most striking example is ∞, the inﬁnite symmetric group consisting of permutations of 1, 2, ··· ﬁxing all G = but ﬁnitely many n ’s, with commutator subgroup the inﬁnite alternating group A∞, which is perfect. In this case a famous theorem of Barratt, Priddy, and Quillen says ∞, 1)+) are the stable homotopy groups of spheres! that the homotopy groups πi(K( There are limits, however, on which subgroups of π1(X) can be killed without aﬀecting the homology of X. For example, for X = S 1 ∨ S 1 it is impossible to kill the commutator subgroup of π1(X) while preserving homology. In fact, by Exercise 23 at the end of this section every space with fundamental group Z× Z must have H2 nontrivial. Σ Σ Elementary Methods of Calculation Section 4.2 375 Fiber Bundles A ‘short exact sequence of spaces’ A ֓ X→X/A gives rise to a long exact sequence of homology groups, but not to a long exact sequence of homotopy groups due to the failure of excision. However, there is a diﬀerent sort of ‘short exact sequence of spaces’ that does give a long exact sequence of homotopy groups. This sort of short p-----→ B, called a ﬁber bundle, is distinguished from the type exact sequence F -→ E A ֓ X→X/A in that it has more homogeneity: All the subspaces p
−1(b) ⊂ E, which are called ﬁbers, are homeomorphic. For example, E could be the product F × B with p : E→B the projection. General ﬁber bundles can be thought of as twisted products. Familiar examples are the M¨obius band, which is a twisted annulus with line segments as ﬁbers, and the Klein bottle, which is a twisted torus with circles as ﬁbers. The topological homogeneity of all the ﬁbers of a ﬁber bundle is rather like the p-----→ H→0 algebraic homogeneity in a short exact sequence of groups 0→K -→ G where the ‘ﬁbers’ p−1(h) are the cosets of K in G. In a few ﬁber bundles F→E→B the space E is actually a group, F is a subgroup (though seldom a normal subgroup), and B is the space of left or right cosets. One of the nicest such examples is the Hopf bundle S 1→S 3→S 2 where S 3 is the group of quaternions of unit norm and S 1 is the subgroup of unit complex numbers. For this bundle, the long exact sequence of homotopy groups takes the form ··· -→ πi(S 1) -→ πi(S 3) -→ πi(S 2) -→ πi−1(S 1) -→ πi−1(S 3) -→ ··· In particular, the exact sequence gives an isomorphism π2(S 2) ≈ π1(S 1) since the two adjacent terms π2(S 3) and π1(S 3) are zero by cellular approximation. Thus we have a direct homotopy-theoretic proof that π2(S 2) ≈ Z. Also, since πi(S 1) = 0 for i > 1 by Proposition 4.1, the exact sequence implies that there are isomorphisms πi(S 3) ≈ πi(S 2) for all i ≥ 3, so in particular π3(S 2) ≈ π3(S 3), and by Corollary 4.25 the latter group is Z. After these preliminary remarks, let us begin by de
ﬁning the property that leads to a long exact sequence of homotopy groups. A map p : E→B is said to have the homotopy lifting property with respect to a space X if, given a homotopy gt : X→B gt : X→E and a map lifting gt. From a formal point of view, this can be regarded as a special case of the lift extension property for a pair (Z, A), which asserts that every map Z→B has a lift Z→E extending a given lift deﬁned on the subspace A ⊂ Z. The case (Z, A) = (X × I, X × {0}) is the homotopy lifting property. g0 = g0, then there exists a homotopy g0 : X→E lifting g0, so p e e e A ﬁbration is a map p : E→B having the homotopy lifting property with respect to all spaces X. For example, a projection B × F→B is a ﬁbration since we can choose lifts of the form gt(x) = (gt(x), h(x)) where g0(x) = (g0(x), h(x)). e e 376 Chapter 4 Homotopy Theory Theorem 4.41. Suppose p : E→B has the homotopy lifting property with respect to disks Dk for all k ≥ 0. Choose basepoints b0 ∈ B and x0 ∈ F = p−1(b0). Then the map p∗ : πn(E, F, x0)→πn(B, b0) is an isomorphism for all n ≥ 1. Hence if B is path-connected, there is a long exact sequence ··· →πn(F, x0)→πn(E, x0) p∗-----→ πn(B, b0)→πn−1(F, x0)→ ··· →π0(E, x0)→0 The proof will use a relative form of the homotopy lifting property. The map p : E→B is said to have the homotopy lifting property for a pair (X, A) if each homotopy ft : X→B lifts to a homotopy g0 and gt : A→E
. In other words, the homotopy lifting property for extending a given lift (X, A) is the lift extension property for (X × I, X × {0} ∪ A× I). gt : X→E starting with a given lift e e The homotopy lifting property for Dk is equivalent to the homotopy lifting property for (Dk, ∂Dk) since the pairs (Dk × I, Dk × {0}) and (Dk × I, Dk × {0}∪∂Dk × I) are homeomorphic. This implies that the homotopy lifting property for disks is equiva- e the skeleta of X it suﬃces to construct a lifting lent to the homotopy lifting property for all CW pairs (X, A). For by induction over gt one cell of X − A at a time. Com: Dk→X of a cell then gives a reduction to the posing with the characteristic map case (X, A) = (Dk, ∂Dk). A map p : E→B satisfying the homotopy lifting property for disks is sometimes called a Serre ﬁbration. Φ e e e e e f0, f = f. G given by f ]) = [f ] since p f : In→E, and this lift satisﬁes Proof: First we show that p∗ is onto. Represent an element of πn(B, b0) by a map f : (In, ∂In)→(B, b0). The constant map to x0 provides a lift of f to E over the subspace J n−1 ⊂ In, so the relative homotopy lifting property for (In−1, ∂In−1) extends this to a lift f represents an element of πn(E, F, x0) with p∗([ f (∂In) ⊂ F since f (∂In) = b0. Then f1]), let G : (In × I, ∂In × I)→(B, b0) be a homotopy from p e f0 on In × {0}, e e f1 : (In, ∂In, J n−1)→(E, F, x0) such that Injectivity of p�
� is similar. Given f0]) = p∗([ f1. p∗([ f1 on In × {1}, and the constant map We have a partial lift e to x0 on J n−1 × I. After permuting the last two coordinates of In × I, the relative hoG : In × I→E. motopy lifting property gives an extension of this partial lift to a full lift f1. So p∗ is injective. e For the last statement of the theorem we plug πn(B, b0) in for πn(E, F, x0) in the long exact sequence for the pair (E, F ). The map πn(E, x0)→πn(E, F, x0) in the exp∗-----→ πn(B, b0), act sequence then becomes the composition πn(E, x0)→πn(E, F, x0) which is just p∗ : πn(E, x0)→πn(B, b0). The 0 at the end of the sequence, surjectivity of π0(F, x0)→π0(E, x0), comes from the hypothesis that B is path-connected since a path in E from an arbitrary point x ∈ E to F can be obtained by lifting a path in B from p(x) to b0. ft : (In, ∂In, J n−1)→(E, F, x0) from This is a homotopy f0 to p f0 to ⊔⊓ ﬁber bundle structure on a space E, with ﬁber F, consists of a projection map p : E→B such that each point of B has a neighborhood U for which there is a Elementary Methods of Calculation Section 4.2 377 homeomorphism h : p−1(U)→U × F making the diagram at the right commute, where the unlabeled map is projection onto the ﬁrst factor. Commutativity of the diagram means that h carries each ﬁber Fb = p−1(b) homeomorphically onto the copy {b}× F of F. Thus the ﬁbers Fb are arranged locally as in the product B × F, though not necessarily globally. An h
as above is called a local trivialization of the bundle. Since the ﬁrst coordinate of h is just p, h is determined by its second coordinate, a map p−1(U)→F which is a homeomorphism on each ﬁber Fb. The ﬁber bundle structure is determined by the projection map p : E→B, but to indicate what the ﬁber is we sometimes write a ﬁber bundle as F→E→B, a ‘short exact sequence of spaces’. The space B is called the base space of the bundle, and E is the total space. Example 4.42. A ﬁber bundle with ﬁber a discrete space is a covering space. Conversely, a covering space whose ﬁbers all have the same cardinality, for example a covering space over a connected base space, is a ﬁber bundle with discrete ﬁber. Example 4.43. One of the simplest nontrivial ﬁber bundles is the M¨obius band, which is a bundle over S 1 with ﬁber an interval. Speciﬁcally, take E to be the quotient of I × [−1, 1] under the identiﬁcations (0, v) ∼ (1, −v), with p : E→S 1 induced by the projection I × [−1, 1]→I, so the ﬁber is [−1, 1]. Glueing two copies of E together by the identity map between their boundary circles produces a Klein bottle, a bundle over S 1 with ﬁber S 1. Example 4.44. Projective spaces yield interesting ﬁber bundles. In the real case we have the familiar covering spaces S n→RPn with ﬁber S 0. Over the complex numbers the analog of this is a ﬁber bundle S 1→S 2n+1→CPn. Here S 2n+1 is the unit sphere in Cn+1 and CPn is viewed as the quotient space of S 2n+1 under the equivalence relation (z0, ···, zn) ∼ λ(z0, ···, zn) for λ ∈ S 1, the unit circle in C. The projection p : S 2n+1→CPn sends (z0
, ···, zn) to its equivalence class [z0, ···, zn], so the ﬁbers are copies of S 1. To see that the local triviality condition for ﬁber bundles is satisﬁed, let Ui ⊂ CPn be the open set of equivalence classes [z0, ···, zn] with zi ≠ 0. Deﬁne hi : p−1(Ui)→Ui × S 1 by hi(z0, ···, zn) = ([z0, ···, zn], zi/\|zi\|). This takes ﬁbers to ﬁbers, and is a homeomorphism since its inverse is the map ([z0, ···, zn], λ) ֏ λ\|zi\|z−1 i (z0, ···, zn), as one checks by calculation. The construction of the bundle S 1→S 2n+1→CPn also works when n = ∞, so there is a ﬁber bundle S 1→S ∞→CP∞. Example 4.45. The case n = 1 is particularly interesting since CP1 = S 2 and the bundle becomes S 1→S 3→S 2 with ﬁber, total space, and base all spheres. This is known as the Hopf bundle, and is of low enough dimension to be seen explicitly. The projection S 3→S 2 can be taken to be (z0, z1) ֏ z0/z1 ∈ C ∪ {∞} = S 2. coordinates we have p(r0eiθ0, r1eiθ1 ) = (r0/r1)ei(θ0−θ1) where r 2 In polar 1 = 1. For a 0 + r 2 378 Chapter 4 Homotopy Theory ﬁxed ratio ρ = r0/r1 ∈ (0, ∞) the angles θ0 and θ1 vary independently over S 1, so the points (r0eiθ0, r1eiθ1 ) form a torus Tρ ⊂ S 3. Letting ρ vary, these disjoint tori Tρ ﬁll up S 3, if we include the limiting cases T

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