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− 1) ( if t ≤ 1/2 if t ≥ 1/2 for x ∈ X and γ ∈ P Y such that γ(1) = f (x). Thus ε is just the adjoint of η. Lemma. Let f : X −→ Y be a map of based spaces. Then the following diagram, in which the top row is the suspension of part of the ﬁber sequence of f and the bottom row is the loops on part of the coﬁber sequence of f, is homotopy commutative: ΣΩF f ΣΩp / ΣΩX ΣΩf / ΣΩY Σι ΣF f Σp / ΣX ι ΩY Cf η π ΣX ΩY / ΩCf Ωi / ΩΣX / ΩΣY ΩΣf Ωπ ΩΣi / ΩΣCf. Proof. Four of the squares commute by naturality and the remaining four squares consist of two pairs that are adjoint to each other. To see that the two bottom left squares commute up to homotopy one need only write down the relevant maps explicitly. Another easily veriﬁed result along the same lines relates the quotient map (M f, X) −→ (Cf, ∗) to η : F f −→ ΩCf. Here in the based context we let M f be the reduced mapping cylinder, in which the line through the basepoint of X is collapsed to a point. Lemma. Let f : X −→ Y be a map of based spaces. Then the following diagram is homotopy commutative, where j : X −→ M f is the inclusion, r : M f −→ Y is the retraction, and π is induced by the quotient map M f −→ Cf : F j = X ×j P M f F r=id ×P r X ×f P Y = F f 'OOOOOOOOOOO π wppppppppppp η ΩCf. (1) Prove the two lemmas stated at the end of §6. PROBLEM / / / / / / / /
/ / / / / / / / / / / /'w CHAPTER 9 Higher homotopy groups The most basic invariants in algebraic topology are the homotopy groups. They are very easy to deﬁne, but very hard to compute. We give the basic properties of these groups here. 1. The deﬁnition of homotopy groups For n ≥ 0 and a based space X, deﬁne πn(X) = πn(X, ∗) = [Sn, X], the set of homotopy classes of based maps Sn −→ X. This is a group if n ≥ 1 and an Abelian group if n ≥ 2. When n = 0 and n = 1, this agrees with our previous deﬁnitions. Observe that πn(X) = πn−1(ΩX) = · · · = π0(ΩnX). For ∗ ∈ A ⊂ X, the (homotopy) ﬁber of the inclusion A −→ X may be identiﬁed with the space P (X; ∗, A) of paths in X that begin at the basepoint and end in A. For n ≥ 1, deﬁne πn(X, A) = πn(X, A, ∗) = πn−1P (X; ∗, A). This is a group if n ≥ 2 and an Abelian group if n ≥ 3. Again, πn(X, A) = π0(Ωn−1P (X; ∗, A)). These are called relative homotopy groups. 2. Long exact sequences associated to pairs With F i = P (X; ∗, A), we have the ﬁber sequence · · · −→ Ω2A −→ Ω2X −→ ΩF i −→ ΩA −→ ΩX ι−→ F i p1−→ A i−→ X associated to the inclusion i : A −→ X, where p1 is the endpoint projection and ι is the inclusion. Applying the functor π0(−) = [S0, −] to this sequence, we obtain the long exact sequence · · · −→
πn(A) −→ πn(X) −→ πn(X, A) ∂−→ πn−1(A) −→ · · · −→ π0(A) −→ π0(X). Deﬁne J n = ∂I n−1 × I ∪ I n−1 × {0} ⊂ I n, with J 1 = {0} ⊂ I. We can write πn(X, A, ∗) = [(I n, ∂I n, J n), (X, A, ∗)], where the notation indicates the homotopy classes of maps of triples: maps and homotopies carry ∂I n into A and J n to the basepoint. Then ∂ : πn(X, A) −→ πn−1(A) 65 66 HIGHER HOMOTOPY GROUPS is obtained by restricting maps to maps (I n, ∂I n, J n) −→ (X, A, ∗) (I n−1 × {1}, ∂I n−1 × {1}) −→ (A, ∗), while πn(A) −→ πn(X) and πn(X) −→ πn(X, A) are induced by the inclusions (A, ∗) ⊂ (X, ∗) and (X, ∗, ∗) ⊂ (X, A, ∗). 3. Long exact sequences associated to ﬁbrations Let p : E −→ B be a ﬁbration, where B is path connected. Fix a basepoint ∗ ∈ B, let F = p−1(∗), and ﬁx a basepoint ∗ ∈ F ⊂ E. The inclusion φ : F −→ F p is a homotopy equivalence, and, being pedantically careful to choose signs appropriately, we obtain the following diagram, in which two out of each three consecutive squares commute and the third commutes up to homotopy: · · · · · · / Ω2E id −Ωι ΩF i −Ωp1 / ΩF −Ωi / Ω
E ι / F i p1 −Ωp Ωφ id −p i F φ E id / Ω2E Ω2p / Ω2B −Ωι / ΩF p −Ωπ / ΩE −Ωp / ΩB ι / F p / E. π Here F i = P (E; ∗, F ), p(ξ) = p ◦ ξ ∈ ΩB for ξ ∈ F i, and the next to last square commutes up to the homotopy h : ι ◦ (−p) ≃ φ ◦ p1 speciﬁed by h(ξ, t) = (ξ(t), p(ξ[1, t])), where ξ[1, t](s) = ξ(1 − s + st). Passing to long exact sequences of homotopy groups and using the ﬁve lemma, together with a little extra argument in the case n = 1, we conclude that p∗ : πn(E, F ) −→ πn(B) is an isomorphism for n ≥ 1. This can also be derived directly from the covering homotopy property. Using φ∗ to identify π∗F with π∗(F p), we may rewrite the long exact sequence of the bottom row of the diagram as · · · −→ πn(F ) −→ πn(E) −→ πn(B) ∂−→ πn−1(F ) −→ · · · −→ π0(E) −→ {∗}. (At the end, a little path lifting argument shows that π0(F ) −→ π0(E) is a surjection.) This is one of the main tools for the computation of homotopy groups. 4. A few calculations We observe some easily derived calculational facts about homotopy groups. Lemma. If X is contractible, then πn(X) = 0 for all n ≥ 0. Lemma. If X is discrete, then πn(X) = 0 for all n > 0. Lemma. If p : E −→ B is a covering, then p∗ :
πn(E) −→ πn(B) is an isomorphism for all n ≥ 2. Lemma. π1(S1) = Z and πn(S1) = 0 if n 6= 1. Lemma. If i ≥ 2, then π1(RP i) = Z2 and πn(RP i) ∼= πn(Si) for n 6= 1. A FEW CALCULATIONS 67 Lemma. For all spaces X and Y and all n, πn(X × Y ) ∼= πn(X) × πn(Y ). Lemma. If i < n, then πi(Sn) = 0. Proof. For any based map f : Si −→ Sn, we can apply smooth (or simplicial) approximation to obtain a based homotopy from f to a map that misses a point p, and we can then deform f to the trivial map by contracting Sn − {p} to the basepoint. There are three standard bundles, called the Hopf bundles, that can be used to obtain a bit more information about the homotopy groups of spheres. Recall that CP 1 is the space of complex lines in C2. That is, CP 1 = (C × C − {0})/(∼), where (z1, z2) ∼ (λz1, λz2) for complex numbers λ, z1, and z2. Write [z1, z2] for the equivalence class of (z1, z2). We obtain a homeomorphism CP 1 −→ S2 by identifying S2 with the one-point compactiﬁcation of C and mapping [z1, z2] to z2/z1 if z1 6= 0 and to the point at ∞ if z1 = 0. The Hopf map η : S3 −→ S2 is speciﬁed by η(z1, z2) = [z1, z2], where S3 is identiﬁed with the unit sphere in the complex plane C2. It is a worthwhile exercise to check that η is a bundle with ﬁber S1. By use of the quaternions and Cayley numbers, we obtain analogous Hopf maps ν : S7 −→ S4 and σ :
S15 −→ S8. Then ν is a bundle with ﬁber S3 and σ is a bundle with ﬁber S7. Since we have complete information on the homotopy groups of S1, the long exact sequence of homotopy groups associated to η has the following direct consequence. Lemma. π2(S2) ∼= Z and πn(S3) ∼= πn(S2) for n ≥ 3. We shall later prove the following more substantial result. Theorem. For all n ≥ 1, πn(Sn) ∼= Z. It is left as an exercise to show that the long exact sequence associated to ν implies that π7(S4) contains an element of inﬁnite order, and σ can be used similarly to show the same for π15(S8). In fact, the homotopy groups πq(Sn) for q > n > 1 are all ﬁnite except for π4n−1(S2n), which is the direct sum of Z and a ﬁnite group. The diﬃculty of computing homotopy groups is well illustrated by the fact that there is no non-contractible simply connected compact manifold (or ﬁnite CW complex) all of whose homotopy groups are known. We shall ﬁnd many non-compact spaces whose homotopy groups we can determine completely. Such computations will rely on the following observation. Lemma. If X is the colimit of a sequence of inclusions Xi −→ Xi+1 of based spaces, then the natural map is an isomorphism for each n. colimi πn(Xi) −→ πn(X) Proof. This follows directly from the point-set topological fact that if K is a compact space, then a map K −→ X has image in one of the Xi. 68 HIGHER HOMOTOPY GROUPS 5. Change of basepoint We shall use our results on change of ﬁbers to generalize our results on change of basepoint from the fundamental group to the higher absolute and relative homotopy groups. In the absolute case, we have the identiﬁcation πn(X, x) = [(Sn, ∗), (X, x)],
where we assume that n ≥ 1. Since the inclusion of the basepoint in Sn is a coﬁbration, evaluation at the basepoint gives a ﬁbration p : X Sn −→ X. We may identify πn(X, x) with π0(Fx) since a path in Fx is just a based homotopy h : Sn × I −→ X with respect to the basepoint x. Another way to see this is to observe that Fx is the nth loop space ΩnX, speciﬁed with respect to the basepoint x. A path class [ξ] : I −→ X from x to x′ induces a homotopy equivalence τ [ξ] : Fx −→ Fx′, and we continue to write τ [ξ] for the induced bijection τ [ξ] : πn(X, x) −→ πn(X, x′). This bijection is an isomorphism of groups. One conceptual way to see this is to observe that addition is induced from the “pinch map” Sn −→ Sn ∨ Sn that is obtained by collapsing an equator to the basepoint. That is, the sum of maps f, g : Sn −→ X is the composite Sn −→ Sn ∨ Sn f ∨g −−→ X ∨ X ▽ −→ X, where ▽ is the folding map, which restricts to the identity map X −→ X on each wedge summand. Evaluation at the basepoint of Sn ∨ Sn gives a ﬁbration X Sn∨Sn −→ X, and the pinch map induces a map of ﬁbrations X Sn∨Sn / X Sn X X. The ﬁber over x in the left-hand ﬁbration is the product Fx × Fx, where Fx is In fact, the induced map of ﬁbers the ﬁber over x in the right-hand ﬁbration. can be identiﬁed as the map ΩnX × ΩnX −→ ΩnX given by composition of loops (using the ﬁrst loop coordinate say). By the naturality of translations of ﬁbers with respect to maps of ﬁbrations
, we have a homotopy commutative diagram Fx × Fx Fx τ [ξ]×τ [ξ] τ [ξ] Fx′ × Fx′ / Fx′ in which the horizontal arrows induce addition on passage to π0. We can argue similarly in the relative case. The triple (I n, ∂I n, J n) is homotopy equivalent to the triple (CSn−1, Sn−1, ∗), as we see by quotienting out J n. Therefore, for a ∈ A, we have the identiﬁcation πn(X, A, a) ∼= [(CSn−1, Sn−1, ∗), (X, A, a)]. Using that the inclusions {∗} −→ Sn−1 and Sn−1 −→ CSn−1 are both coﬁbrations, we can check that evaluation at ∗ speciﬁes a ﬁbration p : (X, A)(CSn−1,Sn−1) −→ A, / / / / 6. n-EQUIVALENCES, WEAK EQUIVALENCES, AND A TECHNICAL LEMMA 69 where the domain is the subspace of X CSn−1 consisting of the indicated maps of pairs. We may identify πn(X, A, a) with π0(Fa). A path class [α] : I −→ A from a to a′ induces a homotopy equivalence τ [α] : Fa −→ Fa′, and we continue to write τ [α] for the induced isomorphism τ [α] : πn(X, A, a) −→ πn(X, A, a′). Our naturality results on change of ﬁbers now directly imply the desired results on change of basepoint. Theorem. If f : (X, A) −→ (Y, B) is a map of pairs and α : I −→ A is a path from a to a′, then the following diagram commutes: πn(X, A, a) τ [α] πn(X, A, a′) f∗ f∗ / πn(Y, B, f (a)) τ [f ◦α] / π
n(Y, B, f (a′)) If h : f ≃ f ′ is a homotopy of maps of pairs and h(a)(t) = h(a, t), then the following diagram commutes: πn(X, A, a) f∗ wnnnnnnnnnnnn f ′ ∗ (PPPPPPPPPPPP πn(Y, B, f (a)) τ [h(a)] / πn(Y, B, f ′(a)). The analogous conclusions hold for the absolute homotopy groups. Therefore, up to non-canonical isomorphism, the homotopy groups of (X, A) are independent of the choice of basepoint in a given path component of A. Corollary. A homotopy equivalence of spaces or of pairs of spaces induces an isomorphism on all homotopy groups. We shall soon show that the converse holds for a quite general class of spaces, namely the class of CW complexes, but we ﬁrst need a few preliminaries. 6. n-Equivalences, weak equivalences, and a technical lemma Definition. A map e : Y −→ Z is an n-equivalence if, for all y ∈ Y, the map e∗ : πq(Y, y) −→ πq(Z, e(y)) is an injection for q < n and a surjection for q ≤ n; e is said to be a weak equivalence if it is an n-equivalence for all n. Thus any homotopy equivalence is a weak equivalence. The following technical lemma will be at the heart of our study of CW complexes, but it will take some getting used to. It gives a useful criterion for determining when a given map is an n-equivalence. It is convenient to take CX to be the unreduced cone X × I/X × {1} here. If f, f ′ : (X, A) −→ (Y, B) are maps of pairs such that f = f ′ on A, then we say that f and f ′ are homotopic relative to A if there is a homotopy h : f ≃ f ′ such that h is constant on A, in the sense that h(a, t) = f (a
) for all a ∈ A and t ∈ I; we write h : f ≃ f ′ rel A. Observe that πn+1(X, x) can be viewed as the set of relative homotopy classes of maps (CSn, Sn) −→ (X, x). / / w ( / 70 HIGHER HOMOTOPY GROUPS Lemma. The following conditions on a map e : Y −→ Z are equivalent. (i) For any y ∈ Y, e∗ : πq(Y, y) −→ πq(Z, e(y)) is an injection for q = n and a surjection for q = n + 1. (ii) Given maps f : CSn −→ Z, g : Sn −→ Y, and h : Sn × I −→ Z such that f \|Sn = h ◦ i0 and e ◦ g = h ◦ i1 in the following diagram, there are maps ˜g and ˜h that make the entire diagram commute. Sn f =zzzzzzzz CSn i0 i0 Z dH / Sn × I h zvvvvvvvvv e H H ˜h H H CSn × I i1 Y i1 Sn g }zzzzzzzz aD D ˜g D D CSn (iii) The conclusion of (ii) holds when f \|Sn = e ◦ g and h is the constant homotopy at this map. Proof. Trivially (ii) implies (iii). We ﬁrst show that (iii) implies (i). If n = 0, (iii) says (in part) that if e(y) and e(y′) can be connected by a path in Z, then y and y′ can be connected by a path in Y. If n > 0, then (iii) says (in part) that if e ◦ g is null homotopic, then g is null homotopic. Therefore πn(e) is injective. If we specialize (iii) by letting g be the constant map at a point y ∈ Y, then f is a map (CSn, Sn) −→ (Z, e(y)), ˜g is a map (CSn, Sn) −→ (Y, y), and ˜h : f ≃ e ◦ ˜
g rel Sn. Therefore πn+1(e) is surjective. Thus assume (i). We must prove (ii), and we assume given f, g, and h making the solid arrow part of the diagram commute. The idea is to use (i) to show that the nth homotopy group of the ﬁber F (e) is zero, to use the given part of the diagram to construct a map Sn −→ F (e), and to use a null homotopy of that map to construct ˜g and ˜h. However, since homotopy groups involve choices of basepoints and the diagram makes no reference to basepoints, the details require careful tracking of basepoints. Thus ﬁx a basepoint ∗ ∈ Sn, let • be the cone point of CSn, and deﬁne y1 = g(∗), z1 = e(y1), z0 = f (∗, 0), and z−1 = f (•). For x ∈ Sn, let fx : I −→ Z and hx : I −→ Z be the paths fx(s) = f (x, s) from f (x, 0) = h(x, 0) to z−1 and hx(t) = h(x, t) from h(x, 0) to h(x, 1) = (e ◦ g)(x). Consider the homotopy ﬁber F (e; y1) = {(y, ζ)\|ζ(0) = z1 and e(y) = ζ(1)} ⊂ Y × Z I. This has basepoint w1 = (y1, cz1). By (i) and the exact sequence e∗−→ πn+1(Z, z1) −→ πn(F (e; y1), w1) −→ πn(Y, y1) we see that πn(F (e; y1), w1) = 0. Deﬁne k0 : Sn −→ F (e; y1) by πn+1(Y, y1) e∗−→ πn(Z, z1), k0(x) = (g(x), hx · f −
1 x · f∗ · h−1 ∗ ). While k0 is not a based map, k0(∗) is connected to the basepoint since h∗·f −1 ∗ ·f∗·h−1 ∗ is equivalent to cz1. By HEP for the coﬁbration {∗} −→ Sn, k0 is homotopic to a based map. This based map is null homotopic in the based sense, hence k0 is null /. n-EQUIVALENCES, WEAK EQUIVALENCES, AND A TECHNICAL LEMMA 71 homotopic in the unbased sense. Let k : Sn × I −→ F (e; y1) be a homotopy from k0 to the trivial map at w1. Write k(x, t) = (˜g(x, t), ζ(x, t)). Then ˜g(x, 1) = y1 for all x ∈ Sn, so that ˜g factors through a map CSn −→ Y, and ˜g = g on Sn. We have a map j : Sn × I × I given by j(x, s, t) = ζ(x, t)(s) that behaves as follows on the boundary of the square for each ﬁxed x ∈ Sn, where ˜gx(t) = ˜g(x, t): cz1/ cz1 O e◦˜gx −1 x ·f∗·h The desired homotopy ˜h, written ˜h(x, s, t) where s is the cone coordinate and t is the interval coordinate, should behave as follows on the boundary of the square: hx·f −1 ∗ e◦˜gx hx O h∗·f −1 ∗ fx Thus we can obtain ˜h by composing j with a suitable reparametrization I 2 −→ I 2 of the square. PROBLEMS (1) Show that, if n ≥ 2, then πn(X ∨ Y ) is isomorphic to πn(X) ⊕ πn(Y ) ⊕ πn+1(X × Y, X ∨ Y ). (2) Compute πn(RP n, RP n−
1) for n ≥ 2. Deduce that the quotient map (RP n, RP n−1) → (RP n/RP n−1, ∗) does not induce an isomorphism of homotopy groups. (3) Compute the homotopy groups of complex projective space CP n in terms of the homotopy groups of spheres. (4) Verify that the “Hopf bundles” are in fact bundles. (5) Show that π7(S4) contains an element of inﬁnite order. (6) Compute all of the homotopy groups of RP ∞ and CP ∞. / CHAPTER 10 CW complexes We introduce a large class of spaces, called CW complexes, between which a weak equivalence is necessarily a homotopy equivalence. Thus, for such spaces, the homotopy groups are, in a sense, a complete set of invariants. Moreover, we shall see that every space is weakly equivalent to a CW complex. 1. The deﬁnition and some examples of CW complexes Let Dn+1 be the unit disk {x \| \|x\| ≤ 1} ⊂ Rn+1 with boundary Sn. Definition. (i) A CW complex X is a space X which is the union of an expanding sequence of subspaces X n such that, inductively, X 0 is a discrete set of points (called vertices) and X n+1 is the pushout obtained from X n by attaching disks Dn+1 along “attaching maps” j : Sn −→ X n. Thus X n+1 is the quotient space obtained from X n ∪ (Jn+1 × Dn+1) by identifying (j, x) with j(x) for x ∈ Sn, where Jn+1 is the discrete set of such attaching maps j. Each resulting map Dn+1 −→ X is called a “cell.” The subspace X n is called the n-skeleton of X. (ii) More generally, given any space A, we deﬁne a relative CW complex (X, A) in the same fashion, but with X 0 replaced by the union of A and a (possibly empty) discrete set of points; we write (X, A)n, or X n when A is clear from the context, for the relative n-
skeleton, and we say that (X, A) has dimension ≤ n if X = X n. (iii) A subcomplex A of a CW complex X is a subspace and a CW complex such that the composite of each cell Dn −→ A of A and the inclusion of A in X is a cell of X. That is, A is the union of some of the cells of X. The pair (X, A) can then be viewed as a relative CW complex. (iv) A map of pairs f : (X, A) −→ (Y, B) between relative CW complexes is said to be “cellular” if f (X n) ⊂ Y n for all n. Of course, pushouts and unions are understood in the topological sense, with the compactly generated topologies. A subspace of X is closed if and only if its intersection with each X n is closed. Examples. (i) A graph is a one-dimensional CW complex. (ii) Via a homeomorphism I × I ∼= D2, the standard presentations of the torus T = S1 × S1, the projective plane RP 2, and the Klein bottle K as quotients of a square display these spaces as CW complexes with one or two vertices, two edges, and one 2-cell: 73 74 CW COMPLEXES T e1/ e1 v e2 O v v v1 e2O e2 O v v2 RP 2 e1/ e1 K e1/ e1 v e2 v v2 v e2 e2 O v1 v (iii) For n ≥ 1, Sn is a CW complex with one vertex {∗} and one n-cell, the attaching map Sn−1 −→ {∗} being the only possible map. Note that this entails a choice of homeomorphism Dn/Sn−1 ∼= Sn. If m < n, then the only cellular map Sm −→ Sn is the trivial map. If m ≥ n, then every based map Sm −→ Sn is cellular. (iv) RP n is a CW complex with m-skeleton RP m and with one m-cell for each m ≤ n. The attaching map j : Sn−1 −→ RP n−1 is the standard double cover. That is, RP n is homeomorphic to RP n−1∪j Dn. Explicitly, write ¯x = [
x1,..., xn+1], i = 1, for a typical point of RP n. Then ¯x is in RP n−1 if and only if xn+1 = x2 0. The required homeomorphism is obtained by identifying Dn and its boundary P sphere with the upper hemisphere En + = {(x1,..., xn+1) \| x2 i = 1 and xn+1 ≥ 0} and its boundary sphere. P (v) CP n is a CW complex whose 2m-skeleton and (2m + 1)-skeleton are both CP m and which has one 2m-cell for each m ≤ n. The attaching map S2n−1 −→ CP n−1 is the standard bundle with ﬁber S1, where S2n−1 is identiﬁed with the unit sphere in Cn. We leave the speciﬁcation of the required homeomorphism as an exercise. 2. Some constructions on CW complexes We need to know that various constructions on spaces preserve CW complexes. We leave most of the proofs as exercises in the meaning of the deﬁnitions. Lemma. If (X, A) is a relative CW complex, then the quotient space X/A is a CW complex with a vertex corresponding to A and one n-cell for each relative n-cell of (X, A). Lemma. For CW complexes Xi with basepoints that are vertices, the wedge of the Xi is a CW complex which contains each Xi as a subcomplex. Lemma. If A is a subcomplex of a CW complex X, Y is a CW complex, and f : A −→ Y is a cellular map, then the pushout Y ∪f X is a CW complex that contains Y as a subcomplex and has one cell for each cell of X that is not in A. The quotient complex (Y ∪f X)/Y is isomorphic to X/A. Lemma. The colimit of a sequence of inclusions of subcomplexes Xn −→ Xn+1 in CW complexes is a CW complex that contains each of the Xi as a subcomplex. HELP AND THE WHITEHEAD THEOREM 75 Lemma. The product X × Y of CW complexes X and Y is a CW complex with an n-cell for each pair consisting of a p-cell of
X and q-cell of Y, where p + q = n. Proof. For p + q = n, there are canonical homeomorphisms (Dn, Sn−1) ∼= (Dp × Dq, Dp × Sq−1 ∪ Sp−1 × Dq). This allows us to deﬁne product cells. We shall look at the general case more closely later, but we point out one important special case for immediate use. Of course, the unit interval is a graph with two vertices and one edge. Lemma. For a CW complex X, X × I is a CW complex that contains X × ∂I as a subcomplex and, in addition, has one (n + 1)-cell for each n-cell of X. A “cellular homotopy” h : f ≃ f ′ between cellular maps X −→ Y of CW complexes is a homotopy that is itself a cellular map X × I −→ Y. 3. HELP and the Whitehead theorem The following “homotopy extension and lifting property” is a powerful organi- zational principle for proofs of results about CW complexes. In the case (X, A) = (Dn, Sn−1) ∼= (CSn−1, Sn−1), it is the main point of the technical lemma proved at the end of the last chapter. Theorem (HELP). Let (X, A) be a relative CW complex of dimension ≤ n and let e : Y −→ Z be an n-equivalence. Then, given maps f : X −→ Z, g : A −→ Y, and h : A × I −→ Z such that f \|A = h ◦ i0 and e ◦ g = h ◦ i1 in the following diagram, there are maps ˜g and ˜h that make the entire diagram commute: A X i0 i0 / A × I h \|xxxxxxxxx bF F F ˜h F e F X × I Z f?~~~~~~~ i1 Y i1 A g ~~~~~~~~~ `@ @ ˜g @ @ X Proof. Proceed by induction over skeleta, applying the case (Dn, Sn−1) one cell at a time to the n-cells of X not in A. In particular, if we take e to be the identity map of Y, we see that
the inclusion A −→ X is a coﬁbration. Observe that, by passage to colimits, we are free to take n = ∞ in the theorem. We write [X, Y ] for homotopy classes of unbased maps in this chapter, and we have the following direct and important application of HELP. Theorem (Whitehead). If X is a CW complex and e : Y −→ Z is an nequivalence, then e∗ : [X, Y ] −→ [X, Z] is a bijection if dim X < n and a surjection if dim X = n 76 CW COMPLEXES Proof. Apply HELP to the pair (X, ∅) to see the surjectivity. Apply HELP to the pair (X × I, X × ∂I), taking h to be a constant homotopy, to see the injectivity. Theorem (Whitehead). An n-equivalence between CW complexes of dimension less than n is a homotopy equivalence. A weak equivalence between CW complexes is a homotopy equivalence. Proof. Let e : Y −→ Z satisfy either hypothesis. Since e∗ : [Z, Y ] −→ [Z, Z] is a bijection, there is a map f : Z −→ Y such that e ◦ f ≃ id. Then e ◦ f ◦ e ≃ e, and, since e∗ : [Y, Y ] −→ [Y, Z] is also a bijection, this implies that f ◦ e ≃ id. If X is a ﬁnite CW complex, in the sense that it has ﬁnitely many cells, and if dim X > 1 and X is not contractible, then it is known that X has inﬁnitely many non-zero homotopy groups. The Whitehead theorem is thus surprisingly strong: in its ﬁrst statement, if low dimensional homotopy groups are mapped isomorphically, then so are all higher homotopy groups. 4. The cellular approximation theorem Cellular maps are under much better algebraic control than general maps, as will become both clear and important later. Fortunately, any map between CW complexes is homotopic to a cellular map. We need a lemma. Definition. A space X is said to be n-connected if πq(X, x) = 0 for
0 ≤ q ≤ n and all x. A pair (X, A) is said to be n-connected if π0(A) −→ π0(X) is surjective and πq(X, A, a) = 0 for 1 ≤ q ≤ n and all a. It is equivalent that the inclusion A −→ X be an n-equivalence. Lemma. A relative CW complex (X, A) with no m-cells for m ≤ n is n- connected. In particular, (X, X n) is n-connected for any CW complex X. Proof. Consider f : (I q, ∂I q, J q) −→ (X, A, a), where q ≤ n. Since the image of f is compact, we may assume that (X, A) has ﬁnitely many cells. By induction on the number of cells, we may assume that X = A ∪j Dr, where r > n. By smooth (or simplicial) approximation, there is a map f ′ : I q −→ X such that f ′ = f on ∂I q, f ′ ≃ f rel ∂I q and f ′ misses a point p in the interior of Dr. Clearly we can deform X − {p} onto A and so deform f ′ to a map into A. Theorem (Cellular approximation). Any map f : (X, A) −→ (Y, B) between relative CW complexes is homotopic relative to A to a cellular map. Proof. We proceed by induction over skeleta. To start the induction, note that any point of Y is connected by a path to a point in Y 0 and apply this to the images of points of X 0 −A to obtain a homotopy of f \|X 0 to a map into Y 0. Assume given gn : X n −→ Y n and hn : X n × I −→ Y such that hn : f \|X n ≃ ιn ◦ gn, where ιn : Y n −→ Y is the inclusion. For an attaching map j : Sn −→ X n of a cell 5. APPROXIMATION OF SPACES BY CW COMPLEXES 77 ˜j : Dn+1 −→ X, we apply HELP to the following diagram: i0 / Sn × I i1 Sn f �
�˜j <zzzzzzzzz Dn+1 hn◦(j×id) zuuuuuuuuuu Y dH H hn+1 H H H ιn+1 Y n+1 dH Sn gn◦j zvvvvvvvvv H gn+1 H H H Dn+1 / Dn+1 × I i0 i1 where gn ◦ j : Sn −→ Y n is composed with the inclusion Y n −→ Y n+1; HELP applies since ιn+1 is an (n + 1)-equivalence. Corollary. For CW complexes X and Y, any map X −→ Y is homotopic to a cellular map, and any two homotopic cellular maps are cellularly homotopic. 5. Approximation of spaces by CW complexes The following result says that there is a functor Γ : hU −→ hU and a natural transformation γ : Γ −→ Id that assign a CW complex ΓX and a weak equivalence γ : ΓX −→ X to a space X. Theorem (Approximation by CW complexes). For any space X, there is a CW complex ΓX and a weak equivalence γ : ΓX −→ X. For a map f : X −→ Y and another such CW approximation γ : ΓY −→ Y, there is a map Γf : ΓX −→ ΓY, unique up to homotopy, such that the following diagram is homotopy commutative: ΓX γ X Γf f ΓY γ / Y. If X is n-connected, n ≥ 1, then ΓX can be chosen to have a unique vertex and no q-cells for 1 ≤ q ≤ n. Proof. The existence and uniqueness up to homotopy of Γf will be immediate since the Whitehead theorem will give a bijection γ∗ : [ΓX, ΓY ] −→ [ΓX, Y ]. Proceeding one path component at a time, we may as well assume that X is path connected, and we may then work with based spaces and based maps. We construct ΓX as the colimit of a sequence of cellular inclusions in / / Xn+1 / · · · · · · i1 i2 / Xn X2
X1 3333333333333 "EEEEEEEEEEEEEEEEEE γ2 γ1 γn {wwwwwwwwwwwwwwwwwww γn+1 X. Let X1 be a wedge of spheres Sq, q ≥ 1, one for each pair (q, j), where j : Sq −→ X represents a generator of the group πq(X). On the (q, j)th wedge summand, the / / / / / / " / / / / / { / 78 CW COMPLEXES map γ1 is the given map j. Clearly γ1 : X1 −→ X induces an epimorphism on all homotopy groups. We give X1 the CW structure induced by the standard CW structures on the spheres Sq. Inductively, suppose that we have constructed CW complexes Xm, cellular inclusions im−1, and maps γm for m ≤ n such that γm ◦ im−1 = γm−1 and (γm)∗ : πq(Xm) −→ πq(X) is a surjection for all q and a bijection for q < m. We construct Xn+1 = Xn ∪ ( (Sn ∧ I+)), _(f,g) where the wedge is taken over cellular representatives f, g : Sn −→ Xn in each pair of homotopy classes [f ], [g] ∈ πn(Xn) such that [f ] 6= [g] but [γn ◦ f ] = [γn ◦ g]. We attach the (f, g)th reduced cylinder Sn ∧ I+ to Xn by identifying (s, 0) with f (s) and (s, 1) with g(s) for s ∈ Sn. Let in : Xn −→ Xn+1 be the inclusion and observe that (in)∗[f ] = (in)∗[g]. Deﬁne γn+1 : Xn+1 −→ X by means of γn on Xn and a chosen homotopy h : Sn ∧ I+ −→ X from γn ◦ f to γn ◦ g on the (f, g)th cylinder. Then (γn+1)∗ : πq(Xn+1) −
→ πq(X) is a surjection for all q, because (γn)∗ is so, and a bijection for q ≤ n by construction. We have not changed the homotopy groups in dimensions less than n since we have not changed the n-skeleton. Since f and g are cellular and since, as is easily veriﬁed, Sn ∧ I+ admits a CW structure with Sn ∧ (∂I)+ as a subcomplex, we conclude from the pushout property of CW complexes that Xn+1 is a CW complex that contains Xn as a subcomplex. Then the colimit ΓX of the Xn is a CW complex that contains all of the Xi as subcomplexes, and the induced map γ : ΓX −→ X induces an isomorphism on all homotopy groups since the homotopy groups of ΓX are the colimits of the homotopy groups of the Xn. If X is n-connected, then we have used no q-cells for q ≤ n in the construction. 6. Approximation of pairs by CW pairs We will need a relative generalization of the previous result, but the reader should not dwell on the details: there are no new ideas. Theorem. For any pair of spaces (X, A) and any CW approximation γ : ΓA −→ A, there is a CW approximation γ : ΓX −→ X such that ΓA is a subcomplex of ΓX and γ restricts to the given γ on ΓA. If f : (X, A) −→ (Y, B) is a map of pairs and γ : (ΓY, ΓB) −→ (Y, B) is another such CW approximation of pairs, there is a map Γf : (ΓX, ΓA) −→ (ΓY, ΓB), unique up to homotopy, such that the following diagram of pairs is homotopy commutative: (ΓX, ΓA) Γf (ΓY, ΓB) γ γ (X, A) / (Y, B). f If (X, A) is n-connected, then (ΓX, ΓA) can be chosen to have no relative q-cells for q ≤ n. / / /
7. APPROXIMATION OF EXCISIVE TRIADS BY CW TRIADS 79 Proof. We proceed as above. We may assume that X has a basepoint in A and that X, but not necessarily A, is path connected. We start with X0 = ΓA ∨ ( Sq), _(q,j) where {(q, j)} runs over q ≥ 1 and based maps j : Sq −→ X that represent generators of πq(X). Here the chosen basepoint is in ΓA. Construct γ0 : X0 −→ X using the maps j and the given map γ : ΓA −→ A. Construct X1 from X0 by attaching 1-cells connecting the vertices in the non-basepoint components of ΓA to the base vertex. Paths in X that connect the images under γ of the non-basepoint vertices to the basepoint of X give γ1 : X1 −→ X extending γ0. From here, the construction continues as in §5. If (X, A) is n-connected, then πq(A) −→ πq(X) is bijective for q < n and surjective for q = n, hence we need only use spheres Sq with q > n to arrange the surjectivity of π∗(X0) −→ π∗(X). To construct Γf, we ﬁrst construct it on ΓA and then use HELP to extend to ΓX: ΓA γ!CCCCCCCC γ ={{{{{{{{ ΓX A X f f ΓA × I h {wwwwwwwww B / Y cG G ˜h G G G / ΓX × I γ γ ΓA Γf \|zzzzzzzz ΓB ΓY bD D Γf D D ΓX The uniqueness up to homotopy of Γf is proved similarly. 7. Approximation of excisive triads by CW triads We will need another, and considerably more subtle, relative approximation theorem. A triad (X; A, B) is a space X together with subspaces A and B. This must not be confused with a triple (X, A, B), which would require B ⊂ A
⊂ X. A triad (X; A, B) is said to be excisive if X is the union of the interiors of A and B. Such triads play a fundamental role in homology and cohomology theory, and some version of the arguments to follow must play a role in any treatment. We prefer to use these arguments to prove a strong homotopical result, rather than its pale homological reﬂection that is seen in standard treatments of the subject. A CW triad (X; A, B) is a CW complex X with subcomplexes A and B such that X = A ∪ B. Theorem. Let (X; A, B) be an excisive triad and let C = A ∩ B. Then there is a CW triad (ΓX; ΓA, ΓB) and a map of triads such that, with ΓC = ΓA ∩ ΓB, the maps γ : (ΓX; ΓA, ΓB) −→ (X; A, B) γ : ΓC −→ C, γ : ΓA −→ A, γ : ΓB −→ B, and γ : ΓX −→ X are all weak equivalences. If (A, C) is n-connected, then (ΓA, ΓC) can be chosen to have no q-cells for q ≤ n, and similarly for (B, C). Up to homotopy, CW approximation of excisive triads is functorial in such a way that γ is natural 80 CW COMPLEXES Proof. Choose a CW approximation γ : ΓC −→ C and use the previous result to extend it to CW approximations γ : (ΓA, ΓC) −→ (A, C) and γ : (ΓB, ΓC) −→ (B, C). We then deﬁne ΓX to be the pushout ΓA ∪ΓC ΓB and let γ : ΓX −→ X be given by the universal property of pushouts. Certainly ΓC = ΓA ∩ ΓB. All of the conclusions except for the assertion that γ : ΓX −→ X is a weak equivalence follow immediately from the result for pairs,
and the lemma and theorem below will complete the proof. A CW triad (X; A, B) is not excisive, since A and B are closed in X, but it is equivalent to an excisive triad. To see this, we describe a simple but important general construction. Suppose that maps i : C −→ A and j : C −→ B are given. Deﬁne the double mapping cylinder M (i, j) = A ∪ (C × I) ∪ B to be the space obtained from C × I by gluing A to C × {0} along i and gluing B to C × {1} along j. Let A ∪C B denote the pushout of i and j and observe that we obtain a natural quotient map q : M (i, j) −→ A ∪C B by collapsing the cylinder, sending (c, t) to the image of c in the pushout. Lemma. For a coﬁbration i : C −→ A and any map j : C −→ B, the quotient map q : M (i, j) −→ A ∪C B is a homotopy equivalence. Proof. Because i is a coﬁbration, the retraction r : M i −→ A is a coﬁber homotopy equivalence. That is, there is a homotopy inverse map and a pair of homotopies under C. These maps and homotopies induce maps of the pushouts that are obtained by gluing B to M i and to C, and q is induced by r. When i is a coﬁbration and j is an inclusion, with X = A ∪ B and C = A ∩ B, we can think of q as giving a map of triads q : (M (i, j); A ∪ (C × [0, 2/3)), (C × (1/3, 1]) ∪ B) −→ (A ∪C B; A, B). The domain triad is excisive, and q restricts to homotopy equivalences from the domain subspaces and their intersection to the target subspaces A, B, and C. This applies when (X; A, B) is a CW triad with C = A ∩ B. Now our theorem on the approximation of excisive tri
ads is a consequence of the following result. Theorem. If e : (X; A, B) −→ (X ′; A′, B′) is a map of excisive triads such that the maps e : C −→ C′, e : A −→ A′, and e : B −→ B′ are weak equivalences, where C = A ∩ B and C′ = A′ ∩ B′, then e : X −→ X ′ is a weak equivalence. Proof. By our technical lemma giving equivalent conditions for a map e to be a weak equivalence, it suﬃces to show that if f \|Sn = e ◦ g in the following diagram, then there exists a map ˜g such that ˜g\|Sn = g and f ≃ e ◦ ˜g rel Sn: cG G X g Sn e / X ′ f ˜g G G G / Dn+1. / O O / c O O 7. APPROXIMATION OF EXCISIVE TRIADS BY CW TRIADS 81 We may assume without loss of generality that Sn ⊂ U ⊂ Dn+1, where U is open in Dn+1 and g is the restriction of a map ˆg : U −→ X such that f \|U = e ◦ ˆg. To see this, deﬁne a deformation d : Dn+1 × I −→ Dn+1 by d(x, t) = 2x/(2 − t) x/\|x\| ( if if \|x\| ≤ (2 − t)/2 \|x\| ≥ (2 − t)/2. Then d(x, 0) = x, d(x, t) = x if x ∈ Sn, and d1 maps the boundary collar {x \| \|x\| ≥ 1/2} onto Sn. Let U be the open boundary collar {x \| \|x\| > 1/2}. Deﬁne ˆg = g ◦ d1 : U −→ X and deﬁne f ′ = f ◦ d1 : Dn+1 −→ X ′. Then ˆg\|Sn = g, e ◦ ˆg = f ′\|U, and f ′ ≃ f rel Sn. Thus the conclusion
will hold for f if it holds with f replaced by f ′. With this assumption on g and f, we claim ﬁrst that the closed sets CA = g−1(X − int A) ∪ f −1(X ′ − A′) and CB = g−1(X − int B) ∪ f −1(X ′ − B′), have empty intersection. Indeed, these sets are contained in the sets ˆCA and ˆCB that are obtained by replacing g by ˆg in the deﬁnitions of CA and CB, and we claim that ˆCA ∩ ˆCB = ∅. Certainly ˆg−1(X − int A) ∩ ˆg−1(X − int B) = ∅ since (X − int A) ∩ (X − int B) = ∅. Similarly, f −1(X ′ − int A′) ∩ f −1(X ′ − int B′) = ∅. Since f −1(X ′ − A′) ⊂ f −1(X ′ − int A′) and similarly for B, this implies that f −1(X ′ − A′) ∩ f −1(X ′ − B′) = ∅. Now suppose that v ∈ ˆCA ∩ ˆCB. In view of the possibilities that we have ruled out, we may assume that v ∈ ˆg−1(X − int A) ∩ f −1(X ′ − B′) ⊂ ˆg−1(int B) ∩ f −1(X ′ − B′). Since ˆg−1(int B) is an open subset of Dn, there must be a point u ∈ ˆg−1(int B) ∩ f −1(X ′ − B′). Then ˆg(u) ∈ int B ⊂ B but f (u) 6∈ B′. This contradicts f \|U = e ◦ ˆg. We can subdivide Dn+1 suﬃciently ﬁnely (as a simplicial or CW complex) that no cell intersects both CA and CB. Let KA be the
union of those cells σ such that g(σ ∩ Sn) ⊂ int A and f (σ) ⊂ int A′ and deﬁne KB similarly. If σ does not intersect CA, then σ ⊂ KA, and if σ does not intersect CB, then σ ⊂ KB. Therefore Dn+1 = KA ∪ KB. By HELP, we can obtain a map ¯g such that the lower triangle in the diagram A ∩ B g e / A′ ∩ B′ ¯g hQ Q Q Q Q Q Q f Sn ∩ (KA ∩ KB) / KA ∩ KB commutes, together with a homotopy ¯h : (KA ∩ KB) × I −→ A′ ∩ B′ such that ¯h : f ≃ e ◦ ¯g rel Sn ∩ (KA ∩ KB). / O O / O O h 82 CW COMPLEXES Deﬁne ¯gA : KA ∩ (Sn ∪ KB) −→ A to be g on KA ∩ Sn and ¯g on KA ∩ KB. Since f = e ◦ g on KA ∩ Sn and ¯h : f ≃ e ◦ ¯g on KA ∩ KB, ¯h induces a homotopy ¯hA : f \|KA ∩ (Sn ∪ B) ≃ e ◦ gA rel Sn ∩ KA. Applying HELP again, we can obtain maps ˜gA and ˜hA such that the following diagram commutes: KA ∩ (Sn ∪ KB) f 8pppppppppppp KA i0 A′ i0 / KA ∩ (Sn ∪ KB) × I ¯hA wooooooooooooo gP P P P P P P ˜hA e KA × I i1 A i1 KA ∩ (Sn ∪ KB) ¯gA xqqqqqqqqqqq fM M M ˜gA M M M KA We have a symmetric diagram with the roles of KA and KB reversed. The maps ˜gA and ˜gB agree on KA ∩ KB and together deﬁne the desired map ˜g : Dn+1 −→ X. The homotopies ˜hA and ˜h
B agree on (KA ∩ KB) × I and together deﬁne the desired homotopy ˜hA : f ≃ e ◦ ˜g rel Sn. PROBLEMS (1) Show that complex projective space CP n is a CW complex with one 2q-cell for each q, 0 ≤ q ≤ n. (2) Let X = {x\|x = 0 or x = 1/n for a positive integer n} ⊂ R. Show that X does not have the homotopy type of a CW complex. (3) Assume given maps f : X −→ Y and g : Y −→ X such that g ◦ f is (We say that Y “dominates” X.) Suppose homotopic to the identity. that Y is a CW complex. Prove that X has the homotopy type of a CW complex. Deﬁne the Euler characteristic χ(X) of a ﬁnite CW complex X to be the (−1)nγn(X), where γn(X) is the number of n-cells of X. Let A alternating sum be a subcomplex of a CW complex X, let Y be a CW complex, let f : A −→ Y be a cellular map, and let Y ∪f X be the pushout of f and the inclusion A −→ X. P 4. Show that Y ∪f X is a CW complex with Y as a subcomplex and X/A as a quotient complex. Formulate and prove a formula relating the Euler characteristics χ(A), χ(X), χ(Y ), and χ(Y ∪f X) when X and Y are ﬁnite. 5. * Think about proving from what we have done so far that χ(X) depends only on the homotopy type of X, not on its decomposition as a ﬁnite CW complex CHAPTER 11 The homotopy excision and suspension theorems The fundamental obstruction to the calculation of homotopy groups is the failure of excision: for an excisive triad (X; A, B), the inclusion (A, A ∩ B) −→ (X, B) fails to induce an isomorphism of homotopy groups in general. It is this that distinguishes homotopy groups from the far more computable hom
ology groups. However, we do have such an isomorphism in a range of dimensions. This implies the Freudenthal suspension theorem, which gives that πn+q(ΣnX) is independent of n if q is small relative to n. We shall rely on the consequence πn(Sn) ∼= Z in our construction of homology groups. 1. Statement of the homotopy excision theorem We shall prove the following theorem later in this chapter, but we ﬁrst explain its consequences. Definition. A map f : (A, C) −→ (X, B) of pairs is an n-equivalence, n ≥ 1, if (f∗)−1(im(π0(B) −→ π0(X))) = im(π0(C) −→ π0(A)) (which holds automatically when A and X are path connected) and, for all choices of basepoint in C, is a bijection for q < n and a surjection for q = n. f∗ : πq(A, C) −→ πq(X, B) Recall that a pair (A, C) is n-connected, n ≥ 0, if π0(C) −→ π0(A) is surjective and πq(A, C) = 0 for q ≤ n. Theorem (Homotopy excision). Let (X; A, B) be an excisive triad such that C = A ∩ B is non-empty. Assume that (A, C) is (m − 1)-connected and (B, C) is (n − 1)-connected, where m ≥ 2 and n ≥ 1. Then the inclusion (A, C) −→ (X, B) is an (m + n − 2)-equivalence. This specializes to give a relationship between the homotopy groups of pairs (X, A) and of quotients X/A and to prove the Freudenthal suspension theorem. Theorem. Let f : X −→ Y be an (n−1)-equivalence between (n−2)-connected spaces, where n ≥ 2; thus πn−1(f ) is an epimorphism. Then the quotient map π : (M f, X) −→ (Cf, ∗) is a (2
n − 2)-equivalence. In particular, Cf is (n − 1)connected. If X and Y are (n − 1)-connected, then π : (M f, X) −→ (Cf, ∗) is a (2n − 1)-equivalence. Proof. We are writing Cf for the unreduced coﬁber M f /X. We have the excisive triad (Cf ; A, B), where A = Y ∪ (X × [0, 2/3]) and B = (X × [1/3, 1])/(X × {1}). 83 84 THE HOMOTOPY EXCISION AND SUSPENSION THEOREMS Thus C ≡ A ∩ B = X × [1/3, 2/3]. It is easy to check that π is homotopic to a composite (M f, X) ≃−→ (A, C) −→ (Cf, B) ≃−→ (Cf, ∗), the ﬁrst and last arrows of which are homotopy equivalences of pairs. The hypothesis on f and the long exact sequence of the pair (M f, X) imply that (M f, X) and therefore also (A, C) are (n − 1)-connected. In view of the connecting isomorphism ∂ : πq+1(CX, X) −→ πq(X) and the evident homotopy equivalence of pairs (B, C) ≃ (CX, X), (B, C) is also (n − 1)-connected, and it is n-connected if X is (n − 1)-connected. The homotopy excision theorem gives the conclusions. We shall later use the following bit of the result to prove the Hurewicz theorem relating homotopy groups to homology groups. Corollary. Let f : X −→ Y be a based map between (n − 1)-connected nondegenerately based spaces, where n ≥ 2. Then Cf is (n − 1)-connected and πn(M f, X) −→ πn(Cf, ∗) is an isomorphism. Moreover, the canonical map η : F f −→ ΩCf induces an isomorphism πn−1(F f ) −
→ πn(Cf ). Proof. Here in the based context, we are thinking of the reduced mapping cylinder and coﬁber, but the maps to them from the unreduced constructions are homotopy equivalences since our basepoints are nondegenerate. Thus the ﬁrst statement is immediate from the theorem. For the second, if j : X −→ M f is the inclusion, then we have a map F r : F j = P (M f ; ∗, X) −→ F f induced by the retraction r : M f −→ Y. By a comparison of long exact sequences, (F r)∗ : πq(M f, X) −→ πq−1(F f ) is an isomorphism for all q. Moreover, η factors through a map F j −→ ΩCf, as we noted at the end of Chapter 8 §7. Thus the second statement follows from the ﬁrst. Specializing f to be a coﬁbration and changing notation, we obtain the following version of the previous theorem. Theorem. Let i : A −→ X be a coﬁbration and an (n − 1)-equivalence between (n − 2)-connected spaces, where n ≥ 2. Then the quotient map (X, A) −→ (X/A, ∗) is a (2n − 2)-equivalence, and it is a (2n − 1)-equivalence if A and X are (n − 1)connected. Proof. The vertical arrows are homotopy equivalences of pairs in the commu- tative diagram (M i, A) π / (Ci, ∗) r ψ (X, A) / (X/A, ∗). / / 2. THE FREUDENTHAL SUSPENSION THEOREM 85 2. The Freudenthal suspension theorem A specialization of the last result gives the Freudenthal suspension theorem. For a based space X, deﬁne the suspension homomorphism Σ : πq(X) −→ πq+1(ΣX) by letting Σf = f ∧ id : Sq+1 ∼= Sq ∧ S1 −→ X ∧ S1 = ΣX. Theorem (Freud
enthal suspension). Assume that X is nondegenerately based and (n − 1)-connected, where n ≥ 1. Then Σ is a bijection if q < 2n − 1 and a surjection if q = 2n − 1. Proof. We give a diﬀerent description of Σ. Consider the “reversed” cone C′X = X ∧ I, where I is given the basepoint 0 rather than 1. Thus C′X = X × I/X × {0} ∪ {∗} × I. For a map f : (I q, ∂I q) −→ (X, ∗), the product f × id : I q+1 −→ X × I passes to quotients to give a map of triples (I q+1, ∂I q+1, J q) −→ (C′X, X, ∗) whose restriction to I q × {1} is f and which induces Σf when we quotient out X × {1}. That is, the following diagram commutes, where ρ : C′X −→ ΣX is the quotient map: πq+1(C′X, X, ∗) ∂ wppppppppppp ρ∗ (PPPPPPPPPPPP πq(X) / πq+1(ΣX). Σ Since C′X is contractible, ∂ is an isomorphism. Since the inclusion X −→ C′X is a coﬁbration and an n-equivalence between (n − 1)-connected spaces, ρ is a 2nequivalence by the last theorem of the previous section. The conclusion follows. This implies the promised calculation of πn(Sn). Theorem. For all n ≥ 1, πn(Sn) = Z and Σ : πn(Sn) −→ πn+1(Sn+1) is an isomorphism. Proof. We saw by use of the Hopf bundle S3 −→ S2 that π2(S2) = Z, and the suspension theorem applies to give the conclusion for n ≥ 2. A little extra argument is needed to check that Σ is an isomorphism for n = 1; one can inspect the connecting homomorphism of the Hopf bundle or
refer ahead to the observation that the Hurewicz homomorphism commutes with the corresponding suspension isomorphism in homology. The dimensional range of the suspension theorem is sharp. We saw before that π3(S2) = π3(S3), which is Z. The suspension theorem applies to show that Σ : π3(S2) −→ π4(S3) is an epimorphism, and it is known that π4(S3) = Z2. Applying suspension repeatedly, we can form a colimit q (X) = colim πq+n(ΣnX). πs w ( / 86 THE HOMOTOPY EXCISION AND SUSPENSION THEOREMS This group is called the qth stable homotopy group of X. For q < n − 1, the maps of the colimit system are isomorphisms and therefore πs q (X) = πq+n(ΣnX) if q < n − 1. The calculation of the stable homotopy groups of spheres, πs q (S0), is one of the deepest and most studied problems in algebraic topology. Important problems of geometric topology, such as the enumeration of the distinct diﬀerential structures on Sq for q ≥ 5, have been reduced to the determination of these groups. 3. Proof of the homotopy excision theorem This is a deep result, and it is remarkable that a direct homotopical proof, in principle an elementary one, is possible. Most standard texts, if they treat this topic at all, give a far more sophisticated proof of a signiﬁcantly weaker result. However, the reader may prefer to skip this argument on a ﬁrst reading. The idea is clear enough. We are trying to show that a certain map of pairs induces an isomorphism in a range of dimensions. We capture the relevant map as part of a long exact sequence, and we prove that the third term in the long exact sequence vanishes in the required range. However, we start with an auxiliary long exact sequence that we shall also need. Recall that a triple (X, A, B) consists of spaces B ⊂ A ⊂ X and must not be confused with a triad. Proposition. For a triple (X, A, B) and any
basepoint in B, the following sequence is exact: · · · −→ πq(A, B) i∗−→ πq(X, B) j∗ −→ πq(X, A) k∗◦∂ −−−→ πq−1(A, B) −→ · · ·. Here i : (A, B) −→ (X, B), j : (X, B) −→ (X, A), and k : (A, ∗) −→ (A, B) are the inclusions. Proof. The proof is a purely algebraic deduction from the long exact se- quences of the various pairs in sight and is left as an exercise for the reader. We now deﬁne the “triad homotopy groups” that are needed to implement the idea of the proof sketched above. Definition. For a triad (X; A, B) with basepoint ∗ ∈ C = A ∩ B, deﬁne πq(X; A, B) = πq−1(P (X; ∗, B), P (A; ∗, C)), where q ≥ 2. More explicitly, πq(X; A, B) is the set of homotopy classes of maps of tetrads (I q; I q−2 × {1} × I, I q−1 × {1}, J q−2 × I ∪ I q−1 × {0}) (X; A, B, ∗), where J q−2 = ∂I q−2 × I ∪ I q−2 × {0} ⊂ I q−1. The long exact sequence of the pair in the ﬁrst form of the deﬁnition is · · · −→ πq+1(X; A, B) −→ πq(A, C) −→ πq(X, B) −→ πq(X; A, B) −→ · · ·. 3. PROOF OF THE HOMOTOPY EXCISION THEOREM 87 Now we return to the homotopy excision theorem. Its conditions m ≥ 1 and n ≥ 1 merely give that π0(C) −→ π0(A)
and π0(C) −→ π0(B) are surjective, and any extraneous components of A or B would not aﬀect the relevant homotopy groups. The condition m ≥ 2 implies that (X, B) is 1-connected. By the long exact sequence just given, the theorem is equivalent to the following one. Theorem. Under the hypotheses of the homotopy excision theorem, πq(X; A, B) = 0 for 2 ≤ q ≤ m + n − 2 and all choices of basepoint ∗ ∈ C. In this form, the conclusion is symmetric in A and B and vacuous if m + n ≤ 3. Thus our hypotheses m ≥ 2 and n ≥ 1 are the minimal ones under which our strategy can apply. In order to have some hope of tackling the problem in direct terms, we ﬁrst reduce it to the case when A and B are each obtained from C by attaching a single cell. We may approximate our given excisive triad by a weakly equivalent CW triad. This does not change the triad homotopy groups. More precisely, by our connectivity hypotheses, we may assume that X is a CW complex that is the union of subcomplexes A and B with intersection C, where (A, C) has no relative q-cells for q < m and (B, C) has no relative q-cells for q < n. Since any map I q −→ X has image contained in a ﬁnite subcomplex, we may assume that X has ﬁnitely many cells. We may also assume that (A, C) and (B, C) each have at least one cell since otherwise the result holds trivially. We claim ﬁrst that, inductively, it suﬃces to prove the result when (A, C) has exactly one cell. Indeed, suppose that C ⊂ A′ ⊂ A, where A is obtained from A′ by attaching a single cell and (A′, C) has one less cell than (A, C). Let X ′ = A′ ∪C B. If the result holds for the triads (X ′; A′, B) and (X; A, X ′), then the result holds for the triad (X; A, B) by application of the ﬁve
lemma to the following diagram: πq+1(A, A′) πq(A′, C) πq(A, C) πq(A, A′) πq−1(A′, C) πq+1(X, X ′) / πq(X ′, B) / πq(X, B) / πq(X, X ′) / πq−1(X ′, B). The rows are the exact sequences of the triples (A, A′, C) and (X, X ′, B). Note for the case q = 1 that all pairs in the diagram are 1-connected. We claim next that, inductively, it suﬃces to prove the result when (B, C) also has exactly one cell. Indeed, suppose that C ⊂ B′ ⊂ B, where B is obtained from B′ by attaching a single cell and (B′, C) has one less cell than (B, C) and let X ′ = A ∪C B′. If the result holds for the triads (X ′; A, B′) and (X; X ′, B), then the result holds for the triad (X; A, B) since the inclusion (A, C) −→ (X, B) factors as the composite (A, C) −→ (X ′, B′) −→ (X, B). / / / / / / / / / / / / 88 THE HOMOTOPY EXCISION AND SUSPENSION THEOREMS Thus we may assume that A = C ∪ Dm and B = C ∪ Dn, where m ≥ 2 and n ≥ 1, and we ﬁx a basepoint ∗ ∈ C. Assume given a map of tetrads (I q; I q−2 × {1} × I, I q−1 × {1}, J q−2 × I ∪ I q−1 × {0}) f (X; A, B, ∗), where 2 ≤ q ≤ m + n − 2. We must prove that f is null homotopic as a map of tetrads. For interior points x ∈ Dm and y ∈ Dn, we have in
clusions of based triads (A; A, A − x) ⊂ (X − {y}; A, X − {x, y}) ⊂ (X; A, X − {x}) ⊃ (X; A, B). The ﬁrst and third of these induce isomorphisms on triad homotopy groups in view of the radial deformation away from y of X −{y} onto A and the radial deformation away from x of X − {x} onto B. It is trivial to check that π∗(A; A, A′) = 0 for any A′ ⊂ A. We shall show that, for well chosen points x and y, f regarded as a map of based triads into (X; A, X − {x}) is homotopic to a map f ′ that has image in (X − {y}; A, X − {x, y}). This will imply that f is null homotopic. Let Dm 1/2 and f (I q α such that f (I q 1/2 ⊂ Dm and Dn α) is contained in the interior of Dn if it intersects Dn 1/2 and whose restriction to the (m − 1)-skeleton of I q does not cover Dm 1/2 ⊂ Dn be the subdisks of radius 1/2. We can cubically α) is contained in the interior of Dm if subdivide I q into subcubes I q it intersects Dm 1/2. By simplicial approximation, f is homotopic as a map of tetrads to a map g whose restriction to the (n − 1)-skeleton of I q with its subdivided cell structure does not cover Dn 1/2. Moreover, we can arrange that the dimension of g−1(y) is at most q − n for a point 1/2 that is not in the image under g of the (n − 1)-skeleton of I q. This is the y ∈ Dn main point of the proof, and to be completely rigorous about it we would have to digress to introduce a bit of dimension theory. Alternatively, we could use smooth approximation to arrive at g and y with appropriate properties. Since the intuition should be clear, we shall content ourselves with showing how the conclusion of the theorem follows. Let π
: I q −→ I q−1 be the projection on the ﬁrst q − 1 coordinates and let K be the prism π−1(π(g−1(y))). Then K can have dimension at most one more than the dimension of g−1(y), so that dim. Therefore g(K) cannot cover Dm 1/2 such that x 6∈ g(K). Since g(∂I q−1 × I) ⊂ A, we see that π(g−1(x)) ∪ ∂I q−1 and π(g−1(y)) are disjoint closed subsets of I q−1. By Uryssohn’s lemma, we may choose a map v : I q−1 −→ I such that 1/2. Choose a point x ∈ Dm v(π(g−1(x)) ∪ ∂I q−1) = 0 and v(π(g−1(y))) = 1. Deﬁne h : I q+1 −→ I q by h(r, s, t) = (r, s − stv(r)) for r ∈ I q−1 and s, t ∈ I. Then let f ′ = g ◦ h1, where h1(r, s) = h(r, s, 1). We claim that f ′ is as desired. Observe that h(r, s, 0) = (r, s), h(r, 0, t) = (r, 0), and h(r, s, t) = (r, s) if r ∈ ∂I q−1. 3. PROOF OF THE HOMOTOPY EXCISION THEOREM 89 Moreover, since r ∈ π(g−1(x)) implies v(r) = 0 and h(r, s, t) = (r, s) if h(r, s, t) ∈ g−1(x) h(r, s, t) = (r, s − st) if h(r, s, t) ∈ g−1(y) since r ∈ π(g−1(y)) implies v(r) = 1. Then g ◦ h is a homotopy of maps of tetr
ads (I q; I q−2 × {1} × I, I q−1 × {1}, J q−2 × I ∪ I q−1 × {0}) from g to f ′, and f ′ has image in (X − {y}; A, X − {x, y}), as required. (X; A, X − {x}, ∗) CHAPTER 12 A little homological algebra Let R be a commutative ring. The main example will be R = Z. We develop some rudimentary homological algebra in the category of R-modules. We shall say more later. For now, we give the minimum that will be needed to develop cellular and singular homology theory. 1. Chain complexes A chain complex over R is a sequence of maps of R-modules · · · −→ Xi+1 di+1−−−→ Xi di−→ Xi−1 −→ · · · such that di ◦ di+1 = 0 for all i. We generally abbreviate d = di. A cochain complex over R is an analogous sequence · · · −→ Y i−1 di−1 −−−→ Y i di −→ Y i+1 −→ · · · with di ◦ di−1 = 0. In practice, we usually require chain complexes to satisfy Xi = 0 for i < 0 and cochain complexes to satisfy Y i = 0 for i < 0. Without these restrictions, the notions are equivalent since a chain complex {Xi, di} can be rewritten as a cochain complex, and vice versa. X −i, d−i An element of the kernel of di is called a cycle and an element of the image of di+1 is called a boundary. We say that two cycles are “homologous” if their diﬀerence is a boundary. We write Bi(X) ⊂ Zi(X) ⊂ Xi for the submodules of boundaries and cycles, respectively, and we deﬁne the ith homology group Hi(X) to be the quotient module Zi(X)/Bi(X). We write H∗(X) for the sequence of Rmodules Hi(X). We understand “graded R-modules” to be sequences of R-modules such as this (and we never take the sum of elements in diﬀerent gradings). 2. Maps and homotop
ies of maps of chain complexes A map f : X −→ X ′ of chain complexes is a sequence of maps of R-modules i ◦ fi = fi−1 ◦ di for all i. That is, the following diagram i such that d′ fi : Xi −→ X ′ commutes for each i: Xi di fi X ′ i ′ i d Xi−1 / X ′ i−1. fi−1 It follows that fi(Bi(X)) ⊂ Bi(X ′) and fi(Zi(X)) ⊂ Zi(X ′). Therefore f induces a map of R-modules f∗ = Hi(f ) : Hi(X) −→ Hi(X ′). 91 / / / 92 A LITTLE HOMOLOGICAL ALGEBRA A chain homotopy s : f ≃ g between chain maps f, g : X −→ X ′ is a sequence of homomorphisms si : Xi −→ X ′ i+1 such that d′ i+1 ◦ si + si−1 ◦ di = fi − gi for all i. Chain homotopy is an equivalence relation since if t : g ≃ h, then s + t = {si + ti} is a chain homotopy f ≃ h. Lemma. Chain homotopic maps induce the same homomorphism of homology groups. Proof. Let s : f ≃ g, f, g : X −→ X ′. If x ∈ Zi(X), then fi(x) − gi(x) = d′ i+1si(x), so that fi(x) and gi(x) are homologous. 3. Tensor products of chain complexes The tensor product (over R) of chain complexes X and Y is speciﬁed by letting (X ⊗ Y )n = Xi ⊗ Yj. i+j=n X When Xi and Yi are zero for i < 0, the sum is ﬁnite, but we don’t need to assume this. The diﬀerential is speciﬁed by d(x ⊗ y) = d(x) ⊗ y + (−1)ix ⊗ d(y) for x ∈ Xi and y ∈ Yj. The sign ensures that
d ◦ d = 0. We may write this as d = d ⊗ id + id ⊗ d. The sign is dictated by the general rule that whenever two entities to which degrees m and n can be assigned are permuted, the sign (−1)mn should be inserted. In the present instance, when calculating (id ⊗ d)(x ⊗ y), we must permute the map d of degree −1 with the element x of degree i. We regard R-modules M as chain complexes concentrated in degree zero, and thus with zero diﬀerential. For a chain complex X, there results a chain complex X ⊗ M ; H∗(X ⊗ M ) is called the homology of X with coeﬃcients in M. Deﬁne a chain complex I by letting I0 be the free Abelian group with two generators [0] and [1], letting I1 be the free Abelian group with one generator [I] such that d([I]) = [0] − [1], and letting Ii = 0 for all other i. Lemma. A chain homotopy s : f ≃ g between chain maps f, g : X −→ X ′ determines and is determined by a chain map h : X ⊗I −→ X ′ such that h(x, [0]) = f (x) and h(x, [1]) = g(x). Proof. Let s correspond to h via (−1)is(x) = h(x ⊗ [I]) for x ∈ Xi. The relation d′ i+1(si(x)) = fi(x) − gi(x) − si−1(di(x)) corresponds to the relation d′h = hd by the deﬁnition of our diﬀerential on I. The sign in the correspondence would disappear if we replaced by X ⊗ I by I ⊗ X. 4. SHORT AND LONG EXACT SEQUENCES 93 4. Short and long exact sequences −→ M A sequence M ′ f g −→ M ′′ of modules is exact if im f = ker g. If M ′ = 0, this means that g is a monomorphism; if M ′′ = 0, it means that f is an epimorphism. A longer sequence
is exact if it is exact at each position. A short exact sequence of chain complexes is a sequence 0 −→ X ′ f −→ X g −→ X ′′ −→ 0 that is exact in each degree. Here 0 denotes the chain complex that is the zero module in each degree. Proposition. A short exact sequence of chain complexes naturally gives rise to a long exact sequence of R-modules · · · −→ Hq(X ′) f∗ −→ Hq(X) g∗ −→ Hq(X ′′) ∂−→ Hq−1(X ′) −→ · · ·. Proof. Write [x] for the homology class of a cycle x. We deﬁne the “connecting homomorphism” ∂ : Hq(X ′′) −→ Hq−1(X ′) by ∂[x′′] = [x′], where f (x′) = d(x) for some x such that g(x) = x′′. There is such an x since g is an epimorphism, and there is such an x′ since gd(x) = dg(x) = 0. It is a standard exercise in “diagram chasing” to verify that ∂ is well deﬁned and the sequence is exact. Naturality means that a commutative diagram of short exact sequences of chain complexes gives rise to a commutative diagram of long exact sequences of R-modules. The essential point is the naturality of the connecting homomorphism, which is easily checked. PROBLEMS For a graded vector space V = {Vn} with Vn = 0 for all but ﬁnitely many n and with all Vn ﬁnite dimensional, deﬁne the Euler characteristic χ(V ) to be (−1)ndim Vn. P (1) Let V ′, V, and V ′′ be such graded vector spaces and suppose there is a long exact sequence · · · −→ V ′ n −→ Vn −→ V ′′ Prove that χ(V ) = χ(V ′) + χ(V ′′). n −→ V ′ n−1 −→ · · ·. (2) If {Vn
, dn} is a chain complex, show that χ(V ) = χ(H∗(V )). (3) Let 0 −→ π g −→ σ −→ 0 be an exact sequence of Abelian groups and let C be a chain complex of ﬂat (= torsion free) Abelian groups. Write H∗(C; π) = H∗(C ⊗ π). Construct a natural long exact sequence f −→ ρ · · · −→ Hq(C; π) f∗ −→ Hq(C; ρ) g∗ −→ Hq(C; σ) β −→ Hq−1(C; π) −→ · · ·. The connecting homomorphism β is called a Bockstein operation. CHAPTER 13 Axiomatic and cellular homology theory Homology groups are the basic computable invariants of spaces. Unlike homotopy groups, these are stable invariants, the same for a space and its suspension, and it is this that makes them computable. In this and the following two chapters, we ﬁrst give both an axiomatic and a cellular description of homology, next revert to an axiomatic development of the properties of homology, and then prove the Hurewicz theorem and use it to prove the uniqueness of homology. 1. Axioms for homology Fix an Abelian group π and consider pairs of spaces (X, A). We shall see that π determines a “homology theory on pairs (X, A).” We say that a map (X, A) −→ (Y, B) of pairs is a weak equivalence if its maps A −→ B and X −→ Y are weak equivalences. Theorem. For integers q, there exist functors Hq(X, A; π) from the homotopy category of pairs of spaces to the category of Abelian groups together with natural transformations ∂ : Hq(X, A; π) −→ Hq−1(A; π), where Hq(X; π) is deﬁned to be Hq(X, ∅; π). These functors and natural transformations satisfy and are characterized by the following axioms. • DIMENSION If X is a point, then H0
(X; π) = π and Hq(X; π) = 0 for all other integers. • EXACTNESS The following sequence is exact, where the unlabeled arrows are induced by the inclusions A −→ X and (X, ∅) −→ (X, A): · · · −→ Hq(A; π) −→ Hq(X; π) −→ Hq(X, A; π) ∂−→ Hq−1(A; π) −→ · · ·. • EXCISION If (X; A, B) is an excisive triad, so that X is the union of the interiors of A and B, then the inclusion (A, A ∩ B) −→ (X, B) induces an isomorphism H∗(A, A ∩ B; π) −→ H∗(X, B; π). • ADDITIVITY If (X, A) is the disjoint union of a set of pairs (Xi, Ai), then the inclusions (Xi, Ai) −→ (X, A) induce an isomorphism iH∗(Xi, Ai; π) −→ H∗(X, A; π). • WEAK EQUIVALENCE If f : (X, A) −→ (Y, B) is a weak equivalence, P then f∗ : H∗(X, A; π) −→ H∗(Y, B; π) is an isomorphism. 95 96 AXIOMATIC AND CELLULAR HOMOLOGY THEORY Here, by a standard abuse, we write f∗ instead of H∗(f ) or Hq(f ). Our approximation theorems for spaces, pairs, maps, homotopies, and excisive triads imply that such a theory determines and is determined by an appropriate theory deﬁned on CW pairs, as spelled out in the following CW version of the theorem. Theorem. For integers q, there exist functors Hq(X, A; π) from the homotopy category of pairs of CW complexes to the category of Abelian groups together with natural transformations ∂ : Hq(X, A) −→ Hq−1(A; π), where Hq(X; �
�) is deﬁned to be Hq(X, ∅; π). These functors and natural transformations satisfy and are characterized by the following axioms. • DIMENSION If X is a point, then H0(X; π) = π and Hq(X; π) = 0 for all other integers. • EXACTNESS The following sequence is exact, where the unlabeled arrows are induced by the inclusions A −→ X and (X, ∅) −→ (X, A): · · · −→ Hq(A; π) −→ Hq(X; π) −→ Hq(X, A; π) ∂−→ Hq−1(A; π) −→ · · ·. • EXCISION If X is the union of subcomplexes A and B, then the inclusion (A, A ∩ B) −→ (X, B) induces an isomorphism H∗(A, A ∩ B; π) −→ H∗(X, B; π). • ADDITIVITY If (X, A) is the disjoint union of a set of pairs (Xi, Ai), then the inclusions (Xi, Ai) −→ (X, A) induce an isomorphism iH∗(Xi, Ai; π) −→ H∗(X, A; π). Such a theory determines and is determined by a theory as in the previous theorem. P Proof. We prove the last statement and return to the rest later. Since a CW triad (which, we recall, was required to be the union of its given subcomplexes) is homotopy equivalent to an excisive triad, it is immediate that the restriction to CW pairs of a theory on pairs of spaces gives a theory on pairs of CW complexes. Conversely, given a theory on CW pairs, we may deﬁne a theory on pairs of spaces by turning the weak equivalence axiom into a deﬁnition. That is, we ﬁx a CW approximation functor Γ from the homotopy category of pairs of spaces to the homotopy category of CW pairs and we deﬁne H∗(X, A; π) = H∗(ΓX, ΓA; π).
Similarly, we deﬁne ∂ for (X, A) to be ∂ for (ΓX, ΓA). For a map f : (X, A) −→ (Y, B) of pairs, we deﬁne f∗ = (Γf )∗. It is clear from our earlier results that this does give a well deﬁned homology theory on pairs of spaces. Clearly, up to canonical isomorphism, this construction of a homology theory on pairs of spaces is independent of the choice of our CW approximation functor Γ. The reader may have seen singular homology before. As we shall explain later, the classical construction of singular homology amounts to a choice of a particularly nice CW approximation functor, one that is actually functorial on the point-set level, before passage to homotopy categories. 2. CELLULAR HOMOLOGY 97 2. Cellular homology We must still construct H∗(X, A; π) on CW pairs. We shall give a seemingly ad hoc construction, but we shall later see that precisely this construction is in fact forced upon us by the axioms. We concentrate on the case π = Z, and we abbreviate notation by setting H∗(X, A) = H∗(X, A; Z). Let X be a CW complex. We shall deﬁne the cellular chain complex C∗(X). We let Cn(X) be the free Abelian group with one generator [j] for each n-cell j. We must deﬁne a diﬀerential dn : Cn(X) −→ Cn−1(X). We shall ﬁrst give a direct deﬁnition in terms of the cell structure and then give a more conceptual description in terms of coﬁber sequences. It will be convenient to work with unreduced cones, coﬁbers, and suspensions in this section; that is, we do not choose basepoints and so we do not collapse out lines through basepoints. (We shall discuss this difference more formally in the next chapter.) We still have the basic result that if i : A −→ X is a coﬁbration, then collapsing the cone on A to a point gives a homotopy equivalence ψ : Ci −→
X/A. We shall use the notation ψ−1 for any chosen homotopy inverse to such a homotopy equivalence. We again obtain π : Ci −→ ΣA by collapsing the base X of the coﬁber to a point. Our ﬁrst deﬁnition of dn involves the calculation of the degrees of maps between spheres. A map f : Sn −→ Sn induces a homomorphism f∗ : πn(Sn) −→ πn(Sn), which is given by multiplication by an integer called the degree of f. As in our discussion earlier for π1, f∗ is deﬁned using a change of basepoint isomorphism, but deg (f ) is independent of the choice of the path connecting ∗ to f (∗). Of course, this only makes sense for n ≥ 1. To deﬁne and calculate degrees, the domain and target of f must both be Sn. However, there are three models of Sn that are needed in our discussion: the standard sphere Sn ⊂ Dn+1, the quotient Dn/Sn−1, and the (unreduced) suspension ΣSn−1. We must ﬁx suitably compatible homeomorphisms relating these “n-spheres.” We deﬁne a homeomorphism νn : Dn/Sn−1 −→ Sn by νn(tx1,..., txn) = (ux1,..., uxn, 2t − 1) for 0 ≤ t ≤ 1 and (x1,..., xn) ∈ Sn−1, where u = (1 − (2t − 1)2)1/2. Thus νn sends the ray from 0 to (x1,..., xn) to the longitude that runs from the south pole (0,..., 0, −1) through the equatorial point (x1,..., xn, 0) to the north pole (0,..., 0, 1). We deﬁne a homeomorphism by ιn : Sn −→ ΣSn−1 ιn(x1,..., xn+1) = (vx1,..., v
xn) ∧ (xn+1 + 1)/2, n i=1 x2 i )1/2. This makes sense since if xi = 0 for 1 ≤ i ≤ n, then where v = 1/( xn+1 = ±1, so that (xn+1 + 1)/2 = 0 or 1 and ιn(x1,..., xn+1) is a cone point. In eﬀect, ιn makes the last coordinate the suspension coordinate. We deﬁne a homeomorphism of pairs P by ξn : (Dn, Sn−1) −→ (CSn−1, Sn−1) ξn(tx1,..., txn) = (x1,..., xn) ∧ (1 − t), 98 AXIOMATIC AND CELLULAR HOMOLOGY THEORY and we continue to write ξn for the induced homeomorphism Dn/Sn−1 ∼= CSn−1/Sn−1 = ΣSn−1. Observe that ιn ◦ νn = −ξn : Dn/Sn−1 −→ ΣSn−1, where the minus is interpreted as the sign map y ∧ t −→ y ∧ (1 − t) on ΣSn−1. We saw in our treatment of coﬁber sequences that, up to homotopy, the maps CSn−1 ∪Sn−1 CSn−1 −→ ΣSn−1 obtained by collapsing out the ﬁrst and second cone also diﬀer by this sign map. By an easy diagram chase, these observations imply the following compatibility statement. It will be used to show that the two deﬁnitions of dn that we shall give are in fact the same. Lemma. The following diagram is homotopy commutative: Dn ∪Sn−1 CSn−1 π / ΣSn−1 ψ Dn/Sn−1 νn ιn / Sn. Returning to our CW complex X, we think of an n-cell j as a map of pairs j : (Dn, Sn−1) −→ (X n, X n−1). There results a homeomorphism α : j Dn/Sn−1 −
→ X n/X n−1 whose restriction to the jth wedge summand is induced by j. Deﬁne W πj : X n/X n−1 −→ Sn to be the composite of α−1 with the map given by νn on the jth wedge summand and the constant map at the basepoint on all other wedge summands. If n = 0, we interpret D0 to be a point and interpret S−1 and X −1 to be empty. With our convention that X/∅ = X+, we see that X 0/X −1 can be identiﬁed with the wedge of one copy of S0 for each vertex j, and πj is still deﬁned. Here we take S0 = {±1}, with basepoint 1. For an n-cell j and an (n − 1)-cell i, where n ≥ 1, we have a composite Sn−1 j −→ X n−1 ρ −→ X n−1/X n−2 πi−→ Sn−1. When n = 1, we interpret ρ to be the inclusion X 0 −→ X 0 be the degree of this composite and deﬁne +. When n ≥ 2, let ai,j dn[j] = iai,j[i]. When n = 1, specify coeﬃcents ai,j implicitly by deﬁning P d1[j] = [j(1)] − [j(−1)]. We claim that dn−1 ◦ dn = 0, and we deﬁne H∗(X) = H∗(C∗(X)). / / O O 2. CELLULAR HOMOLOGY 99 To see that dn−1 ◦ dn = 0, we use the theory of coﬁber sequences to obtain a more conceptual description of dn. We deﬁne the “topological boundary map” ∂n : X n/X n−1 −→ Σ(X n−1/X n−2) to be the composite X n/X n−1 ψ −−−→ Ci π−→ ΣX n−1 Σρ −−→ Σ(X n−1/X n−2), −1 where
i : X n−1 −→ X n is the inclusion. We claim that ∂n induces dn upon application of a suitable functor, and we need some preliminaries to show this. For certain based spaces X, we adopt the following provisional deﬁnition of the “reduced nth homology group” of X. Definition. Let X be a based (n − 1)-connected space. Deﬁne ˜H ′ n(X) as follows. n = 0: The free Abelian group generated by the set π0(X)− {∗} of non-basepoint components of X. n = 1: The Abelianization π1(X)/[π1(X), π1(X)] of the fundamental group of X. n ≥ 2: The nth homotopy group of X. Up to canonical isomorphism, ˜H ′ n(X) is independent of the choice of basepoint in its path component. In fact, we can deﬁne ˜H ′ n(X) in terms of unbased homotopy classes of maps Sn −→ X. We also need a suspension homomorphism, and we adopt another provisional deﬁnition. Definition. Let X be a based (n − 1)-connected space. Deﬁne n(X) −→ ˜H ′ by letting Σ[f ] = [Σf ◦ ιn+1] for f : Sn −→ X; that is, Σ[f ] is represented by Sn+1 ιn+1−−−→ ΣSn Σf n+1(ΣX) −−→ ΣX. Σ : ˜H ′ This only makes sense for n ≥ 1. For n = 0, and a point x ∈ X that is not in the component of the basepoint, we deﬁne Σ[x] = [f −1 · fx], where ∗ ∈ X is the basepoint and fx is the path t −→ x ∧ t from one cone point to the other in the unreduced suspension ΣX. ∗ Lemma. If X is a wedge of n-spheres, then Σ : ˜H ′ n(X) −→
˜H ′ n+1(ΣX) is an isomorphism. Proof. We claim ﬁrst that ˜H ′ n(X) is the free Abelian group with generators given by the inclusions of the wedge summands. Since maps and homotopies of maps Sn −→ X have images in compact subspaces, it suﬃces to check this on ﬁnite wedges, and, when n ≥ 2, we can give the pair (×iSn, ∨iSn) the structure of a CW pair with no relative q-cells for q < 2n − 1. The claim follows by cellular approximation of maps and homotopies. Now the conclusion of the lemma follows from the case of a single sphere in view of the canonical direct sum decompositions. Returning to our CW complex X, we take the homotopy classes [j ◦ ν−1 n ] of the composites −1 Sn ν n−−→ Dn/Sn−1 j −→ X n/X n−1 100 AXIOMATIC AND CELLULAR HOMOLOGY THEORY as canonical basis elements of ˜H ′ n(X n/X n−1). Lemma. The diﬀerential dn : Cn(X) −→ Cn−1(X) can be identiﬁed with the composite −1 (∂n)∗ −−−→ ˜H ′ n(X n/X n−1) n(Σ(X n−1/X n−2)) Σ n : ˜H ′ d′ Proof. The identiﬁcation of the groups is clear: we let the basis element [j] of Cn(X) correspond to the basis element [j ◦ ν−1 n(X n/X n−1). For an ncell j and an (n − 1)-cell i, the following diagram is homotopy commutative by the naturality of ψ and π, the deﬁnition of ai,j, and our lemma relating diﬀerent models of the n-sphere: n−1(X n−1/X n−2). n ] of ˜H ′ −−−→ ˜H ′ ai,j
Sn −1 n ν Dn/Sn−1 +WWWWWWWWWWWWWWWWWWWWWWWWWWW −1 ιn ψ Dn ∪ CSn−1 π / ΣSn−1 j j∪Cj Σj ai,j Sn ιn ΣSn−1 Σπi X n/X n−1 −1 ψ / X n ∪ CX n−1 π / / ΣX n−1 Σρ / Σ(X n−1/X n−2). An inspection of circles shows that this diagram homotopy commutes even when n = 1. The bottom composite is the topological boundary map. Write d′ n in matrix form, n[j ◦ ν−1 d′ Then, since the composite n ] = Σ−1(∂n)∗[j ◦ ν−1 n ] = ia′ i,j[i ◦ ν−1 n−1]. P πi ◦ (i ◦ ν−1 n−1) : Sn−1 −→ Sn−1 is the identity map for each i, a′ i,j is the degree of the map Sn −→ Sn that we obtain by traversing the diagram counterclockwise from the top left to the top right. The diagram implies that a′ i,j = ai,j. Lemma. dn−1 ◦ dn = 0. Proof. The composite Σ∂n−1 ◦ ∂n is homotopic to the trivial map since the following diagram is homotopy commutative and the composite Σπ ◦Σi is the trivial map: X n ∪ CX n−1 π / ΣX n−1 Σi / Σ(X n−1 ∪ CX n−2) Σπ / Σ2X n−2 ψ Σρ Σψ Σ2ρ X n/X n−1 ∂n / Σ(X n−1/X n−2) Σ(X n−1/X n−2) / Σ2X n−2/X n−3. Σ∂n−1 The conclusion follows from our identiﬁcation of dn and the naturality of Σ
. 3. Veriﬁcation of the axioms For a CW complex X with base vertex ∗, deﬁne ˜C∗(X) = C∗(X)/C∗(∗) and deﬁne ˜H∗(X) = H∗( ˜C∗(X)). This is the reduced homology of X. For a subcomplex A of X, deﬁne C∗(X, A) = C∗(X)/C∗(A) ∼= ˜C∗(X/A. THE CELLULAR CHAINS OF PRODUCTS 101 and deﬁne H∗(X, A) = H∗(C∗(X, A)) ∼= ˜H∗(X/A). The long exact homology sequence of the exact sequence of chain complexes 0 −→ C∗(A) −→ C∗(X) −→ C∗(X, A) −→ 0 gives the connecting homorphisms ∂ : Hq(X, A) −→ Hq−1(A) and the long exact sequence called for in the exactness axiom. If f : X −→ Y is a cellular map, it induces maps X n/X n−1 −→ Y n/Y n−1 that commute up to homotopy with the topological boundary maps and so induce homomorphisms fn : Cn(X) −→ Cn(Y ) that commute with the diﬀerentials. That is, f∗ is a chain map, and it induces a homomorphism H∗(X) −→ H∗(Y ). For any CW complex X, X × I is a CW complex whose cellular chains are isomorphic to C∗(X) ⊗ C∗(I), as we shall verify in the next section. Here C∗(I) ∼= I has basis elements [0] and [1] of degree zero and [I] of degree one such that d([I]) = [0] − [1]. We have observed that, for chain complexes C and D, a chain map C ⊗ I −→ D can be identiﬁed with a chain homotopy between its restrictions to C ⊗ Z[0] and
C ⊗ Z[1]. A cellular homotopy h : X × I −→ Y induces just such a chain map, hence cellularly homotopic maps induce the same homomorphism on homology. The analogous conclusion for pairs follows by consideration of quotient complexes. The dimension and additivity axioms are obvious. If X is a point, then C∗(X) = Z, concentrated in degree zero. The cellular chain complex of ∐Xi is the direct sum of the chain complexes C∗(Xi), and similarly for pairs. Excision is also obvious. If X = A ∪ B, then the inclusion A/A ∩ B −→ X/B is an isomorphism of CW complexes. We have dealt so far with the case of integral homology. For more general coeﬃcient groups π, we deﬁne C∗(X, A; π) = C∗(X, A) ⊗ π and proceed in exactly the same fashion to deﬁne homology groups and verify the axioms. Observe that a homomorphism of groups π −→ ρ induces a natural transformation that commutes with the connecting homomorphisms. Hq(X, A; π) −→ Hq(X, A; ρ) 4. The cellular chains of products A nice fact about cellular homology is that the deﬁnition leads directly to an algebraic procedure for the calculation of the homology of Cartesian products. We explain the topological point here and return to the algebra later, where we discuss the K¨unneth theorem for the computation of the homology of tensor products of chain complexes. Theorem. If X and Y are CW complexes, then X × Y is a CW complex such that C∗(X × Y ) ∼= C∗(X) ⊗ C∗(Y ). Proof. We have already seen that X × Y is a CW complex. Its n-skeleton is (X × Y )n = X p × Y q, p+q=n [ 102 AXIOMATIC AND CELLULAR HOMOLOGY THEORY and it has one n-cell, denoted i × j, for each p-cell i and q-cell j. We deﬁne an isomorphism of graded Abelian groups κ : C
∗(X) ⊗ C∗(Y ) −→ C∗(X × Y ) by setting κ([i] ⊗ [j]) = (−1)pq[i × j]. It is clear from the deﬁnition of the product cell structure that κ commutes up to sign with the diﬀerentials, and the insertion of the coeﬃcient (−1)pq ensures that the signs work out. As we shall see, the coeﬃcient appears because we write suspension coordinates on the right rather than on the left. To be precise about this veriﬁcation, we ﬁx homeomorphisms (Dn, Sn−1) ∼= (I n, ∂I n) by radial contraction to the unit cube centered at 0 followed by translation to the unit cube centered at 1/2. This ﬁxes homeomorphisms ιp,q : (Dn, Sn−1) ∼= (I n, ∂I n) = (I p × I q, I p × ∂I q ∪ ∂I p × I q) ∼= (Dp × Dq, Dp × Sq−1 ∪ Sp−1 × Dq) and thus ﬁxes the product cells i × j. For each p and q, the following diagrams are homotopy commutative, where n = p + q and where t : (ΣSp−1) ∧ Sq = Sp−1 ∧ S1 ∧ Sq −→ Sp−1 ∧ Sq ∧ S1 = Σ(Sp−1 ∧ Sq) is the transposition map. Note that, by a quick check when q = 1 and induction, t has degree (−1)q. Sn −1 n ν Dn/Sn−1 ιp,q ιn ΣSn−1 Σν −1 n−1 Σ(Dn−1/Sn−2) Σιp,q−1 (Dp/Sp−1) ∧ (Dq/Sq−1) Σ((Dp/Sp−1) ∧ (Dq−1/Sq−2)) νp∧νq Sp ∧ Sq / Sp ∧ (ΣSq−1)
id ∧ιq Σ(νp∧νq−1) Σ(Sp ∧ Sq−1) / / / 5. SOME EXAMPLES: T, K, AND RP n 103 and Sn −1 n ν Dn/Sn−1 ιp,q ιn ΣSn−1 Σν −1 n−1 Σ(Dn−1/Sn−2) Σιp−1,q (Dp/Sp−1) ∧ (Dq/Sq−1) Σ((Dp−1/Sp−2) ∧ (Dq/Sq−1)) νp∧νq Σ(νp−1∧νq) Sp ∧ Sq / (ΣSp−1) ∧ Sq ιp∧id (−1)qt / Σ(Sp−1 ∧ Sq). The homotopy commutativity would be clear if we worked only with cubes and replaced the maps ιn and ιp,q with the evident identiﬁcations. The only point at issue then would be which copy of I/∂I is to be interpreted as the suspension coordinate. The homotopy commutativity of the diagrams as written follows directly. Now comparison of our description of the cellular diﬀerential in terms of the topological boundary map and the algebraic description of the diﬀerential on tensor products shows that κ is an isomorphism of chain complexes. 5. Some examples: T, K, and RP n Cellular chains make some computations quite trivial. For example, since Sn is a CW complex with one vertex and one n-cell, we see immediately that ˜Hn(Sn; π) ∼= π and ˜Hq(Sn; π) = 0 for q 6= n. A little less obviously, if we look back at the CW decompositions of the torus T, the projective plane RP 2, and the Klein bottle K and if we let j denote the unique 2-cell in each case, then we ﬁnd the following descriptions of the cellular chains and integral homologies by quick direct inspections. We agree to write Zn for the cyclic group Z/nZ. Examples. (i) The cell complex C∗(T ) has one basis element
[v] in degree zero, two basis elements [e1] and [e2] in degree one, and one basis element [j] in degree two. All basis elements are cycles, hence H∗(T ; Z) = C∗(T ). (ii) The cell complex C∗(RP 2) has two basis elements [v1] and [v2] in degree zero, two basis elements [e1] and [e2] in degree one, and one basis element [j] in degree two. The diﬀerentials are given by d([e1]) = [v1] − [v2], d([e2]) = [v2] − [v1], and d([j]) = 2[e1] + 2[e2]. Therefore H0(RP 2; Z) = Z with basis element the homology class of [v1] (or [v2]), H1(RP 2; Z) = Z2 with non-zero element the homology class of e1 + e2, and Hq(RP 2; Z) = 0 for q ≥ 2. (iii) The cell complex C∗(K) has one basis element [v] in degree zero, two basis elements [e1] and [e2] in degree one, and one basis element [j] in degree two. The only non-zero diﬀerential is d([j]) = 2[e2]. Therefore H0(K; Z) = Z with basis element the homology class of [v], H1(K; Z) = Z ⊕ Z2 with Z generated by the class of [e1] and Z2 generated by the class of [e2], and Hq(K; Z) = 0 for q ≥ 2. / / / / 104 AXIOMATIC AND CELLULAR HOMOLOGY THEORY However, these examples are misleading. While homology groups are far easier to compute than homotopy groups, direct chain level calculation is seldom the method of choice. Rather, one uses chains as a tool for developing more sophisticated algebraic techniques, notably spectral sequences. We give an illustration that both shows that chain level calculations are sometimes practicable even when there are many non-zero diﬀerentials to determine and indicates why one might not wish to attempt such calculations for really complicated spaces. We shall use cellular chains to compute
the homology of RP n. We think of RP n as the quotient of Sn obtained by identifying antipodal points, and we need to know the degree of the antipodal map. Lemma. The degree of the antipodal map an : Sn −→ Sn is (−1)n+1. Proof. Since a1 ≃ id via an obvious rotation, the result is clear for n = 1. The homeomorphism ιn : Sn −→ ΣSn−1 satisﬁes ιn(−x1,..., −xn+1) = (−vx1,..., −vxn) ∧ (1 − xn+1)/2, i )1/2. That is, ιn ◦ an = −(Σan−1) ◦ ιn. The conclusion follows x2 where v = 1/( by induction on n. We shall give RP n a CW structure with one q-cell for 0 ≤ q ≤ n by passage to quotients from a CW structure on Sn with two q-cells for 0 ≤ q ≤ n. (Note that this cell structure on RP 2 will be more economical than the one used in the calculation above.) The q-skeleton of Sn will be Sq, which we identify with the subspace of Sn whose points have last n − q coordinates zero. We denote the two q-cells of Sn by jq ±. The two vertices are the points ±1 of S0. Let Eq ± be the upper and lower hemispheres in Sq, so that Sq = Eq + ∪ Eq − and Sq−1 = Eq + ∩ Eq −. We shall write π± : Sq −→ Eq ±/Sq−1 for the quotient maps that identify the lower or upper hemispheres to the basepoint. Of course, these are homotopy equivalences. We deﬁne homeomorphisms ± : Dq −→ Eq jq ± ⊂ Sq by jq ±(x1,..., xq) = (±x1,..., ±xq, ±(1 − i )1/2). x2 This decomposes Sq as the the union of the images of two q-cells. The intersection of these images is Sq−1 since
P (x1,..., xq, (1 − i )1/2) = (−y1,..., −yq, −(1 − x2 i )1/2) y2 P if and only if xi = −yi for each i and P jq + \| Sq−1 = id and P x2 i = 1. Clearly P jq − \| Sq−1 = aq−1. Inspection of deﬁnitions shows that the following diagram commutes: π+ Sq−1 Eq−1 + /Sq−2 jq−1 + Dq−1/Sq−2 aq−1 aq−1 Sq−1 π− / / Eq−1 − /Sq−2 Dq−1/Sq−2. jq−1 − / / o o o o 5. SOME EXAMPLES: T, K, AND RP n 105 Since a2 q−1 = id, it follows that we also obtain a commutative diagram if we interchange + and −. If we invert the homeomorphisms jq−1 ± and compose on the right with the homeomorphism iq−1 : Dq−1/Sq−2 −→ Sq−1, then the degrees of the four resulting composite homotopy equivalences give the coeﬃcients of the diﬀerential dq. By composing iq−1 with a homeomorphism of degree −1 if necessary, we can arrange that the degree of iq−1 ◦(jq−1 + )−1 ◦π+ is 1. We then deduce from the lemma and the deﬁnition of the diﬀerential on the cellular chains that +] = (−1)qdq[jq + ] + (−1)q[jq−1 − ] −] = [jq−1 dq[jq for all q ≥ 1. Now, identifying antipodal points, we obtain the promised CW decomposition of RP n. If p : Sn −→ RP n is the quotient map, then + = p ◦ jq − : (Dq, Sq−1) −→ (RP q, RP q−1). p ◦ jq We call this map jq and see that these maps give RP n
a CW structure. Therefore Cq(RP n) = Z with basis element [jq] for q ≥ 0. Moreover, it is immediate from the calculation just given that d[jq] = (1 + (−1)q)[jq−1] for all q ≥ 1. This is zero if q is odd and multiplication by 2 if q is even, and we read oﬀ that Z if q = 0 Z2 if 0 < q < n and q is odd Z if q = n is odd otherwise. 0 Hq(RP n; Z) =    If we work mod 2, taking Z2 as coeﬃcient group, then the answer takes a nicer form, namely Hq(RP n; Z2) = Z2 if 0 ≤ q ≤ n if q > n. 0 ( This calculation well illustrates general facts about the homology of compact connected closed n-manifolds M that we shall prove later. The nth integral homology group of such a manifold M is Z if M is orientable and zero if M is not orientable. The nth mod 2 homology group of M is Z2 whether or not M is orientable. PROBLEMS (1) If X is a ﬁnite CW complex, show that χ(X) = χ(H∗(X; k)) for any ﬁeld k. (2) Let A be a subcomplex of a CW complex X, let Y be a CW complex, and let f : A −→ Y be a cellular map. What is the relationship between H∗(X, A) and H∗(Y ∪f X, Y )? Is there a similar relationship between π∗(X, A) and π∗(Y ∪f X, Y )? If not, give a counterexample. (3) Fill in the details of the computation of the diﬀerentials on the cellular chains in the examples in §5. (4) Compute H∗(Sm × Sn) for m ≥ 1 and n ≥ 1. Convince yourself that you can do this by use of CW structures, by direct deduction from the axioms, and by the K¨unneth theorem (for which see Chapter
17). 106 AXIOMATIC AND CELLULAR HOMOLOGY THEORY (5) Let p be an odd prime number. Regard the cyclic group π of order p as the group of pth roots of unity contained in S1. Regard S2n−1 as the unit sphere in Cn, n ≥ 2. Then π ⊂ S1 acts freely on S2n−1 via ζ(z1,..., zn) = (ζz1,..., ζzn). Let Ln = S2n−1/π be the orbit space; it is called a lens space and is an odd primary analogue of RP n. The obvious quotient map S2n−1 −→ Ln is a universal covering. (a) Compute the integral homology of Ln, n ≥ 2, by mimicking the calculation of H∗(RP n). (b) Compute H∗(Ln; Zp), where Zp = Z/pZ. CHAPTER 14 Derivations of properties from the axioms Returning to the axiomatic approach to homology, we assume given a theory on pairs of spaces and make some deductions from the axioms. We abbreviate notations by setting Eq(X, A) = Hq(X, A; π). However, the arguments in this chapter make no use whatever of the dimension axiom. A “generalized homology theory” E∗ is deﬁned to be a system of functors Eq(X, A) and natural transformations ∂ : Eq(X, A) −→ Eq−1(A) that satisfy all of our axioms except for the dimension axiom. Similarly, we have the notion of a generalized homology theory on CW pairs, and the results of the ﬁrst section of the previous chapter generalize directly to give the following result. Theorem. A homology theory E∗ on pairs of spaces determines and is deter- mined by its restriction to a homology theory E∗ on pairs of CW complexes. The study of such generalized homology theories pervades modern algebraic topology, and we shall describe some examples later on. The brave reader may be willing to think of E∗ in such generality in this chapter. The timorous reader may well prefer to think of E�
�(X, A) concretely, following our proposal that E∗(X, A) be taken as an alternative notation for H∗(X, A; π). 1. Reduced homology; based versus unbased spaces One of the themes of this chapter is the relationship between homology theories on pairs of spaces and reduced homology theories on based spaces. The latter are more convenient in most advanced work in algebraic topology. For a based space X, we deﬁne the reduced homology of X to be ˜Eq(X) = Eq(X, ∗). Since the basepoint is a retract of X, there results a direct sum decomposition E∗(X) ∼= ˜E∗(X) ⊕ E∗(∗) that is natural with respect to based maps. For ∗ ∈ A ⊂ X, the summand E∗(∗) maps isomorphically under the map E∗(A) −→ E∗(X), and the exactness axiom implies that there is a reduced long exact sequence · · · −→ ˜Eq(A) −→ ˜Eq(X) −→ Eq(X, A) ∂−→ ˜Eq−1(A) −→ · · ·. We can obtain the unreduced homology groups as special cases of the reduced ones. For an unbased space X, we deﬁne a based space X+ by adjoining a disjoint basepoint to X. By the additivity axiom, we see immediately that E∗(X) = ˜E∗(X+). 107 108 DERIVATIONS OF PROPERTIES FROM THE AXIOMS Similarly, a map f : X −→ Y of unbased spaces induces a map f+ : X+ −→ Y+ of based spaces, and the map f∗ on unreduced homology coincides with the map (f+)∗ on reduced homology. We shall make considerable use of coﬁber sequences in this chapter. To be consistent about this, we should always work with reduced cones and coﬁbers. However, it is more convenient to make the convention that we work with unreduced cones and coﬁbers when we apply unreduced homology theories, and we work with reduced cones
and coﬁbers when we apply reduced homology theories. In fact, the unreduced cone on a space Y coincides with the reduced cone on Y+: the line through the disjoint basepoint is identiﬁed to the cone point when constructing the reduced cone on Y+. Therefore the unreduced coﬁber of an unbased map f coincides with the reduced coﬁber of the based map f+. Our convention really means that we are always working with reduced coﬁbers, but when we are studying unreduced homology theories we are implicitly applying the functor (−)+ to put ourselves in the based context before constructing cones and coﬁbers. The observant reader will have noticed that the unreduced suspension of X is not the reduced suspension on X+. Rather, under either interpretation of suspension, Σ(X+) is homotopy equivalent to the wedge of Σ(X) and a circle. 2. Coﬁbrations and the homology of pairs We use coﬁbrations to show that the homology of pairs of spaces is in principle a special case of the reduced homology of spaces. Theorem. For any coﬁbration i : A −→ X, the quotient map q : (X, A) −→ (X/A, ∗) induces an isomorphism E∗(X, A) −→ E∗(X/A, ∗) = ˜E∗(X/A). Proof. Consider the (unreduced) coﬁber Ci = X ∪i CA = X ∪i A × I/A × {1}. We have an excisive triad (Ci; X ∪i A × [0, 2/3], A × [1/3, 1]/A × {1}). The excision axiom gives that the top inclusion in the following commutative diagram induces an isomorphism on passage to homology: (X ∪i A × [0, 2/3], A × [1/3, 2/3]) (Ci, A × [1/3, 1]/A × {1}) r (X, A) ψ / (X/A, ∗) q The map r is obtained by restriction of the retraction M i −→ X and is a homot
opy equivalence of pairs. The map ψ collapses CA to a point and is also a homotopy equivalence of pairs. The conclusion follows. As in our construction of cellular homology, we choose a homotopy inverse ψ−1 : X/A −→ Ci and consider the composite X/A −1 ψ −−−→ Ci π−→ ΣA / / / 3. SUSPENSION AND THE LONG EXACT SEQUENCE OF PAIRS 109 to be a topological boundary map ∂ : X/A −→ ΣA. Observe that we may replace any inclusion i : A −→ X by the canonical coﬁbration A −→ M i and then apply the result just given to obtain E∗(X, A) ∼= ˜E∗(Ci). 3. Suspension and the long exact sequence of pairs We have a fundamentally important consequence of the results of the previous section, which should be contrasted with what happened with homotopy groups. Recall that a basepoint ∗ ∈ X is nondegenerate if the inclusion {∗} −→ X is a coﬁbration. This ensures that the inclusion of the line through the basepoint in the unreduced suspension of X is a coﬁbration, so that the map from the unreduced suspension to the suspension that collapses out the line through the basepoint is a homotopy equivalence. We apply reduced homology here, so we use reduced cones and suspensions. Theorem. For a nondegenerately based space X, there is a natural isomor- phism Σ : ˜Eq(X) ∼= ˜Eq+1(ΣX). Proof. Since CX is contractible, its reduced homology is identically zero. By the reduced long exact sequence, there results an isomorphism ˜Eq+1(ΣX) ∼= ˜Eq+1(CX/X) ∂−→ ˜Eq(X). An easy diagram chase gives the following consequence, which describes the axiomatically given connecting homomorphism of the pair (X, A) in terms of the topological boundary map ∂ : X/A −→ ΣA and the suspension isomorphism. Corollary. Let ∗ ∈ A ⊂ X, where
i : A −→ X is a coﬁbration between nondegenerately based spaces. In the long exact sequence · · · −→ ˜Eq(A) −→ ˜Eq(X) −→ ˜Eq(X/A) ∂−→ ˜Eq−1(A) −→ · · · of the pair (X, A), the connecting homomorphism ∂ is the composite ˜Eq(X/A) ∂∗−→ ˜Eq(ΣA) Σ −−−→ ˜Eq−1(A). −1 Since S0 consists of two points, ˜E∗(S0) = E∗(∗). Since Sn is the suspension of Sn−1, we have the following special case of the suspension isomorphism. Corollary. For any n and q, ˜Eq(Sn) ∼= Eq−n(∗). Of course, for the theory H∗(X; π), this was immediate from our construction in terms of cellular chains. 110 DERIVATIONS OF PROPERTIES FROM THE AXIOMS 4. Axioms for reduced homology In the study of generalized homology theories, it is most convenient to restrict attention to reduced homology theories deﬁned on nondegenerately based spaces. The results of the previous sections imply that we can do so without loss of generality. Again the reader has the choice of bravery or timorousness in interpreting E∗, but we opt for bravery: Definition. A reduced homology theory ˜E∗ consists of functors ˜Eq from the homotopy category of nondegenerately based spaces to the category of Abelian groups that satisfy the following axioms. • EXACTNESS If i : A −→ X is a coﬁbration, then the sequence ˜Eq(A) −→ ˜Eq(X) −→ ˜Eq(X/A) is exact. • SUSPENSION For each integer q, there is a natural isomorphism Σ : ˜Eq(X) ∼= ˜Eq+1(ΣX). • ADDITIVITY If X is the wedge of a set of nondegenerately based spaces Xi, then the inclusions Xi −
→ X induce an isomorphism ˜E∗(Xi) −→ ˜E∗(X). • WEAK EQUIVALENCE If f : X −→ Y is a weak equivalence, then P f∗ : ˜E∗(X) −→ ˜E∗(Y ) i is an isomorphism. The reduced form of the dimension axiom would read ˜H0(S0) = π and ˜Hq(S0) = 0 for q 6= 0. Theorem. A homology theory E∗ on pairs of spaces determines and is deter- mined by a reduced homology theory ˜E∗ on nondegenerately based spaces. Proof. Given a theory on pairs, we deﬁne ˜E∗(X) = E∗(X, ∗) and deduce the new axioms. For additivity, the speciﬁed wedge is the quotient (∐Xi)/(∐{∗i}), where ∗i is the basepoint of Xi, and our result on quotients of coﬁbrations applies to compute its homology. Conversely, assume given a reduced homology theory ˜E∗, and deﬁne E∗(X) = ˜E∗(X+) and E∗(X, A) = ˜E∗(C(i+)), where C(i+) is the coﬁber of the based inclusion i+ : A+ −→ X+. Equivalently, C(i+) is the unreduced coﬁber of i : A −→ X with its cone point as basepoint. We must show that the suspension axiom and our restricted exactness axiom imply the original, seemingly much stronger, exactness and excision axioms. We have the long exact coﬁber sequence associated to the based inclusion i+ : A+ −→ X+, in which each consecutive pair of maps is equivalent to a coﬁbration and the associated quotient map. Noting that X+/A+ = X/A, we deﬁne the connecting homomorphism ∂q : Eq(X, A) −→ Eq−1(A) to be the composite ∂∗−→ ˜Eq
(ΣA+) Σ and ﬁnd that the exactness and suspension axioms for ˜E∗ imply the exactness axiom for E∗. For excision, we could carry out a similarly direct homotopical argument, −−−→ ˜Eq−1(A+) ˜Eq(X+/A+) −1 4. AXIOMS FOR REDUCED HOMOLOGY 111 but it is simpler to observe that this follows from the equivalence of theories on pairs of spaces with theories on pairs of CW complexes together with the next two theorems. For the additivity axiom, we note that the coﬁber of a disjoint union of maps is the wedge of the coﬁbers of the given maps. Corollary. For nondegenerately based spaces X, E∗(X) is naturally isomor- phic to ˜E∗(X) ⊕ E∗(∗). Proof. The long exact sequence in E∗ of the pair (X, ∗) is naturally split in each degree by means of the homomorphism induced by the projection X −→ {∗}. We require of based CW complexes that the basepoint be a vertex. It is certainly a nondegenerate basepoint. We give the circle its standard CW structure and so deduce a CW structure on the suspension of a based CW complex. Definition. A reduced homology theory ˜E∗ on based CW complexes consists of functors ˜Eq from the homotopy category of based CW complexes to the category of Abelian groups that satisfy the following axioms. • EXACTNESS If A is a subcomplex of X, then the sequence ˜Eq(A) −→ ˜Eq(X) −→ ˜Eq(X/A) is exact. • SUSPENSION For each integer q, there is a natural isomorphism Σ : ˜Eq(X) ∼= ˜Eq+1(ΣX). • ADDITIVITY If X is the wedge of a set of based CW complexes Xi, then the inclusions Xi −→ X induce an isomorphism ˜E∗(Xi) −→ ˜E∗(X). i Theorem. A reduced homology theory ˜E∗ on nondeg
enerately based spaces determines and is determined by its restriction to a reduced homology theory on based CW complexes. P Proof. This is immediate by CW approximation of based spaces. Theorem. A homology theory E∗ on CW pairs determines and is determined by a reduced homology theory ˜E∗ on based CW complexes. Proof. Given a theory on pairs, we deﬁne ˜E∗(X) = E∗(X, ∗) and deduce the new axioms directly. Conversely, given a reduced theory on based CW complexes, we deﬁne E∗(X) = ˜E∗(X+) and E∗(X, A) = ˜E∗(X/A). Of course X/A is homotopy equivalent to C(i+), where i+ : A+ −→ X+ is the inclusion. The arguments for exactness and additivity are the same as those given in the analogous result for nondegenerately based spaces, but now excision is obvious since if (X; A, B) is a CW triad, then the inclusion A/A ∩ B −→ X/B is an isomorphism of based CW complexes. 112 DERIVATIONS OF PROPERTIES FROM THE AXIOMS 5. Mayer-Vietoris sequences The Mayer-Vietoris sequences are long exact sequences associated to excisive triads that will play a fundamental role in our later proof of the Poincar´e duality theorem. We need two preliminaries, both of independent interest. The ﬁrst is the long exact sequence of a triple (X, A, B) of spaces B ⊂ A ⊂ X, which is just like its analogue for homotopy groups. Proposition. For a triple (X, A, B), the following sequence is exact: j∗ −→ Eq(X, A) ∂−→ Eq−1(A, B) −→ · · ·. i∗−→ Eq(X, B) · · · −→ Eq(A, B) Here i : (A, B) −→ (X, B) and j : (X, B) −→ (X, A) are inclusions and ∂ is the composite Eq(X, A) ∂−→ Eq−1(A
) −→ Eq−1(A, B). Proof. There are two easy arguments. One can either use diagram chasing from the various long exact sequences of pairs or one can apply CW approximation to replace (X, A, B) by a triple of CW complexes. After the replacement, we have that X/A ∼= (X/B)/(A/B) as a CW complex, and the desired sequence is the reduced exact sequence of the pair (X/B, A/B). Lemma. Let (X; A, B) be an excisive triad and set C = A ∩ B. The map E∗(A, C) ⊕ E∗(B, C) −→ E∗(X, C) induced by the inclusions of (A, C) and (B, C) in (X, C) is an isomorphism. Proof. Again, there are two easy proofs. One can either pass to homology from the diagram excision (X, C) excision (B, C) (A, C) $IIIIIIIII zuuuuuuuuu zuuuuuuuuu $IIIIIIIII (X, A) (X, B) and use algebra or one can approximate (X; A, B) by a CW triad, for which X/C ∼= A/C ∨ B/C as a CW complex. Theorem (Mayer-Vietoris sequence). Let (X; A, B) be an excisive triad and set C = A ∩ B. The following sequence is exact: · · · −→ Eq(C) ψ −→ Eq(A) ⊕ Eq(B) φ −→ Eq(X) ∆−→ Eq−1(C) −→ · · ·. Here, if i : C −→ A, j : C −→ B, k : A −→ X, and ℓ : B −→ X are the inclusions, then ψ(c) = (i∗(c), j∗(c)), φ(a, b) = k∗(a) − ℓ∗(b), and ∆ is the composite Eq(X) −→ Eq(X, B) ∼= Eq(A, C) ∂−→ Eq−
1(C). $ z $ z 5. MAYER-VIETORIS SEQUENCES 113 Proof. Note that the deﬁnition of φ requires a sign in order to make φ◦ψ = 0. The proof of exactness is algebraic diagram chasing and is left as an exercise. The following diagram may help: Eq(B) i∗ Eq(A) Eq(C) Eq(X) xqqqqqqqqqq &MMMMMMMMMM xqqqqqqqqqq fMMMMMMMMMM 8qqqqqqqqqq &MMMMMMMMMM ∂ &MMMMMMMMMM xqqqqqqqqqq &MMMMMMMMMM 8qqqqqqqqqq fMMMMMMMMMM xqqqqqqqqqq ∂ Eq−1(C) Eq(X, A) j∗ Eq(X, B) −∆ ∼= Eq(X, C) ∼= ∆ Eq(B, C) ∂ Eq(A, C) Here the arrow labeled “−∆” is in fact −∆ by an algebraic argument from the direct sum decomposition of Eq(X, C). Alternatively, one can use CW approximation. For a CW triad, there is a short exact sequence 0 −→ C∗(C) −→ C∗(A) ⊕ C∗(B) −→ C∗(X) −→ 0 whose associated long exact sequence is the Mayer-Vietoris sequence. We shall also need a relative analogue, but the reader may wish to ignore this for now. It will become important when we study manifolds with boundary. Theorem (Relative Mayer-Vietoris sequence). Let (X; A, B) be an excisive triad and set C = A ∩ B. Assume that X is contained in some ambient space Y. The following sequence is exact: · · · −→ Eq(Y, C) ψ −→ Eq(Y, A) ⊕ Eq(Y, B) φ −→ Eq(Y, X) ∆−→ Eq−1(Y, C) −→ · · ·. Here, if i : (Y, C) −→ (Y, A), j : (Y, C) −→ (
Y, B), k : (Y, A) −→ (Y, X), and ℓ : (Y, B) −→ (Y, X) are the inclusions, then ψ(c) = (i∗(c), j∗(c)), φ(a, b) = k∗(a) − ℓ∗(b), and ∆ is the composite Eq(Y, X) ∂−→ Eq−1(X, B) ∼= Eq−1(A, C) −→ Eq−1(Y, C). 114 DERIVATIONS OF PROPERTIES FROM THE AXIOMS Proof. This too is left as an exercise, but it is formally the same exercise. The relevant diagram is the following one: Eq(Y, B) Eq(Y, A) ∂ wooooooooooo 'OOOOOOOOOOO wooooooooooo gOOOOOOOOOOO 7ooooooooooo 'OOOOOOOOOOO Eq(Y, C) Eq(Y, X) ∂ 'OOOOOOOOOOO wooooooooooo 'OOOOOOOOOOO 7ooooooooooo gOOOOOOOOOOO wooooooooooo Eq−1(Y, C) Eq−1(X, A) ∂ Eq−1(X, B) −∆ ∼= Eq−1(X, C) ∼= ∆ Eq−1(B, C) Eq−1(A, C) Alternatively, one can use CW approximation. For a CW triad (X; A, B), with X a subcomplex of a CW complex Y, there is a short exact sequence 0 −→ C∗(Y /C) −→ C∗(Y /A) ⊕ C∗(Y /B) −→ C∗(Y /X) −→ 0 whose associated long exact sequence is the relative Mayer-Vietoris sequence. A comparison of deﬁnitions gives a relationship between these sequences. Corollary. The absolute and relative Mayer-Vietoris sequences are related by the following commutative diagram: Eq(Y, C) ∂ Eq−1(C) ψ ψ Eq(Y, A) ⊕ Eq(Y, B) ∂+∂ / Eq−1(A) ⊕ Eq−1(B) φ
φ Eq(Y, X) ∆ / Eq−1(Y, C) ∂ ∂ / Eq−1(X) / Eq−2(C). ∆ 6. The homology of colimits In this section, we let X be the union of an expanding sequence of subspaces Xi, i ≥ 0. We have seen that the compactness of spheres Sn and cylinders Sn × I implies that, for any choice of basepoint in X0, the natural map colim π∗(Xi) −→ π∗(X) is an isomorphism. We shall use the additivity and weak equivalence axioms and the Mayer-Vietoris sequence to prove the analogue for homology. THE HOMOLOGY OF COLIMITS 115 Theorem. The natural map colim E∗(Xi) −→ E∗(X) is an isomorphism. We record an algebraic description of the colimit of a sequence for use in the proof. Lemma. Let fi : Ai −→ Ai+1 be a sequence of homomorphisms of Abelian groups. Then there is a short exact sequence iAi where α(ai) = ai − fi(ai) for ai ∈ Ai and the restriction of β to Ai is the canonical P map given by the deﬁnition of a colimit. 0 −→ iAi P β −→ colim Ai −→ 0, α−→ By the additivity axiom, we may as well assume that X and the Xi are path connected. The proof makes use of a useful general construction called the “telescope” of the Xi, denoted tel Xi. Let ji : Xi −→ Xi+1 be the given inclusions and consider the mapping cylinders Mi+1 = (Xi × [i, i + 1]) ∪ Xi+1 that are obtained by identifying (x, i + 1) with ji(x) for x ∈ Xi. Inductively, let Y0 = X0 × {0} and suppose that we have constructed Yi ⊃ Xi × {i}. Deﬁne Yi+1 to be the double mapping cylinder Yi ∪ Mi+1 obtained by identifying (x, i) ∈ Yi with (x, i) ∈ Mi+1 for x ∈ Xi. Deﬁ
ne tel Xi to be the union of the Yi, with the colimit topology. Thus with the evident identiﬁcations at the ends of the cylinders. i [ tel Xi = Xi × [i, i + 1], Using the retractions of the mapping cylinders, we obtain composite retractions ri : Yi −→ Xi such that the following diagrams commute Yi ri Xi ⊂ Yi+1 ri+1 / Xi+1 ji Since the ri are homotopy equivalences and since homotopy groups commute with colimits, it follows that we obtain a weak equivalence r : tel Xi −→ X on passage to colimits. By the weak equivalence axiom, r induces an isomorphism on homology. It therefore suﬃces to prove that the natural map colim E∗(Xi) ∼= colim E∗(Yi) −→ E∗(tel Xi) is an isomorphism. We deﬁne subspaces A and B of tel Xi by choosing ε < 1 and letting A = X0 × [0, 1] i≥1 X2i−1 × [2i − ε, 2i] ∪ X2i × [2i, 2i + 1] and ` ` B = i≥0 X2i × [2i + 1 − ε, 2i + 1] ∪ X2i+1 × [2i + 1, 2i + 2]. ` / / / 116 DERIVATIONS OF PROPERTIES FROM THE AXIOMS We let C = A ∩ B and ﬁnd that C = i≥0 Xi × [i + 1 − ε, i + 1]. This gives an excisive triad, and a quick inspection shows that we have canonical homotopy equivalences ` A ≃ i≥0X2i, B ≃ i≥0X2i+1, and C ≃ i≥0Xi. Moreover, under these equivalences the inclusion C −→ A has restrictions ` and j2i+1 : X2i+1 −→ X2i+2, id : X2i −→ X2i ` ` while the inclusion C −→ B has restrictions j2i : X2i −→ X2i+1 and id : X2i+1
−→ X2i+1. By the additivity axiom, E∗(A) = iE∗(X2i), E∗(B) = iE∗(X2i+1), and E∗(C) = iE∗(Xi). We construct the following commutative diagram, whose top row is the MayerVietoris sequence of the triad (tel Xi; A, B) and whose bottom row is a short exact sequence as displayed in our algebraic description of colimits: P P P · · · · · · / Eq(C) Eq(A) ⊕ Eq(B) Eq(tel Xi) ∼= i Eq(Xi) ′ α ∼= i Eq(Xi) ′ β P P(−1)i P Pi(−1)i ∼= Eq(X Eq(Xi) i Eq(Xi) By the deﬁnition of the maps in the Mayer-Vietoris sequence, α′(xi) = xi +(ji)∗(xi) i(xi) = (−1)i(ki)∗(xi) for xi ∈ Eq(Xi), where ki : Xi −→ X is the inclusion. and β′ The commutativity of the lower left square is just the relation i(−1)i)α′(xi) = (−1)i(xi − (ji)∗(xi)). / colim Eq(Xi) / 0. P ( The diagram implies the required isomorphism ξ. P Remark. There is a general theory of “homotopy colimits,” which are up to homotopy versions of colimits. The telescope is the homotopy colimit of a sequence. The double mapping cylinder that we used in approximating excisive triads by CW triads is the homotopy pushout of a diagram of the shape • ←− • −→ •. We implicitly used homotopy coequalizers in constructing CW approximations of spaces. (1) Complete the proof that the Mayer-Vietoris sequence is exact. PROBLEM / / / / / / / / / / / / / / / / / / / / / CHAPTER 15 The Hurewicz and uniqueness theorems We now return to
the context of “ordinary homology theories,” namely those that satisfy the dimension axiom. We prove a fundamental relationship, called the Hurewicz theorem, between homotopy groups and homology groups. We then use it to prove the uniqueness of ordinary homology with coeﬃcients in π. 1. The Hurewicz theorem Although the reader may prefer to think in terms of the cellular homology theory already constructed, the proof of the Hurewicz theorem depends only on the axioms. It is this fact that will allow us to use the result to prove the uniqueness of homology theories in the next section. We take π = Z and delete it from the notation. The dimension axiom implicitly ﬁxes a generator i0 of ˜H0(S0), and we choose generators in of ˜Hn(Sn) inductively by setting Σin = in+1. Definition. For based spaces X, deﬁne the Hurewicz homomorphism by h : πn(X) −→ ˜Hn(X) h([f ]) = f∗(in). Lemma. If n ≥ 1, then h is a homomorphism for all X. Proof. For maps f, g : Sn −→ X, [f + g] is represented by the composite Sn p −→ Sn ∨ Sn f ∨g −−→ X ∨ X ▽ −→ X, where p is the pinch map and ▽ is the codiagonal map; that is, ▽ restricts to the identity on each wedge summand. Since p∗(in) = in + in and ▽ induces addition on ˜H∗(X), the conclusion follows. Lemma. The Hurewicz homomorphism is natural and the following diagram commutes for n ≥ 0: πn(X) Σ / ˜Hn(X) Σ πn+1(ΣX) / ˜Hn+1(ΣX). h Proof. The naturality of h is clear, and the naturality of Σ on homology implies the commutativity of the diagram: (h ◦ Σ)([f ]) = (Σf )∗(Σin) = Σ(f∗(in)) = Σ(h([f ])). 117
/ / 118 THE HUREWICZ AND UNIQUENESS THEOREMS Lemma. Let X be a wedge of n-spheres. Then h : πn(X) −→ ˜Hn(X) is the Abelianization homomorphism if n = 1 and is an isomorphism if n > 1. Proof. When X is a single sphere, h[id] = in and the conclusion is obvious. In general, πn(X) is the free group if n = 1 or the free Abelian group if n ≥ 2 with generators given by the inclusions of the wedge summands. Since h maps these generators to the canonical generators of the free Abelian group ˜Hn(X), the conclusion follows. That is all that we shall need in the next section, but we can generalize the lemma to arbitrary (n − 1)-connected based spaces X. Theorem (Hurewicz). Let X be any (n − 1)-connected based space. Then h : πn(X) −→ ˜Hn(X) is the Abelianization homomorphism if n = 1 and is an isomorphism if n > 1. Proof. We can assume without loss of generality that X is a CW complex with a single vertex, based attaching maps, and no q-cells for 1 ≤ q < n. The inclusion of the (n + 1)-skeleton in X induces an isomorphism on πn by the cellular approximation theorem and induces an isomorphism on ˜Hn by our cellular construction of homology or by a deduction from the axioms that will be given in the next section. Thus we may assume without loss of generality that X = X n+1. Then X is the coﬁber of a map f : K −→ L, where K and L are both wedges of n-spheres. We have the following commutative diagram: πn(K) πn(L) πn(X) 0 ˜Hn(K) / ˜Hn(L) / ˜Hn(X) / 0. The lemma gives the conclusion for the two left vertical arrows. Since X/L is a wedge of (n + 1)-spheres, the bottom row is exact by our long exact homology sequences and the known homology of wedges of spheres. When
n = 1, a corollary of the van Kampen theorem gives that π1(X) is the quotient of π1(L) by the normal subgroup generated by the image of π1(K). An easy algebraic exercise shows that the sequence obtained from the top row by passage to Abelianizations is therefore exact. If n > 1, the homotopy excision theorem implies that the top row is exact. To see this, factor f as the composite of the inclusion K −→ M f and the deformation retraction r : M f −→ L. Since X = Cf, we have the following commutative diagram, in which the top row is exact: πn(K) / πn(M f ) πn(M f, K) 0 r∗ πn(K) / πn(L) / πn(X) / 0. Since K and L are (n − 1)-connected and n > 1, a corollary of the homotopy excision theorem gives that X is (n − 1)-connected and πn(M f, K) −→ πn(X) is an isomorphism. THE UNIQUENESS OF THE HOMOLOGY OF CW COMPLEXES 119 2. The uniqueness of the homology of CW complexes We assume given an ordinary homology theory on CW pairs and describe how it must be computed. We focus on integral homology, taking π = Z and deleting it from the notation. With a moment’s reﬂection on the case n = 0, we see that the Hurewicz theorem gives a natural isomorphism ˜H ′ n(X) −→ ˜Hn(X) for (n − 1)-connected based spaces X. Here the groups on the left are deﬁned in terms of homotopy groups and were used in our construction of cellular chains, while the groups on the right are those of our given homology theory. We use the groups on the right to construct cellular chains in our given theory, and we ﬁnd that the isomorphism is compatible with diﬀerentials. From here, to prove uniqueness, we only need to check from the axioms that our given theory is computable from the homology groups of these cellular chain complexes. Thus let X be a CW complex. For
each integer n, deﬁne Cn(X) = Hn(X n, X n−1) ∼= ˜Hn(X n/X n−1). Deﬁne to be the composite d : Cn(X) −→ Cn−1(X) Hn(X n, X n−1) ∂−→ Hn−1(X n−1) −→ Hn−1(X n−1, X n−2). It is not hard to check that d ◦ d = 0. Theorem. C∗(X) is isomorphic to the cellular chain complex of X. Proof. Since X n/X n−1 is the wedge of an n-sphere for each n-cell of X, we see by the additivity axiom that Cn(X) is the free Abelian group with one generator [j] for each n-cell j. We must compare the diﬀerential with the one that we deﬁned earlier. Let i : X n−1 −→ X n be the inclusion. We see from our proof of the suspension isomorphism that d coincides with the composite ˜Hn(X n/X n−1) ∼= ˜Hn(Ci) → ˜Hn(ΣX n−1) Σ By the naturality of the Hurewicz homomorphism and its commutation with sus- pension, this coincides with the diﬀerential that we deﬁned originally. −−−→ ˜Hn−1(X n−1) → ˜Hn−1(X n−1/X n−2). −1 Similarly, if we start with a homology theory H∗(−; π), we can use the axioms to construct a chain complex C∗(X; π), and a comparison of deﬁnitions then gives an isomorphism of chain complexes C∗(X; π) ∼= C∗(X) ⊗ π. We have identiﬁed our axiomatically derived chain complex of X with the cellular chain complex of X, and we again adopt the notation C∗(X, A) = ˜C∗(X/A). Theorem. There is a natural isomorph
ism H∗(X, A) ∼= H∗(C∗(X, A)) under which the natural transformation ∂ agrees with the natural transformation induced by the connecting homomorphisms associated to the short exact sequences 0 −→ C∗(A) −→ C∗(X) −→ C∗(X, A) −→ 0. 120 THE HUREWICZ AND UNIQUENESS THEOREMS Proof. In view of our comparison of theories on pairs of spaces and theories on pairs of CW complexes and our comparison of theories on pairs with reduced theories, it suﬃces to obtain a natural isomorphism of reduced theories on based CW complexes X. By the additivity axiom, we may as well assume that X is connected. More precisely, we must obtain a system of natural isomorphisms ˜Hn(X) ∼= H∗( ˜Cn(X)) that commute with the suspension isomorphisms. By the dimension and additivity axioms, we know the homology of wedges of spheres. Since X n/X n−1 is a wedge of n-spheres, the long exact homology sequence associated to the coﬁber sequence X n−1 −→ X n −→ X n/X n−1 and an induction on n imply that ˜Hq(X n−1) −→ ˜Hq(X n) is an isomorphism for q < n − 1 and ˜Hq(X n) = 0 for q > n. Of course, the analogues for cellular homology are obvious. Note in particular that ˜Hn(X n+1) ∼= ˜Hn(X n+i) for all i > 1. Since homology commutes with colimits on sequences of inclusions, this implies that the inclusion X n+1 −→ X induces an isomorphism ˜Hn(X n+1) −→ ˜Hn(X). Using these facts, we easily check from the exactness axiom that the rows and columns are exact in the following commutative diagram: ˜Hn+1(X n+1/X n) 0 dn+1 )SSSSSSSSSSSSSS 0 ∂ / ˜Hn(X n) i∗ ˜Hn(X) ∼= ˜Hn
(X n+1) 0 ˜Hn(X n/X n−1) ρ∗ ∂ ˜Hn−1(X n−1) )SSSSSSSSSSSSSS dn ˜Hn−1(X n−1/X n−2). Deﬁne α : ˜Hn(X) −→ Hn( ˜C∗(X)) by letting α(x) be the homology class of ρ∗(y) for any y such that i∗(y) = x. It is an exercise in diagram chasing and the deﬁnition of the homology of a chain complex to check that α is a well deﬁned isomorphism. The reduced chain complex of ΣX can be identiﬁed with the suspension of the reduced chain complex of X. That is, ˜Cn+1(ΣX) ∼= ˜Cn(X), compatibly with the diﬀerential. All maps in the diagram commute with suspension, and this implies that the isomorphisms α commute with the suspension isomorphisms. ) / / / / / ) 2. THE UNIQUENESS OF THE HOMOLOGY OF CW COMPLEXES 121 PROBLEMS (1) Let π be any group. Construct a connected CW complex K(π, 1) such that π1(K(π, 1)) = π and πq(K(π, 1)) = 0 for q 6= 1. (2) * In Problem 1, it is rarely the case that K(π, 1) can be constructed as a compact manifold. What is a necessary condition on π for this to happen? (3) Let n ≥ 1 and let π be an Abelian group. Construct a connected CW complex M (π, n) such that ˜Hn(X; Z) = π and ˜Hq(X; Z) = 0 for q 6= n. (Hint: construct M (π, n) as the coﬁber of a map between wedges of spheres.) The spaces M (π, n) are called Moore spaces. (4) Let n ≥ 1 and let π be an Abelian group. Construct a connected CW complex K(π, n) such that πn(X
) = π and πq(X) = 0 for q 6= n. (Hint: start with M (π, n), using the Hurewicz theorem, and kill its higher homotopy groups.) The spaces K(π, n) are called Eilenberg-Mac Lane spaces. (5) There are familiar spaces that give K(Z, 1), K(Z2, 1), and K(Z, 2). Name them. (6) Let X be any connected CW complex whose only non-vanishing homotopy group is πn(X) ∼= π. Construct a homotopy equivalence K(π, n) −→ X, where K(π, n) is the Eilenberg-Mac Lane space that you have constructed. (7) * For groups π and ρ, compute [K(π, n), K(ρ, n)]; here [−, −] means based homotopy classes of based maps. CHAPTER 16 Singular homology theory We explain, without giving full details, how the standard approach to singular homology theory ﬁts into our framework. We also introduce simplicial sets and spaces and their geometric realization. These notions play a fundamental role in modern algebraic topology. 1. The singular chain complex The standard topological n-simplex is the subspace ∆n = {(t0,..., tn)\|0 ≤ ti ≤ 1, ti = 1} of Rn+1. There are “face maps” P δi : ∆n−1 −→ ∆n, 0 ≤ i ≤ n, speciﬁed by δi(t0,..., tn−1) = (t0,..., ti−1, 0, ti,..., tn−1) and “degeneracy maps” speciﬁed by σi : ∆n+1 −→ ∆n, 0 ≤ i ≤ n, σi(t0,..., tn+1) = (t0,..., ti−1, ti + ti+1,..., tn+1). For a space X, deﬁne SnX to be the set of continuous maps f : ∆n −→ X. In particular,
regarding a point of X as the map that sends 1 to x, we may identify the underlying set of X with S0X. Deﬁne the ith face operator di : SnX −→ Sn−1X, 0 ≤ i ≤ n, by where u ∈ ∆n−1, and deﬁne the ith degeneracy operator di(f )(u) = f (δi(u)), si : SnX −→ Sn+1X, 0 ≤ i ≤ n, by where v ∈ ∆n+1. The following identities are easily checked: si(f )(v) = f (σi(v)), di ◦ dj = dj−1 ◦ di if i < j sj−1 ◦ di di ◦ sj =  id  sj ◦ di−1 si ◦ sj = sj+1 ◦ si  if i < j if i = j or i = j + 1 if i > j + 1. if i ≤ j. 123 124 SINGULAR HOMOLOGY THEORY A map f : ∆n −→ X is called a singular n-simplex. It is said to be nondegenerate if it is not of the form si(g) for any i and g. Let Cn(X) be the free Abelian group generated by the nondegenerate n-simplexes, and think of Cn(X) as the quotient of the free Abelian group generated by all singular n-simplexes by the subgroup generated by the degenerate n-simplexes. Deﬁne n d = (−1)idi : Cn(X) −→ Cn−1(X). The identities ensure that C∗(X) is then a well deﬁned chain complex. In fact, i=0 X n−1 n d ◦ d = (−1)i+jdi ◦ dj, i=0 X j=0 X and, for i < j, the (i, j)th and (j − 1, i)th summands add to zero. This gives that d ◦ d = 0 before quotienting out the degenerate simplexes, and the degenerate simplexes span a subcomplex. The singular homology of X is usually deﬁ
ned in terms of this chain complex: H∗(X; π) = H∗(C∗(X) ⊗ π). 2. Geometric realization One can prove the compatibility of this deﬁnition with our deﬁnition by checking the axioms and quoting the uniqueness of homology. We instead describe how the new deﬁnition ﬁts into our original deﬁnition in terms of CW approximation and cellular chain complexes. We deﬁne a space ΓX, called the “geometric realization of the total singular complex of X,” as follows. As a set where the equivalence relation ∼ is generated by ` ΓX = n≥0(SnX × ∆n)/(∼), (f, δiu) ∼ (di(f ), u) for f : ∆n −→ X and u ∈ ∆n−1 and (f, σiv) ∼ (si(f ), v) for f : ∆n −→ X and v ∈ ∆n+1. Topologize ΓX by giving 0≤n≤q(SnX × ∆n)/(∼) the quotient topology and then giving ΓX the topology of the union. Deﬁne γ : ΓX −→ X by ` γ\|f, u\| = f (u) for f : ∆n −→ X and u ∈ ∆n, where \|f, u\| denotes the equivalence class of (f, u). Now the following two theorems imply that that this construction provides a canonical way of realizing our original construction of homology. Theorem. For any space X, ΓX is a CW complex with one n-cell for each nondegenerate singular n-simplex, and the cellular chain complex C∗(ΓX) is naturally isomorphic to the singular chain complex C∗(X). Theorem. For any space X, the map γ : ΓX −→ X is a weak equivalence. 3. PROOFS OF THE THEOREMS 125 Thus the singular chain complex of X is the cellular chain complex of a functorial CW approximation of X, and this shows that our original construction of homology coincides with the classical construction in terms of singular
chains. Each approach has its mathematical and pedagogical advantages. 3. Proofs of the theorems We give a detailed outline of how the required CW decomposition of ΓX is obtained and sketch the proof that γ is a weak equivalence. Let ¯X = n≥0 SnX × ∆n. Deﬁne functions ` λ : ¯X −→ ¯X and ρ : ¯X −→ ¯X by if f = sjp · · · sj1 g where g is nondegenerate and 0 ≤ j1 < · · · < jp and λ(f, u) = (g, σj1 · · · σjpu) ρ(f, u) = (di1 · · · diq f, v) if u = δiq · · · δi1 v where v is an interior point and 0 ≤ i1 < · · · < iq. Note that the unique point of ∆0 is interior. Say that a point (f, u) is nondegenerate if f is nondegenerate and u is interior. A combinatorial argument from the deﬁnitions gives the following observation. Lemma. The composite λ ◦ ρ carries each point of ¯X to the unique nondegen- erate point to which it is equivalent. Let (ΓX)n be the image in ΓX of m≤n SmX × ∆m. Then (ΓX)n − (ΓX)n−1 = {nondegenerate n-simplexes} × {∆n − ∂∆n}. ` This implies that ΓX is a CW complex whose n-cells are the maps ˜f : (∆n, ∂∆n) −→ ((ΓX)n, (ΓX)n−1) speciﬁed by ˜f (u) = \|f, u\| for a nondegenerate n-simplex f. Here we think of (∆n, ∂∆n) as the domains of cells via oriented homeomorphisms with (Dn, Sn−1). To compute d on the cellular chains C∗(ΓX), we must compute the degrees of the composites Sn−1 ∼= ∂∆n ˜
f −→ (ΓX)n−1/(ΓX)n−2 π˜g−→ ∆n−1/∂∆n−1 ∼= Sn−1 for nondegenerate n-simplexes f and (n − 1)-simplexes g. The only relevant g are the dif since f traverses these g on the various faces of ∂∆n. Let ¯δi : ∂∆n −→ ∆n−1/∂∆n−1 be the map that collapses all faces of ∂∆n other than δi∆n−1 to the basepoint and is δ−1 on δi∆n−1. Then, with g = dif, the composite above reduces to the map i Sn−1 ∼= ∂∆n ¯δi−→ ∆n−1/∂∆n−1 ∼= Sn−1. It is not hard to check that the degree of this map is (−1)i (provided that we choose our homeomorphisms sensibly). If n = 2, the three maps δi are given by δ0(1 − t, t) = (0, 1 − t, t) δ1(1 − t, t) = (1 − t, 0, t) δ2(1 − t, t) = (1 − t, t, 0) 126 SINGULAR HOMOLOGY THEORY and we can visualize the maps ¯δi as follows:???? _???? ¯δi???? δ2_ _???? wooooooo δ1 δ0 ooooooo The alternation of orientations and thus of signs should be clear. This shows that C∗(ΓX) = C∗(X), as claimed. We must still explain why γ : ΓX −→ X is a weak equivalence. In fact, it is tautologically obvious that γ induces an epimorphism on all homotopy groups: a map of pairs determines the map of pairs f : (∆n, ∂∆n) −→ (X, x) ˜f : (∆n, ∂∆n) −→ (ΓX, \|x, 1\|) speciﬁed by ˜f
(u) = \|f, u\|, and γ ◦ ˜f = f. Injectivity is more delicate, and we shall only give a sketch. Given a map g : (∆n, ∂∆n) −→ (ΓX, \|x, 1\|), we may ﬁrst apply cellular approximation to obtain a homotopy of g with a cellular map and we may then subdivide the domain and apply a further homotopy so as to obtain a map g′ ≃ g such that g′ is simplicial, in the sense that g′ ◦ e is a cell of ΓX for every cell e of the subdivision of ∆n. Suppose that γ ◦ g and thus γ ◦ g′ is homotopic to the constant map cx at the point x. We may view a homotopy h : γ ◦ g′ ≃ cx as a map h : (∆n × I, ∂∆n × I ∪ ∆n × {1}) −→ (X, x). We can simplicially subdivide ∆n × I so ﬁnely that our subdivided ∆n = ∆n × {0} is a subcomplex. We can then lift h simplex by simplex to a simplicial map ˜h : (∆n × I, ∂∆n × I ∪ ∆n × {1}) −→ (ΓX, \|x, 1\|) such that ˜h restricts to ˜g′ on ∆n × {0} and γ ◦ ˜h = h. 4. Simplicial objects in algebraic topology A simplicial set K∗ is a sequence of sets Kn, n ≥ 0, connected by face and degeneracy operators di : Kn −→ Kn−1 and si : Kn −→ Kn+1, 0 ≤ i ≤ n, that satisfy the commutation relations that we displayed for the total singular complex S∗X = {SnX} of a space X. Thus S∗ is a functor from spaces to simplicial sets. We may deﬁne the geometric realization \|K∗\| of general simplicial sets exactly as we deﬁned the geometric realization ΓX = \|S∗X\| of the total singular complex of a topological space. In fact,
the total singular complex and geometric realization functors are adjoint. If S S is the category of simplicial sets and U the category of spaces, then U (\|K∗\|, X) ∼= S S (K∗, S∗X).. SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY 127 The identity map of S∗X on the right corresponds to γ : \|S∗X\| −→ X on the left. In general, for a map f∗ : K∗ −→ S∗X of simplicial sets, the corresponding map of spaces is the composite \|K∗\| \|f∗\| −−→ \|S∗X\| γ −→ X. In fact, one can develop homotopy theory and homology in the category of simplicial sets in a fashion parallel to and, in a suitable sense, equivalent to the development that we have here given for topological spaces. For example, we have the chain complex C∗(K∗) deﬁned exactly as we deﬁned the singular chain complex, using the alternating sum of the face maps, and there result homology groups H∗(K∗; π) = H∗(C∗(K∗) ⊗ π). Exactly as in the case of S∗X, \|K∗\| is a CW complex and C∗(K∗) is naturally isomorphic to the cellular chain complex C∗(\|K∗\|). Singular homology is the special case obtained by taking K∗ = S∗X for spaces X. The passage back and forth between simplicial sets and topological spaces plays a major role in many applications. The ideas generalize. One can deﬁne a simplicial object in any category C as a sequence of objects Kn of C connected by face and degeneracy maps in C that satisfy the commutation relations that we have displayed. Thus we have simplicial groups, simplicial Abelian groups, simplicial spaces, and so forth. We can think of simplicial sets as discrete simplicial spaces, and we then see that geometric realization generalizes directly to a functor \| − \| from the category S U of simplicial spaces to the category U of spaces. This provides a very useful way of constructing spaces with desirable properties. We note one of the principal features of
geometric realization. Deﬁne the product X∗ × Y∗ of simplicial spaces X∗ and Y∗ to be the simplicial space whose space of n-simplexes is Xn × Yn, with faces and degeneracies di × di and si × si. The projections induce maps of simplicial spaces from X∗ × Y∗ to X∗ and Y∗. On passage to geometric realization, these give the coordinates of a map \|X∗ × Y∗\| −→ \|X∗\| × \|Y∗\|. It turns out that this map is always a homeomorphism. Now restrict attention to simplicial sets K∗ and L∗. Then the homeomorphism just speciﬁed is a map between CW complexes. However, it is not a cellular map; rather, it takes the n-skeleton of \|K∗ × L∗\| to the 2n-skeleton of \|K∗\| × \|L∗\|. It is homotopic to a cellular map, no longer a homeomorphism, and there results a chain homotopy equivalence C∗(\|K∗ × L∗\|) −→ C∗(\|K∗\|) ⊗ C∗(\|L∗\|). In particular, for spaces X and Y, there is a natural chain homotopy equivalence from the singular chain complex C∗(X × Y ) to the tensor product C∗(X) ⊗ C∗(Y ). One can be explicit about this but, pedagogically, one technical advantage of approaching homology via CW complexes is that it leaves us free to work directly with the natural cell structures on Cartesian products of CW complexes and to postpone the introduction of chain homotopy equivalences such as these to a later stage of the development. 128 SINGULAR HOMOLOGY THEORY 5. Classifying spaces and K(π, n)s We illustrate these ideas by deﬁning the “classifying spaces” and “universal bundles” associated to topological groups G and describing how this leads to a beautiful conceptual construction of the Eilenberg-Mac Lane spaces K(π, n) associated to discrete Abelian groups π. Recall that these are spaces such that πq(K(π, n)) = π if q
= n 0 if q 6= n. ( We deﬁne a map p∗ : E∗(G) −→ B∗(G) of simplicial topological spaces. Let En(G) = Gn+1 and Bn(G) = Gn, and let pn : Gn+1 −→ Gn be the projection on the ﬁrst n coordinates. The faces and degeneracies are deﬁned on En(G) by di(g1,..., gn+1) = (g2,..., gn+1) (g1,..., gi−1, gigi+1, gi+2,..., gn+1) if 1 ≤ i ≤ n if i = 0 ( and si(g1,..., gn+1) = (g1,..., gi−1, e, gi,..., gn+1) if 0 ≤ i ≤ n. The faces and degeneracies on Bn(G) are deﬁned in the same way, except that the last coordinate gn+1 is omitted and the last face operation dn takes the form dn(g1,..., gn) = (g1,..., gn−1). Certainly p∗ is a map of simplicial spaces. If we let G act from the right on En(G) by multiplication on the last coordinate, (g1,..., gn, gn+1)g = (g1,..., gn, gn+1g), then E∗(G) is a simplicial G-space. That is, the action of G commutes with the face and degeneracy maps. We may view Bn(G) as the orbit space En(G)/G. We deﬁne E(G) = \|E∗(G)\|, B(G) = \|B∗(G)\|, and p = \|p∗(G)\| : E(G) −→ B(G). Then E(G) inherits a free right action by G, and B(G) is the orbit space E(G)/G. The space BG is called the classifying space of G. The space E(G) is the union of the images E(G)n of the
spaces m≤n Gm+1 × ∆m, and ` E(G)n − E(G)n−1 = (Gn − W ) × G × (∆n − ∂∆n), where W ⊂ Gn is the “fat wedge” consisting of those points at least one of whose coordinates is the identity element e. Similarly, we have subspaces B(G)n such that B(G)n − B(G)n−1 = (Gn − W ) × (∆n − ∂∆n). The map p restricts to the projection between these subspaces. Intuitively, it looks as if p should be a bundle with ﬁber G, and this is indeed the case if the identity element of G is a nondegenerate basepoint. This condition is enough to ensure local triviality as we glue together over the ﬁltration {B(G)n}. It is less intuitive, but true, that the space E(G) is contractible. By the long exact homotopy sequence, these facts imply that πq+1(BG) ∼= πq(G) for all q ≥ 0. For topological groups G and H, the obvious shuﬄe homeomorphisms (G × H)n ∼= Gn × H n 5. CLASSIFYING SPACES AND K(π, n)S 129 specify isomorphisms of simplicial spaces E∗(G × H) ∼= E∗(G) × E∗(H) and B∗(G × H) ∼= B∗(G) × B∗(H) that are compatible with the projections. Since geometric realization commutes with products, we conclude that B(G × H) is homeomorphic to B(G) × B(H). Thus B is a product-preserving functor on the category of topological groups. Now suppose that G is a commutative topological group. Then its multiplication G× G −→ G and inverse map G −→ G are homomorphisms. We conclude that B(G) and E(G) are again commutative topological groups. The multiplication on B(G) is determined by the multiplication on G as the composite B(G) × B(G) ∼= B(G × G) −→
B(G). Moreover, the map p : E(G) −→ B(G) and the inclusion of G in E(G) as the ﬁber over the basepoint (the unique point in B0(G)) are homomorphisms. This allows us to iterate the construction, setting B0(G) = G and Bn(G) = B(Bn−1(G)) for n ≥ 1. Specializing to a discrete Abelian group π, we deﬁne As promised, we have K(π, n) = Bn(π). πq(K(π, n)) = πq−1(K(π, n − 1)) = · · · = πq−n(K(π, 0)) = π if q = n 0 if q 6= n. ( PROBLEMS (1) Let X be a space that satisﬁes the hypotheses used to construct a universal cover ˜X. Let π = π1(X) and consider the action of the group π on the space ˜X given by the isomorphism of π with Aut( ˜X). Let A be an Abelian group and let Z[π] act trivially on A, a · σ = a for σ ∈ π and a ∈ A. Do one or both of the following, and convince yourself that the other choice also works. (a) [Cellular chains] Assume that X is a CW complex. Show that ˜X is a CW complex such that the action of π on ˜X induces an action of the group ring Z[π] on the cellular chain complex C∗( ˜X) such that each Cq( ˜X) is a free Z[π]-module and C∗(X; A) ∼= A ⊗Z[π] C∗( ˜X). (b) [Singular chains] Show that the action of π on ˜X induces an action of Z[π] on the singular chain complex C∗( ˜X) such that each Cq( ˜X) is a free Z[π]-module and C∗(X; A) ∼= A ⊗Z[π] C∗( ˜X). (2) Let π be
a group and let K(π, 1) be a connected CW complex such that π1(K(π, 1)) = π and πq(K(π, 1)) = 0 for q 6= 1. Use Problem 1 to show that there is an isomorphism Z[π] H∗(K(π, 1); A) ∼= Tor ∗ (A, Z). (3) Let p : Y −→ X be a covering space with ﬁnite ﬁbers, say of cardinality n. Using singular chains, construct a homomorphism t : H∗(X; A) −→ H∗(Y ; A) such that the composite p∗ ◦ t : H∗(X; A) −→ H∗(X; A) is multiplication by n; t is called a “transfer homomorphism.” CHAPTER 17 Some more homological algebra The reader will by now appreciate that the calculation of homology groups, although far simpler than the calculation of homotopy groups, can still be a diﬃcult task. In practice, one seldom uses chains explicitly; rather, one uses them to prove algebraic theorems that simplify topological calculations. Indeed, if one focuses on singular chains, then one eschews chain level computations in principle as well as in practice. We here recall some classical results in homological algebra that explain how to calculate H∗(X; π) from H∗(X) ≡ H∗(X; Z) and how to calculate H∗(X × Y ) from H∗(X) ⊗ H∗(Y ). We then say a little about cochain complexes in preparation for the deﬁnition of cohomology groups. We again work over a general commutative ring R, although the main example will be R = Z. Tensor products are understood to be taken over R. 1. Universal coeﬃcients in homology Let X and Y be chain complexes over R. We think of H∗(X) ⊗ H∗(Y ) as a graded R-module which, in degree n, is p+q=n Hp(X) ⊗ Hq(Y ). We deﬁne P α : H∗(X) ⊗ H∗(Y
) −→ H∗(X ⊗ Y ) by α([x] ⊗ [y]) = [x ⊗ y] for cycles x and y that represent homology classes [x] and [y]. As a special case, for an R-module M we have α : H∗(X) ⊗ M −→ H∗(X ⊗ M ). We omit the proof of the following standard result, but we shall shortly give the quite similar proof of a cohomological analogue. Recall that an R-module M is said to be ﬂat if the functor M ⊗ N is exact (that is, preserves exact sequences in the variable N ). We say that a graded R-module is ﬂat if each of its terms is ﬂat. We assume that the reader has seen torsion products, which measure the failure of tensor products to be exact functors. For a principal ideal domain (PID) R, the only torsion product is the ﬁrst one, denoted TorR 1 (M, N ). It can be computed by constructing a short exact sequence 0 −→ F1 −→ F0 −→ M −→ 0 and tensoring with N to obtain an exact seqence 0 −→ TorR 1 (M, N ) −→ F1 ⊗ N −→ F0 ⊗ N −→ M ⊗ N −→ 0, where F1 and F0 are free R-modules. That is, we choose an epimorphism F0 −→ M and note that, since R is a PID, its kernel F1 is also free. 131 132 SOME MORE HOMOLOGICAL ALGEBRA Theorem (Universal coeﬃcient). Let R be a P ID and let X be a ﬂat chain complex over R. Then, for each n, there is a natural short exact sequence 0 −→ Hn(X) ⊗ M α−→ Hn(X ⊗ M ) β −→ TorR 1 (Hn−1(X), M ) −→ 0. The sequence splits, so that Hn(X ⊗ M ) ∼= (Hn(X) ⊗ M ) ⊕ TorR 1 (Hn−1(X), M ), but the splitting is not natural. In Chapter 20 §
3, we shall see an important class of examples in which the splitting is very far from being natural. Corollary. If R is a ﬁeld, then is a natural isomorphism. α : H∗(X) ⊗ M −→ H∗(X; M ) 2. The K¨unneth theorem The universal coeﬃcient theorem in homology is a special case of the K¨unneth theorem. Theorem (K¨unneth). Let R be a P ID and let X be a ﬂat chain complex and Y be any chain complex. Then, for each n, there is a natural short exact sequence 0 −→ Hp(X)⊗Hq(Y ) α−→ Hn(X⊗Y ) β −→ TorR 1 (Hp(X), Hq(Y )) −→ 0. p+q=n X The sequence splits, so that p+q=n−1 X Hn(X ⊗ Y ) ∼= ( Hp(X) ⊗ Hq(Y )) ⊕ ( TorR 1 (Hp(X), Hq(Y ))), but the splitting is not natural. p+q=n X p+q=n−1 X Returning to topology for a moment, observe that this applies directly to the computation of the homology of the Cartesian product of CW complexes X and Y in view of the isomorphism C∗(X × Y ) ∼= C∗(X) ⊗ C∗(Y ). Corollary. If R is a ﬁeld, then α : H∗(X) ⊗ H∗(Y ) −→ H∗(X ⊗ Y ) is a natural isomorphism. We prove the corollary to give the idea. The general case is proved by an elaboration of the argument. In fact, in practice, algebraic topologists carry out the vast majority of their calculations using a ﬁeld of coeﬃcients, and it is then the corollary that is relevant to the study of the homology of Cartesian products. There is a simple but important technical point to make about this. Let us for the moment remember to indicate the ring over which we are taking tensor products. For chain complexes X and Y over Z,
we have (X ⊗Z R) ⊗R (Y ⊗Z R) ∼= (X ⊗Z Y ) ⊗Z R. We can therefore use the corollary to compute H∗(X ⊗Z Y ; R) from H∗(X; R) and H∗(Y ; R). 3. HOM FUNCTORS AND UNIVERSAL COEFFICIENTS IN COHOMOLOGY 133 Proof of the corollary. Assume ﬁrst that Xi = 0 for i 6= p, so that X = Xp is just an R-module with no diﬀerential. The square commutes and the row and column are exact in the diagram 0 0 / Bq(Y ) / Zq(Y ) / Hq(Y ) / 0. dq+1 Yq+1 dq+1 Yq dq Yq−1 Since all modules over a ﬁeld are free and thus ﬂat, this remains true when we tensor the diagram with Xp. This proves that if n = p + q, then Zn(Xp ⊗ Y ) = Xp ⊗ Zq(Y ), Bn(Xp ⊗ Y ) = Xp ⊗ Bq(Y ), and therefore Hn(X ⊗ Y ) = Xp ⊗ Hq(Y ). In the general case, regard the graded modules Z(X) and B(X) as chain complexes with zero diﬀerential. The exact sequences 0 −→ Zp(X) −→ Xp dp−→ Bp−1(X) −→ 0 of R-modules deﬁne a short exact seqence of chain complexes since dp−1 ◦ dp = 0. Deﬁne the suspension of a graded R-module N by (ΣN )n+1 = Nn. Tensoring with Y, we obtain a short exact sequence of chain complexes 0 −→ Z(X) ⊗ Y −→ X ⊗ Y −→ ΣB(X) ⊗ Y −→ 0. It follows from the ﬁrst part and additivity that H∗(Z(X) ⊗ Y ) = Z(X) ⊗ H∗(Y )
and H∗(ΣB(X) ⊗ Y ) = ΣB(X) ⊗ H∗(Y ). Moreover, by inspection of deﬁnitions, the connecting homomorphism of the long exact sequence of homology modules associated to our short exact sequence of chain complexes is just the inclusion B ⊗ H∗(Y ) −→ Z ⊗ H∗(Y ). In particular, the long exact sequence breaks up into short exact sequences 0 −→ B(X) ⊗ H∗(Y ) −→ Z(X) ⊗ H∗(Y ) −→ H∗(X ⊗ Y ) −→ 0. However, since tensoring with H∗(Y ) is an exact functor, the cokernel of the inclu- sion B ⊗ H∗(Y ) −→ Z ⊗ H∗(Y ) is H∗(X) ⊗ H∗(Y ). The conclusion follows. 3. Hom functors and universal coeﬃcients in cohomology For a chain complex X = X∗, we deﬁne the dual cochain complex X ∗ by setting X q = Hom(Xq, R) and dq = (−1)q Hom(dq+1, id). As with tensor products, we understand Hom to mean HomR when R is clear from the context. On elements, for an R-map f : Xq −→ R and an element x ∈ Xq+1, (dqf )(x) = (−1)qf (dq(x)). / / / / / / O O 134 SOME MORE HOMOLOGICAL ALGEBRA More generally, for an R-module M, we deﬁne a cochain complex Hom(X, M ) in the same way. The sign is conventional and is designed to facilitate the deﬁnition of Hom(X, Y ) for a chain complex X and cochain complex Y ; however, we shall not have occasion to use the latter deﬁnition. In analogy with the notation H∗(X; M ) = H∗(X ⊗ M ), we write H ∗(X; M ) = H ∗(Hom(X, M )). We have a cohom
ological version of the universal coeﬃcient theorem. We assume that the reader has seen Ext modules, which measure the failure of Hom to be an exact functor. For a PID R, the only Ext module is the ﬁrst one, denoted Ext1 R(M, N ). It can be computed by constructing a short exact sequence and applying Hom to obtain an exact seqence 0 −→ F1 −→ F0 −→ M −→ 0 0 −→ Hom(M, N ) −→ Hom(F0, N ) −→ Hom(F1, N ) −→ Ext1 R(M, N ) −→ 0, where F1 and F0 are free R-modules. For each n, deﬁne α : H n(Hom(X, M )) −→ Hom(Hn(X), M ) by letting α[f ]([x]) = f (x) for a cohomology class [f ] represented by a “cocycle” f : Xn −→ M and a homology class [x] represented by a cycle x. It is easy to check that f (x) is independent of the choices of f and x since x is a cycle and f is a cocycle. Theorem (Universal coeﬃcient). Let R be a P ID and let X be a free chain complex over R. Then, for each n, there is a natural short exact sequence 0 −→ Ext1 R(Hn−1(X), M ) β −→ H n(X; M ) α−→ Hom(Hn(X), M ) −→ 0. The sequence splits, so that H n(X; M ) ∼= Hom(Hn(X), M ) ⊕ Ext1 R(Hn−1(X), M ), but the splitting is not natural. Corollary. If R is a ﬁeld, then α : H ∗(X; M ) −→ Hom(H∗(X), M ) is a natural isomorphism. Again, there is a technical point to be made here. If X is a complex of free Abelian groups and M is an R-module, such as R itself, then HomZ(X, M ) ∼= HomR(X ⊗Z R, M ). One way to see this is to observe that, if B is
a basis for a free Abelian group F, then HomZ(F, M ) and HomR(F ⊗Z R, M ) are both in canonical bijective correspondence with maps of sets B −→ M. More algebraically, a homomorphism f : F −→ M of Abelian groups determines the corresponding map of R-modules as the composite of f ⊗ id and the action of R on M : F ⊗Z R −→ M ⊗Z R −→ M. 4. PROOF OF THE UNIVERSAL COEFFICIENT THEOREM 135 4. Proof of the universal coeﬃcient theorem We need two properties of Ext in the proof. First, Ext1 R-module F. Second, when R is a PID, a short exact sequence R(F, M ) = 0 for a free 0 −→ L′ −→ L −→ L′′ −→ 0 of R-modules gives rise to a six-term exact sequence 0 −→ Hom(L′′, M ) −→ Hom(L, M ) −→ Hom(L′, M ) δ−→ Ext1 R(L′′, M ) −→ Ext1 R(L, M ) −→ Ext1 R(L′, M ) −→ 0. Proof of the universal coefficient theorem. We write Bn = Bn(X), Zn = Zn(X), and Hn = Hn(X) to abbreviate notation. Since each Xn is a free R-module and R is a PID, each Bn and Zn is also free. We have short exact sequences and 0 / Bn in / Zn πn / / Hn / 0 0 / Zn jn / Xn dn / o_ _ _ σn / Bn−1 / 0; we choose a splitting σn of the second. Writing f ∗ = Hom(f, M ) consistently, we obtain a commutative diagram with exact rows and columns 0 0 0 / Hom(Hn, M ) · · · / Hom(Xn−1, M ) π ∗ n ∗ n d / Hom(Zn, M ) ∗ j n Hom(Bn, M ) d ∗ n+1 Hom(Xn, M ) Hom(Xn+1, M ) ∗ i n d ∗ n+
1 ∗ j n−1 σ ∗ n ∗ n d 6lllllll 0 δ Hom(Zn−1, M ) ∗ i n−1 Hom(Bn−1, M ) Ext1 R(Hn−1 By inspection of the diagram, we see that the canonical map α coincides with the composite H n(X; M ) = ker d∗ n+1/ im d∗ n = ker i∗ nj∗ n/ im d∗ ni∗ n−1 ∗ j n−→ im π∗ n −1 ∗ n) (π −−−−→ Hom(Hn, M ). n is an epimorphism, so is α. The kernel of α is im d∗ Since j∗ δ(d∗ induces the required splitting. n)−1 maps this group isomorphically onto Ext1 n/ im d∗ n−1, and R(Hn−1, M ). The composite δσ∗ n ni 136 SOME MORE HOMOLOGICAL ALGEBRA 5. Relations between ⊗ and Hom We shall need some observations about cochain complexes and tensor products, and we ﬁrst recall some general facts about the category of R-modules. For Rmodules L, M, and N, we have an adjunction Hom(L ⊗ M, N ) ∼= Hom(L, Hom(M, N )). We also have a natural homomorphism Hom(L, M ) ⊗ N −→ Hom(L, M ⊗ N ), and this is an isomorphism if either L or N is a ﬁnitely generated projective Rmodule. Again, we have a natural map Hom(L, M ) ⊗ Hom(L′, M ′) −→ Hom(L ⊗ L′, M ⊗ M ′), which is an isomorphism if L and L′ are ﬁnitely generated and projective or if L is ﬁnitely generated and projective and M = R. We can replace L and L′ by chain complexes and obtain similar maps, inserting signs where needed. For example, a chain homotopy X ⊗ I −→ X ′ between chain maps f, g : X −→ X ′ induces a chain map Hom(
X ′, M ) −→ Hom(X ⊗ I, M ) ∼= Hom(I, Hom(X, M )) ∼= Hom(X, M ) ⊗ I ∗, where I ∗ = Hom(I, R). It should be clear that this implies that our original chain homotopy induces a homotopy of cochain maps f ∗ ≃ g∗ : Hom(X ′, M ) −→ Hom(X, M ). If Y and Y ′ are cochain complexes, then we have the natural homomorphism α : H ∗(Y ) ⊗ H ∗(Y ′) −→ H ∗(Y ⊗ Y ′) given by α([y]⊗[y′]) = [y⊗y′], exactly as for chain complexes. (In fact, by regrading, we may view this as a special case of the map for chain complexes.) The K¨unneth theorem applies to this map. For its ﬂatness hypothesis, it is useful to remember that, for any Noetherian ring R, the dual Hom(F, R) of a free R-module is a ﬂat R-module. As indicated above, if Y = Hom(X, M ) and Y ′ = Hom(X ′, M ′) for chain complexes X and X ′ and R-modules M and M ′, then we also have the map of cochain complexes ω : Hom(X, M ) ⊗ Hom(X ′, M ′) −→ Hom(X ⊗ X ′, M ⊗ M ′) speciﬁed by the formula ω(f ⊗ f ′)(x ⊗ x′) = (−1)(deg f ′ )(deg x)f (x) ⊗ f ′(x′). We continue to write ω for the map it induces on cohomology, and we then have the composite ω ◦ α : H ∗(X; M ) ⊗ H ∗(X ′; M ′) −→ H ∗(X ⊗ X ′; M ⊗ M ′). When M = M ′ = A is a commutative R-algebra,

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