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0 and T∞ where the radii r0 and r1 are zero, making the tori T0 and T∞ degenerate to circles. These two circles are the unit circles in the two C factors of C2, so under stereographic projection of S 3 from the point (0, 1) onto R3 they correspond to the unit circle in the xy plane and the z axis. The concentric tori Tρ are then arranged as in the following ﬁgure. Each torus Tρ is a union of circle ﬁbers, the pairs (θ0, θ1) with θ0 − θ1 constant. These ﬁber circles have slope 1 on the torus, winding around once longitudinally and once meridionally. With respect to the ambient space it might be more accurate to say they have slope ρ. As ρ goes to 0 or ∞ the ﬁber circles approach the circles T0 and T∞, which are also ﬁbers. The ﬁgure shows four of the tori decomposed into ﬁbers. Example 4.46. Replacing the ﬁeld C by the quaternions H, the same constructions yield ﬁber bundles S 3→S 4n+3→HPn over quaternionic projective spaces HPn. Here the ﬁber S 3 is the unit quaternions, and S 4n+3 is the unit sphere in Hn+1. Taking n = 1 gives a second Hopf bundle S 3→S 7→S 4 = HP1. Example 4.47. Another Hopf bundle S 7→S 15→S 8 can be deﬁned using the octonion algebra O. Elements of O are pairs of quaternions (a1, a2) with multiplication given by (a1, a2)(b1, b2) = (a1b1 − b2a2, a2b1 + b2a1). Regarding S 15 as the unit sphere in the 16 dimensional vector space O2, the projection map p : S 15→S 8 = O ∪ {∞} is (z0, z1)֏ z0z−1 1, just as for the other Hopf bundles, but because O is not associative, a little care is needed to show this is a �
�ber bundle with ﬁber S 7, the unit octonions. Let U0 and U1 be the complements of ∞ and 0 in the base space O ∪ {∞}. Deﬁne hi : p−1(Ui)→Ui × S 7 and gi : Ui × S 7→p−1(Ui) by h0(z0, z1) = (z0z−1 h1(z0, z1) = (z0z−1 1, z1/\|z1\|), 1, z0/\|z0\|), g0(z, w) = (zw, w)/\|(zw, w)\| g1(z, w) = (w, z−1w)/\|(w, z−1w)\| Elementary Methods of Calculation Section 4.2 379 If one assumes the known fact that any subalgebra of O generated by two elements is associative, then it is a simple matter to check that gi and hi are inverse homeomorphisms, so we have a ﬁber bundle S 7→S 15→S 8. Actually, the calculation that gi and hi are inverses needs only the following more elementary facts about octonions z, w, where the conjugate z of z = (a1, a2) is deﬁned by the expected formula z = (a1, −a2) : (1) r z = zr for all r ∈ R and z ∈ O, where R ⊂ O as the pairs (r, 0). (2) \|z\|2 = zz = zz, hence z−1 = z/\|z\|2. (3) \|zw\| = \|z\|\|w\|. (4) zw = w z, hence (zw)−1 = w −1z−1. (5) z(zw) = (zz)w and (zw)w = z(ww), hence z(z−1w) = w and (zw)w −1 = z. These facts can be checked by somewhat tedious direct calculation. More elegant derivations can be found in Chapter 8 of [Ebbinghaus 1991]. There is an octonion projective plane OP2 obtained by attaching a cell e16 to S 8 via the
Hopf map S 15→S 8, just as CP2 and HP2 are obtained from the other Hopf maps. However, there is no octonion analog of RPn, CPn, and HPn for n > 2 since associativity of multiplication is needed for the relation (z0, ···, zn) ∼ λ(z0, ···, zn) to be an equivalence relation. There are no ﬁber bundles with ﬁber, total space, and base space spheres of other dimensions than in these Hopf bundle examples. This is discussed in an exercise for §4.D, which reduces the question to the famous ‘Hopf invariant one’ problem. Proposition 4.48. A ﬁber bundle p : E→B has the homotopy lifting property with respect to all CW pairs (X, A). A theorem of Huebsch and Hurewicz proved in §2.7 of [Spanier 1966] says that ﬁber bundles over paracompact base spaces are ﬁbrations, having the homotopy lift- ing property with respect to all spaces. This stronger result is not often needed in algebraic topology, however. Proof: As noted earlier, the homotopy lifting property for CW pairs is equivalent to the homotopy lifting property for disks, or equivalently, cubes. Let G : In × I→B, G(x, t) = gt(x), be a homotopy we wish to lift, starting with a given lift g0 of g0. Choose an open cover {Uα} of B with local trivializations hα : p−1(Uα)→Uα × F. Using compactness of In × I, we may subdivide In into small cubes C and I into intervals Ij = [tj, tj+1] so that each product C × Ij is mapped by G into a single Uα. We may gt has already been constructed over ∂C for each of assume by induction on n that gt over a cube C we may proceed in stages, constructe gt for t in each successive interval Ij. This in eﬀect reduces us to the case that ing no subdivision of In × I is necessary, so G maps all of In × I to a single Uα. Then we e G with the local trivialization have G
(In × {0} ∪ ∂In × I) ⊂ p−1(Uα), and composing the subcubes C. To extend this e e e e 380 Chapter 4 Homotopy Theory hα reduces us to the case of a product bundle Uα × F. In this case the ﬁrst coordinate gt is just the given gt, so only the second coordinate needs to be constructed. of a lift This can be obtained as a composition In × I→In × {0} ∪ ∂In × I→F where the ﬁrst ⊔⊓ map is a retraction and the second map is what we are given. e Example 4.49. Applying this theorem to a covering space p : E→B with E and B path-connected, and discrete ﬁber F, the resulting long exact sequence of homotopy groups yields Proposition 4.1 that p∗ : πn(E)→πn(B) is an isomorphism for n ≥ 2. We also obtain a short exact sequence 0→π1(E)→π1(B)→π0(F )→0, consistent with the covering space theory facts that p∗ : π1(E)→π1(B) is injective and that the ﬁber F can be identiﬁed, via path-lifting, with the set of cosets of p∗π1(E) in π1(B). Example 4.50. From the bundle S 1→S ∞→CP∞ we obtain πi(CP∞) ≈ πi−1(S 1) for all i since S ∞ is contractible. Thus CP∞ is a K(Z, 2). In similar fashion the bundle S 3→S ∞→HP∞ gives πi(HP∞) ≈ πi−1(S 3) for all i, but these homotopy groups are far more complicated than for CP∞ and S 1. In particular, HP∞ is not a K(Z, 4). Example 4.51. The long exact sequence for the Hopf bundle S 1→S 3→S 2 gives isomorphisms π2(S 2) ≈ π1(S 1) and πn(S
3) ≈ πn(S 2) for all n ≥ 3. Taking n = 3, we see that π3(S 2) is inﬁnite cyclic, generated by the Hopf map S 3→S 2. From this example and the preceding one we see that S 2 and S 3 × CP∞ are simplyconnected CW complexes with isomorphic homotopy groups, though they are not homotopy equivalent since they have quite diﬀerent homology groups. Example 4.52: Whitehead Products. Let us compute π3( free abelian with basis consisting of the Hopf maps S 3→S 2 attaching maps S 3→S 2 β ⊂ all unordered pairs α ≠ β. α of the cells e2 α S 2 W α ∨ S 2 α × e2 α S 2 α ⊂ W α), showing that it is α S 2 α together with the β in the products S 2 β for W α × S 2 Suppose ﬁrst that there are only ﬁnitely many summands S 2 α Xα)→πn( α Xα of path-connected spaces, the map πn( α Xα) ≈ W α. For a ﬁnite prodα Xα) induced by uct α πn(Xα) is generated by the inclusion is surjective since the group πn( subgroups πn(Xα). Thus the long exact sequence of homotopy groups for the pair ( α Xα) breaks up into short exact sequences α Xα, L Q Q Q Q W α Xα, 0 -→ πn+1( α Xα) -→ πn( W These short exact sequences split since the inclusions Xα ֓ πn(Xα)→πn( α Xα) and hence a splitting homomorphism Taking Xα = S 2 α and n = 3, we get an isomorphism W α Xα) -→ πn( W L Q Q α Xα) -→ 0 α Xα induce maps α Xα). α πn(Xα)→πn( W W α) ≈ π4( α S 2 W α S 2 α π3(S 2 π3( α, α) α) is free with basis the
Hopf maps S 3→S 2 α S 2 W α) ⊕ L Q α π3(S 2 The factor ample. For the other factor we have π4( tion 4.28. The quotient L α S 2 α/ α S 2 α, α S 2 α has 5 skeleton a wedge of spheres S 4 W α S 2 W α) ≈ π4( Q Q α by the preceding exα S 2 α) by Proposiα/ αβ for α ≠ β, α S 2 W Q Elementary Methods of Calculation Section 4.2 381 Q α S 2 α S 2 α/ α S 2 α, W αβ S 4 W α S 2 α, α) ≈ π4( so π4( α S 2 Hence π4( α)→π3( Via the injection ∂ : π4( Q W of the cells e2 α × e2 β form a basis for the summand Im ∂ of π3( Q proof for the case of ﬁnitely many summands S 2 follows immediately since any map S 3→ union of summands, and similarly for any homotopy between such maps. αβ S 4 αβ) is free with basis the inclusions S 4 αβ. α × e2 α) is free with basis the characteristic maps of the 4 cells e2 β. W α) this means that the attaching maps α). This ﬁnishes the α. The case of inﬁnitely many S 2 α ’s α has compact image, lying in a ﬁnite αβ ֓ The maps S 3→S 2 α ∨ S 2 β in this example are expressible in terms of a product in homotopy groups called the Whitehead product, deﬁned as follows. Given basepointpreserving maps f : S k→X and g : S ℓ→X, let [f, g] : S k+ℓ−1→X be the composition S k+ℓ−1 -→ S k ∨ S ℓ f ∨g------------→ X where the ﬁrst map is the attaching map of the (k + ℓ) cell of S k × S ℓ with its usual CW structure. Since homotopies of f or g give rise to hom
otopies of [f, g], we have a well-deﬁned product πk(X)× πℓ(X)→πk+ℓ−1(X). The notation [f, g] is used since for k = ℓ = 1 this is just the commutator product in π1(X). It is an exercise to show that when k = 1 and ℓ > 1, [f, g] is the diﬀerence between g and its image under the π1 action of f. In these terms the map S 3→S 2 β in the preceding example is the Whitehead product [iα, iβ] of the two inclusions of S 2 into S 2 β. Another example of a Whitehead product we have encountered previously is [11, 11] : S 2n−1→S n, which is the attaching map of the 2n cell of the space J(S n) considered in §3.2. α ∨ S 2 α ∨ S 2 The calculation of π3( α) is the ﬁrst nontrivial case of a more general theorem of Hilton calculating all the homotopy groups of any wedge sum of spheres in α S 2 W terms of homotopy groups of spheres, using Whitehead products. A further general- ization by Milnor extends this to wedge sums of suspensions of arbitrary connected CW complexes. See [Whitehead 1978] for an exposition of these results and further information on Whitehead products. Example 4.53: Stiefel and Grassmann Manifolds. The ﬁber bundles with total space a sphere and base space a projective space considered above are the cases n = 1 of families of ﬁber bundles in each of the real, complex, and quaternionic cases: O(n) -→Vn(Rk) -→ Gn(Rk) U(n) -→Vn(Ck) -→ Gn(Ck) Sp(n) -→Vn(Hk) -→ Gn(Hk) O(n) -→ Vn(R∞) -→ Gn(R∞) U(n) -→ Vn(C∞) -→ Gn(C∞) Sp(n) -→ Vn(H∞) -→ Gn(H�
�) Taking the real case ﬁrst, the Stiefel manifold Vn(Rk) is the space of n frames in Rk, that is, n tuples of orthonormal vectors in Rk. This is topologized as a subspace of the product of n copies of the unit sphere in Rk. The Grassmann manifold Gn(Rk) is the space of n dimensional vector subspaces of Rk. There is a natural surjection p : Vn(Rk)→Gn(Rk) sending an n frame to the subspace it spans, and Gn(Rk) is topologized as a quotient space of Vn(Rk) via this projection. The ﬁbers of the map 382 Chapter 4 Homotopy Theory p are the spaces of n frames in a ﬁxed n plane in Rk and so are homeomorphic to Vn(Rn). An n frame in Rn is the same as an orthogonal n× n matrix, regarding the columns of the matrix as an n frame, so the ﬁber can also be described as the orthogonal group O(n). There is no diﬃculty in allowing k = ∞ in these deﬁnitions, and in fact Vn(R∞) = k Vn(Rk) and Gn(R∞) = k Gn(Rk). S The complex and quaternionic Stiefel manifolds and Grassmann manifolds are deﬁned in the same way using the usual Hermitian inner products in Ck and Hk. The unitary group U(n) consists of n× n matrices whose columns form orthonormal bases for Cn, and the symplectic group Sp(n) is the quaternionic analog of this. S We should explain why the various projection maps Vn→Gn are ﬁber bundles. Let us take the real case for concreteness, though the argument is the same in all cases. If we ﬁx an n plane P ∈ Gn(Rk) and choose an orthonormal basis for P, then we obtain continuously varying orthonormal bases for all n planes P ′ in a neighborhood U of P by projecting the basis for P orthogonally onto P ′ to obtain a nonorthonormal basis for P ′
, then applying the Gram–Schmidt process to this basis to make it orthonormal. The formulas for the Gram–Schmidt process show that it is continuous. Having orthonormal bases for all n planes in U, we can use these to identify these n planes with Rn, hence n frames in these n planes are identiﬁed with n frames in Rn, and so p−1(U) is identiﬁed with U × Vn(Rn). This argument works for k = ∞ as well as for ﬁnite k. In the case n = 1 the total spaces V1 are spheres, which are highly connected, and the same is true in general: Vn(Rk) is (k − n − 1) connected. Vn(Ck) is (2k − 2n) connected. Vn(Hk) is (4k − 4n + 2) connected. Vn(R∞), Vn(C∞), and Vn(H∞) are contractible. The ﬁrst three statements will be proved in the next example. For the last statement the argument is the same in the three cases, so let us consider the real case. Deﬁne a homotopy ht : R∞→R∞ by ht(x1, x2, ···) = (1−t)(x1, x2, ···)+t(0, x1, x2, ···). This is linear for each t, and its kernel is easily checked to be trivial. So if we apply ht to an n frame we get an n tuple of independent vectors, which can be made orthonormal by the Gram–Schmidt formulas. Thus we have a deformation retraction, in the weak sense, of Vn(R∞) onto the subspace of n frames with ﬁrst coordinate zero. Iterating this n times, we deform into the subspace of n frames with ﬁrst n coordinates zero. For such an n frame (v1, ···, vn) deﬁne a homotopy (1−t)(v1, ···, vn)+t(e1, ···, en) where ei is the ith standard basis vector in R∞. This homotopy preserves linear
independence, so after again applying Gram–Schmidt we have a deformation through n frames, which ﬁnishes the construction of a contraction of Vn(R∞). Since Vn(R∞) is contractible, we obtain isomorphisms πiO(n) ≈ πi+1Gn(R∞) for all i and n, and similarly in the complex and quaternionic cases. Elementary Methods of Calculation Section 4.2 383 Example 4.54. For m < n ≤ k there are ﬁber bundles Vn−m(Rk−m) -→ Vn(Rk) p-----→ Vm(Rk) where the projection p sends an n frame onto the m frame formed by its ﬁrst m vectors, so the ﬁber consists of (n − m) frames in the (k − m) plane orthogonal to a given m frame. Local trivializations can be constructed as follows. For an m frame F, choose an orthonormal basis for the (k − m) plane orthogonal to F. This determines orthonormal bases for the (k − m) planes orthogonal to all nearby m frames by orthogonal projection and Gram–Schmidt, as in the preceding example. This allows us to identify these (k−m) planes with Rk−m, and in particular the ﬁbers near p−1(F ) are identiﬁed with Vn−m(Rk−m), giving a local trivialization. There are analogous bundles in the complex and quaternionic cases as well, with local triviality shown in the same way. Restricting to the case m = 1, we have bundles Vn−1(Rk−1)→Vn(Rk)→S k−1 whose associated long exact sequence of homotopy groups allows us deduce that Vn(Rk) is (k − n − 1) connected by induction on n. In the complex and quaternionic cases the same argument yields the other connectivity statements in the preceding example. Taking k = n we obtain ﬁber bundles O(k − m)→O(k)→Vm(Rk). The ﬁbers are in fact just the cosets αO(k − m) for α ∈ O(k), where O(
k − m) is regarded as the subgroup of O(k) ﬁxing the ﬁrst m standard basis vectors. So we see that Vm(Rk) is identiﬁable with the coset space O(k)/O(k − m), or in other words the orbit space for the free action of O(k −m) on O(k) by right-multiplication. In similar fashion one can see that Gm(Rk) is the coset space O(k)/ the subgroup O(m)× O(k − m) ⊂ O(k) consists of the orthogonal transformations O(m)× O(k − m) where taking the m plane spanned by the ﬁrst m standard basis vectors to itself. The corresponding observations apply also in the complex and quaternionic cases, with the unitary and symplectic groups. Example 4.55: Bott Periodicity. Specializing the preceding example by taking m = 1 and k = n we obtain bundles O(n − 1) -→ O(n) U(n − 1) -→ U(n) Sp(n − 1) -→ Sp(n) p-----→ S n−1 p-----→ S 2n−1 p-----→ S 4n−1 The map p can be described as evaluation of an orthogonal, unitary, or symplectic transformation on a ﬁxed unit vector. These bundles show that computing homotopy groups of O(n), U(n), and Sp(n) should be at least as diﬃcult as computing homo- topy groups of spheres. For example, if one knew the homotopy groups of O(n) and O(n − 1), then from the long exact sequence of homotopy groups for the ﬁrst bundle one could say quite a bit about the homotopy groups of S n−1. 384 Chapter 4 Homotopy Theory The bundles above imply a very interesting stability property. In the real case, the inclusion O(n−1)֓O(n) induces an isomorphism on πi for i < n−2, from the long exact sequence of the ﬁrst bundle. Hence the groups πiO(n) are independent of n if n is suﬃciently large
, and the same is true for the groups πiU(n) and πiSp(n) via the other two bundles. One of the most surprising results in all of algebraic topology is the Bott Periodicity Theorem which asserts that these stable groups repeat periodically, with a period of eight for O and Sp and a period of two for U. Their values are given in the following table: i mod 8 πiO(n) πiU(n) πiSp(n Z2 Z2 Z2 Z2 0 Z 0 5 0 Z Z 0 Stable Homotopy Groups We showed in Corollary 4.24 that for an n connected CW complex X, the suspension map πi(X)→πi+1(SX) is an isomorphism for i < 2n + 1. In particular this holds for i ≤ n so SX is (n + 1) connected. This implies that in the sequence of iterated suspensions πi(X) -→ πi+1(SX) -→ πi+2(S 2X) -→ ··· all maps are eventually isomorphisms, even without any connectivity assumption on X itself. The resulting stable homotopy group is denoted π s i (X). An especially interesting case is the group π s n > i + 1. This stable homotopy group is often abbreviated to π s stable i stem. It is a theorem of Serre which we prove in [SSAT] that π s i i (S 0), which equals πi+n(S n) for i and called the is always ﬁnite for i > 0. These stable homotopy groups of spheres are among the most fundamental ob- jects in topology, and much eﬀort has gone into their calculation. At the present time, complete calculations are known only for i up to around 60 or so. Here is a table for i ≤ 19, taken from [Toda 1962]: Patterns in this apparent chaos begin to emerge only when one projects π s i onto its p components, the quotient groups obtained by factoring out all elements of order relatively prime to the prime p. For i > 0 the p component pπ s morphic to the subgroup of π s quotient viewpoint is in some ways preferable. is of course isoi consisting of elements
of order a power of p, but the i Elementary Methods of Calculation Section 4.2 385 The ﬁgure below is a schematic diagram of the 2 components of π s i for i ≤ 60. A vertical chain of n dots in the ith column represents a Z2n summand of π s i. The bottom dot of such a chain denotes a generator of this summand, and the vertical segments denote multiplication by 2, so the second dot up is twice a generator, the next dot is four times a generator, and so on. The three generators η, ν, and σ in dimensions 1, 3, and 7 are represented by the Hopf bundle maps S 3→S 2, S 7→S 4, S 15→S 8 deﬁned in Examples 4.45, 4.46, and 4.47. Some of the other elements also have standard names indicated by the Greek letter labels. The other line segments in the diagram provide some information about compoi+j deﬁned sitions of maps between spheres. Namely, there are products π s by compositions S i+j+k→S j+k→S k. j →π s i × π s Proposition 4.56. The composition products π s i π s structure on π s α ∈ π s ∗ = i and β ∈ π s j. L i+j induce a graded ring i satisfying the commutativity relation αβ = (−1)ij βα for j →π s i × π s This will be proved at the end of this subsection. It follows that pπ s of the p components pπ s In 2π s property. ∗, the direct sum i, is also a graded ring satisfying the same commutativity ∗ many of the compositions with suspensions of the Hopf maps η and ν are nontrivial, and these nontrivial compositions are indicated in the diagram by segments extending 1 unit to the right and diagonally upward for η or 3 units to the right, usually horizontally, for ν. Thus for example we see the relation η3 = 4ν in 2π s 3 ≈ Z24, where the actual relation is η3 = 12ν since 2η = 0 implies 2η3 = 0, so η3 is the unique element of order two in this Z24. 3 �
� Z8 is a quotient of π s 3. Remember that 2π s Across the bottom of the diagram there is a repeated pattern of pairs of ‘teeth’. This pattern continues to inﬁnity, though with the spikes in dimensions 8k − 1 not all of the same height, namely, the spike in dimension 2m(2n + 1) − 1 has height m + 1. i for i ≤ 100. Here vertical segments denote multiplication by 3 and the other solid segments denote composition with elements α1 ∈ 3π s 10. The meaning of the dashed lines will be The next diagram shows the 3 components of π s 3 and β1 ∈ 3π s 386 Chapter 4 Homotopy Theory explained later. The most regular part of the diagram is the ‘triadic ruler’ across the bottom. This continues in the same pattern forever, with spikes of height m + 1 in dimension 4k − 1 for 3m the highest power of 3 dividing 4k. Looking back at the p = 2 diagram, one can see that the vertical segments of the ‘teeth’ form a ‘dyadic ruler’. The case p = 5 is shown in the next diagram. Again one has the inﬁnite ruler, this time a ‘pentadic’ ruler. The four dots with question marks below them near the right edge of the diagram are hypothetical since it is still undecided whether these potential elements of 5π s i for i = 932, 933, 970, and 971 actually exist. These three diagrams are based on calculations described in [Isaksen 2014] for p = 2 and [Ravenel 1986] for p = 3, 5. Elementary Methods of Calculation Section 4.2 387 4k−1 known as Im J, the image of a homomorphism J : π4k−1(O)→π s For each p there is a similar inﬁnite ‘ p adic ruler’, corresponding to cyclic subgroups of order pm+1 in pπ s 2j(p−1)−1 for all j, where pm is the highest power of p dividing j. These subgroups are the p components of a certain cyclic subgroup of π s 4k−1. There are also Z2 subgroups of π s i for i = 8k, 8k + 1 forming Im J in
these dimensions. In the diagram of 2π s ∗ these are the parts of the teeth connected to the spike in dimension 8k − 1. The J homomorphism will be studied in some detail in [VBKT]. ∗ include classes ηn ∈ 2π s 2n for n ≥ 4, βn ∈ pπ s 2(p3−1)n−2p2−2p+1 for p ≥ 7. The element βn appears in the diagram for p = 5 as the dot in the upper part of the diagram labeled by the number n. These βn ’s generate the strips along the upward diagonal, except when n is a multiple of 5 and the strip is generated by β2βn−1 rather than βn. There are also elements βn for certain fractional values of n. The element γ2 generates the long strip starting in dimension 437, but γ3 = 0. The element γ4 in dimension 933 is one of the question marks. The other known inﬁnite families in π s 2(p2−1)n−2p for p ≥ 5, and γn ∈ pπ s In π s ∗ there are many compositions which are zero. One can get some idea of this from the diagrams above, where all sequences of edges break oﬀ after a short time. As a special instance of the vanishing of products, the commutativity formula in Proposition 4.56 implies that the square of an odd-dimensional element of odd order is zero. More generally, a theorem of Nishida says that every positive-dimensional ∗ is nilpotent, with αn = 0 for some n. For example, for the element element α ∈ π s β1 ∈ 5π s 38 the smallest such n is 18. ∗. Suppose one has maps W The widespread vanishing of products in π s ∗ can be seen as limiting their usefulness in describing the structure of π s ∗. But it can also be used to construct new h-----→ Z such that the comelements of π s positions gf and hg are both homotopic to constant maps. A nullhomotopy of gf gives an extension of gf to a map F : CW→Y, and a nullhomotopy of hg gives an extension of hg to a map G : CX→Z. Regarding the
suspension SW as the union of two cones CW, deﬁne the Toda bracket hf, g, hi : SW→Z to be the composition G(Cf ) on one cone and hF on the other. f-----→ X g-----→ Y The map hf, g, hi is not uniquely determined by f, g, and h since it depends on the choices of the nullhomotopies. In the case of π s ∗, the various choices of hf, g, hi range over a coset of a certain subgroup, described in an exercise at the end of the section. 388 Chapter 4 Homotopy Theory There are also higher-order Toda brackets hf1, ···, fni deﬁned in somewhat similar fashion. The dashed lines in the diagrams of 3π s ∗ join an element x to a bracket element hα1, ···, α1, xi. Many of the elements in all three diagrams can be expressed in terms of brackets. For example, in 2π s ∗ the 8 dimensional element ν is hν, η, νi. This element is also equal to ησ + ε where ε = hν 2, 2, ηi = h2, η, ν, η2i. ∗ are η4 = hσ 2, 2, ηi, ν4 = h2σ, σ, νi = −hσ, ν, σ i, Some other bracket formulas in 2π s σ = hν, σ, ησ i, θ4 = hσ, 2σ, σ, 2σ i, and η5 = hη, 2, θ4i. ∗ and 5π s Proof of 4.56: Only distributivity and commutativity need to be checked. One distributivity law is easy: Given f, g : S i+j+k→S j+k and h : S j+k→S k, then h(f + g) = hf + hg since both expressions equal hf and hg on the two hemispheres of S i+j+k. The other distributivity law will follow from this one and the commutativity relation. To prove the commutativity relation it will be convenient to express suspension in terms of smash product. The
smash product S n ∧ S 1 can be regarded as the quotient space of S n × I with S n × ∂I ∪ {x0}× I collapsed to a point. This is the same as the quotient of the suspension S n+1 of S n obtained by collapsing to a point the suspension of x0. Collapsing this arc in S n+1 to a point again yields S n+1, so we obtain in this way a homeomorphism identifying S n ∧ S 1 with S n+1. Under this identiﬁcation the suspension Sf of a basepoint-preserving map f : S n→S n becomes the smash product f ∧ 11 : S n ∧ S 1→S n ∧ S 1. By iteration, the k fold suspension S kf then corresponds to f ∧ 11 : S n ∧ S k→S n ∧ S k. Now we verify the commutativity relation. Let f : S i+k→S k and g : S j+k→S k be given. We may assume k is even. Consider the commutative diagram below, where σ and τ transpose the two factors. Thinking of S j+k and S k as smash products of circles, σ is the composition of k(j + k) transpositions of adjacent circle factors. Such a transposition has degree −1 since it is realized as a reﬂection of the S 2 = S 1 ∧ S 1 involved. Hence σ has degree (−1)k(j+k), which is +1 since k is even. Thus σ is homotopic to the identity. Similarly, τ is homotopic to the identity. Hence f ∧g = (11∧g)(f ∧11) is homotopic to the composition (g ∧11)(f ∧11), which is stably equivalent to the composition gf. Symmetrically, f g is stably homotopic to g ∧ f. So it suﬃces to show f ∧ g ≃ (−1)ij g ∧ f. This we do by the commutative diagram at the right, where σ and τ are again the transpositions of the two factors. As before, τ is homotopic to the identity, but now σ has degree (−1)(i+k
)(j+k), which equals (−1)ij since k is even. The composition (g ∧ f )σ is homotopic to (−1)ij (g∧f ) since additive inverses in homotopy groups are obtained by precomposing with a reﬂection, of degree −1. Thus from the commutativity of the diagram we obtain the relation f ∧ g ≃ (−1)ij g ∧ f. ⊔⊓ Elementary Methods of Calculation Section 4.2 389 Exercises 1. Use homotopy groups to show there is no retraction RPn→RPk if n > k > 0. 2. Show the action of π1(RPn) on πn(RPn) ≈ Z is trivial for n odd and nontrivial for n even. k πn( be the map which is the identity on the S 2 3. Let X be obtained from a lens space of dimension 2n + 1 by deleting a point. Compute π2n(X) as a module over Z[π1(X)]. 4. Let X ⊂ Rn+1 be the union of the inﬁnite sequence of spheres S n k of radius 1/k and center (1/k, 0, ···, 0). Show that πi(X) = 0 for i < n and construct a homomorphism from πn(X) onto β→S 2 5. Let summand and which on the S 2 β summand is the sum of the identity map and a homeomorphism β→S 2 S 2 β)× I under the identiﬁcations (x, 0) ∼ (f (x), 1). The mapping torus of the restriction of f to S 2 α α ⊂ X. Show that the maps π2(A)→π2(X)→π2(X, A) forms a subspace A = S 1 × S 2 form a short exact sequence 0→Z→Z ⊕ Z→Z→0, and compute the action of π1(A) on these three groups. In particular, show the action of π1(A) is trivial on π2(A) and π2(X, A) but is nontrivial on π2(X). α.
Let X be the mapping torus of f, the quotient space of (S 2 α ∨ S 2 6. Show that the relative form of the Hurewicz theorem in dimension n implies the absolute form in dimension n − 1 by considering the pair (CX, X) where CX is the cone on X. 7. Construct a CW complex X with prescribed homotopy groups πi(X) and prescribed actions of π1(X) on the πi(X) ’s. 8. Show the suspension of an acyclic CW complex is contractible. 9. Show that a map between simply-connected CW complexes is a homotopy equiva- lence if its mapping cone is contractible. Use the preceding exercise to give an example p(t). Show π ′ where this fails in the nonsimply-connected case. 10. Let the CW complex X be obtained from S 1 ∨ S n, n ≥ 2, by attaching a cell en+1 by a map representing the polynomial p(t) ∈ Z[t, t−1] ≈ πn(S 1 ∨ S n), so πn(X) ≈ Z[t, t−1]/ n(X) is cyclic and compute its order in terms of p(t). Give examples showing that the group πn(X) can be ﬁnitely generated or not, independently of whether π ′ 11. Let X be a connected CW complex with 1 skeleton X 1. Show that π2(X, X 1) ≈ π2(X)× K where K is the kernel of π1(X 1)→π1(X), a free group. Show also that the 2(X)→π ′ map π ′ 2(X, X 1) need not be injective by considering the case X = RP2 with its standard CW structure. 12. Show that a map f : X→Y of connected CW complexes is a homotopy equivalence if it induces an isomorphism on π1 and if a lift Y to the universal covers induces an isomorphism on homology. [The latter condition can be restated in terms of n(X) is ﬁnite or inﬁnite. X→ f : e e e 390 Chapter 4 Hom
otopy Theory homology with local coeﬃcients as saying that f∗ : H∗(X; Z[π1X])→H∗(Y ; Z[π1Y ]) is an isomorphism; see §3.H.] 13. Show that a map between connected n dimensional CW complexes is a homotopy equivalence if it induces an isomorphism on πi for i ≤ n. [Pass to universal covers and use homology.] 14. If an n dimensional CW complex X contains a subcomplex Y homotopy equivalent to S n, show that the map πn(Y )→πn(X) induced by inclusion is injective. [Use the Hurewicz homomorphism.] 15. Show that a closed simply-connected 3 manifold is homotopy equivalent to S 3. [Use Poincar´e duality, and also the fact that closed manifolds are homotopy equiva- lent to CW complexes, from Corollary A.12 in the Appendix. The stronger statement that a closed simply-connected 3 manifold is homeomorphic to S 3 is the Poincar´e conjecture, ﬁnally proved by Perelman. The higher-dimensional analog, that a closed n manifold homotopy equivalent to S n is homeomorphic to S n, had been proved earlier for all n ≥ 4.] 16. Show that the closed surfaces with inﬁnite fundamental group are K(π, 1) ’s by showing that their universal covers are contractible, via the Hurewicz theorem and results of §3.3. 17. Show that the map hX, Y i→Hom, [f ] ֏ f∗, is a bijection if X is an (n−1) connected CW complex and Y is a path-connected space with πi(Y ) = 0 for i > n. Deduce that CW complex K(G, n) ’s are uniquely determined, up to homotopy πn(X), πn(Y ) type, by G and n. 18. If X and Y are simply-connected CW complexes such that Hj(Y ) are ﬁnite and of relatively prime orders for all pairs (i, j), show that the inclusion X ∨ Y ֓ X × Y is a homotopy equivalence and X ∧
Y is contractible. [Use the K¨unneth formula.] 19. If X is a K(G, 1) CW complex, show that πn(X n) is free abelian for n ≥ 2. 20. Let G be a group and X a simply-connected space. Show that for the product K(G, 1)× X the action of π1 on πn is trivial for all n > 1. Hi(X) and e e 21. Given a sequence of CW complexes K(Gn, n), n = 1, 2, ···, let Xn be the CW complex formed by the product of the ﬁrst n of these K(Gn, n) ’s. Via the inclusions Xn−1 ⊂ Xn coming from regarding Xn−1 as the subcomplex of Xn with nth coordinate equal to a basepoint 0 cell of K(Gn, n), we can then form the union of all the Xn ’s, a CW complex X. Show πn(X) ≈ Gn for all n. 22. Show that Hn+1(K(G, n); Z) = 0 if n > 1. [Build a K(G, n) from a Moore space M(G, n) by attaching cells of dimension > n + 1.] 23. Extend the Hurewicz theorem by showing that if X is an (n − 1) connected CW complex, then the Hurewicz homomorphism h : πn+1(X)→Hn+1(X) is surjective Elementary Methods of Calculation Section 4.2 391 K(π1(X), 1) when n > 1, and when n = 1 show there is an isomorphism H2(X)/h ≈. [Build a K(πn(X), n) from X by attaching cells of dimension n + 2 H2 and greater, and then consider the homology sequence of the pair (Y, X) where Y is X with the (n + 2) cells of K(πn(X), n) attached. Note that the image of the boundary map Hn+2(Y, X)→Hn+1(X) coincides with the image of h, and Hn+1(Y ) ≈ Hn+1.
The previous exercise is needed for the case n > 1.] π2(X) ≈ G iﬀ H2(K(G, 1); Z) = 0. 24. Show there is a Moore space M(G, 1) with π1 [Use the preceding problem. Build such an M(G, 1) from the 2 skeleton K2 of a K(G, 1) by attaching 3 cells according to a basis for the free group H2(K2; Z).] In particular, there is no M(Zn, 1) with fundamental group Zn, free abelian of rank n, if n ≥ 2. M(G, 1) K(πn(X), n) ≈ Hn X, then X and πn(X), where h is the Hurewicz map. 25. For X a connected CW complex with πi(X) = 0 for 1 < i < n for some n ≥ 2, K(π1(X), 1) show that Hn(X)/h 26. Generalizing the example of RP2 and S 2 × RP∞, show that if X is a connected X × K(π1(X), 1) ﬁnite-dimensional CW complex with universal cover have isomorphic homotopy groups but are not homotopy equivalent if π1(X) contains elements of ﬁnite order. 27. Show that the image of the map π2(X, x0)→π2(X, A, x0) lies in the center of π2(X, A, x0). (This exercise should be in §4.1.) 28. Show that the group Zp × Zp with p prime cannot act freely on any sphere S n, by ﬁlling in details of the following argument. Such an action would deﬁne a covering space S n→M with M a closed manifold. When n > 1, build a K(Zp × Zp, 1) from M by attaching a single (n + 1) cell and then cells of higher dimension. Deduce that H n+1(K(Zp × Zp, 1); Zp) is Zp or 0, a contradiction. (The case n = 1 is more elementary.) e e 1 ··· ℓ′ 1, ···, �
��′ 29. Finish the homotopy classiﬁcation of lens spaces begun in Exercise 2 of §3.E by showing that two lens spaces Lm(ℓ1, ···, ℓn) and Lm(ℓ′ n) are homotopy equivalent if ℓ1 ··· ℓn ≡ ±knℓ′ n mod m for some integer k, via the following steps: (a) Reduce to the case k = 1 by showing that Lm(ℓ′ n) if k is relatively prime to m. [Rechoose the generator of the Zm action on S 2n−1.] (b) Let f : L→L′ be a map constructed as in part (b) of the exercise in §3.E. Construct a map g : L→L′ as a composition L -→ L ∨ S 2n−1 -→ L ∨ S 2n−1 -→ L′ where the ﬁrst map collapses the boundary of a small ball to a point, the second map is the wedge of the identity on L and a map of some degree d on S 2n−1, and the third map is f on L and the projection S 2n−1→L′ on S 2n−1. Show that g has degree k1 ··· kn + dm, that is, g induces multiplication by k1 ··· kn + dm on [Show ﬁrst that a lift of g to the universal cover S 2n−1 has this H2n−1(−; Z). degree.] n) = Lm(kℓ′ 1, ···, kℓ′ 1, ···, ℓ′ 392 Chapter 4 Homotopy Theory (c) If ℓ1 ··· ℓn ≡ ±ℓ′ 1 ··· ℓ′ n mod m, choose d so that k1 ··· kn + dm = ±1 and show this implies that g induces an isomorphism on all homotopy groups, hence is a homotopy equivalence. [For πi with i > 1, consider a lift of g to the universal cover.] 30. Let E be a subspace of R2 obtained by deleting a subspace of {0}× R. For which such spaces E is the projection E→
R, (x, y) ֏ x, a ﬁber bundle? 31. For a ﬁber bundle F→E→B such that the inclusion F ֓ E is homotopic to a constant map, show that the long exact sequence of homotopy groups breaks up into split short exact sequences giving isomorphisms πn(B) ≈ πn(E) ⊕ πn−1(F ). In particular, for the Hopf bundles S 3→S 7→S 4 and S 7→S 15→S 8 this yields isomorphisms πn(S 4) ≈ πn(S 7) ⊕ πn−1(S 3) πn(S 8) ≈ πn(S 15) ⊕ πn−1(S 7) Thus π7(S 4) and π15(S 8) contain Z summands. 32. Show that if S k→Sm→S n is a ﬁber bundle, then k = n − 1 and m = 2n − 1. [Look at the long exact sequence of homotopy groups.] 33. Show that if there were ﬁber bundles S n−1→S 2n−1→S n for all n, then the groups πi(S n) would be ﬁnitely generated free abelian groups computable by induction, and nonzero for i ≥ n ≥ 2. 34. Let p : S 3→S 2 be the Hopf bundle and let q : T 3→S 3 be the quotient map collapsing the complement of a ball in the 3 dimensional torus to a point. Show that pq : T 3→S 2 induces the trivial map on π∗ and H∗, but is not homotopic to a constant map. 35. Show that the ﬁber bundle S 3→S 4n+3→HPn gives rise to a quotient ﬁber bundle S 2→CP2n+1→HPn by factoring out the action of S 1 on S 4n+3 by complex scalar multiplication. 36. For basepoint-preserving maps f : S 1→X and g : S n→X with n > 1, show that the Whitehead product [f, g] is ±(g − f g)
, where f g denotes the action of f on g. e 37. Show that all Whitehead products in a path-connected H–space are trivial. 38. Show π3(S 1 ∨S 2) is not ﬁnitely generated as a module over Z[π1(S 1 ∨S 2)] by considering Whitehead products in the universal cover, using the results in Example 4.52. Generalize this to πi+j−1(S 1 ∨ S i ∨ S j) for i, j > 1. 39. Show that the indeterminacy of a Toda bracket hf, g, hi with f ∈ π s h ∈ π s k is the subgroup f π s j+k+1 + h π s i+j+1 of +j+k+1. Connections with Cohomology Section 4.3 393 The Hurewicz theorem provides a strong link between homotopy groups and ho- mology, and hence also an indirect relation with cohomology. But there is a more di- rect connection with cohomology of a quite diﬀerent sort. We will show that for every CW complex X there is a natural bijection between H n(X; G) and the set hX, K(G, n)i of basepoint-preserving homotopy classes of maps from X to a K(G, n). We will also deﬁne a natural group structure on hX, K(G, n)i that makes the bijection a group iso- morphism. The mere fact that there is any connection at all between cohomology and homotopy classes of maps is the ﬁrst surprise here, and the second is that Eilenberg– MacLane spaces are involved, since their deﬁnition is entirely in terms of homotopy groups, which on the face of it have nothing to do with cohomology. After proving this basic isomorphism H n(X; G) ≈ hX, K(G, n)i and describing a few of its immediate applications, the later parts of this section aim toward a further study of Postnikov towers, which were introduced brieﬂy in §4.1. These provide a general theoretical method for realizing an arbitrary CW complex as a sort of twisted product of E
ilenberg–MacLane spaces, up to homotopy equivalence. The most ge- ometric interpretation of the phrase ‘twisted product’ is the notion of ﬁber bundle introduced in the previous section, but here we need the more homotopy-theoretic notion of a ﬁbration, so before we begin the discussion of Postnikov towers we ﬁrst take a few pages to present some basic constructions and results about ﬁbrations. As we shall see, Postnikov towers can be expressed as sequences of ﬁbrations with ﬁbers Eilenberg–MacLane spaces, so we can again expect close connections with cohomology. One such connection is provided by k invariants, which describe, at least in principle, how Postnikov towers for a broad class of spaces are determined by a sequence of cohomology classes. Another application of these ideas, described at the end of the section, is a technique for factoring basic extension and lifting prob- lems in homotopy theory into a sequence of smaller problems whose solutions are equivalent to the vanishing of certain cohomology classes. This technique goes under the somewhat grandiose title of Obstruction Theory, though it is really quite a simple idea when expressed in terms of Postnikov towers. The Homotopy Construction of Cohomology The main result of this subsection is the following fundamental relationship be- tween singular cohomology and Eilenberg–MacLane spaces: Theorem 4.57. There are natural bijections T : hX, K(G, n)i→H n(X; G) for all CW complexes X and all n > 0, with G any abelian group. Such a T has the form T ([f ]) = f ∗(α) for a certain distinguished class α ∈ H n(K(G, n); G). 394 Chapter 4 Homotopy Theory In the course of the proof we will deﬁne a natural group structure on hX, K(G, n)i such that the transformation T is an isomorphism. A class α ∈ H n(K(G, n); G) with the property stated in the theorem is called a fundamental class. The proof of the theorem will yield an explicit fundamental class, namely the element of H n(K(G, n); G) ≈ Hom(Hn(
K; Z), G) given by the inverse of the Hurewicz isomorphism G = πn(K(G, n))→Hn(K; Z). Concretely, if we choose K(G, n) to be a CW complex with (n − 1) skeleton a point, then a fundamental class is represented by the cellular cochain assigning to each n cell of K(G, n) the element of πn(K(G, n)) deﬁned by a characteristic map for the n cell. The theorem also holds with hX, K(G, n)i replaced by [X, K(G, n)], the non- basepointed homotopy classes. This is easy to see when n > 1 since every map X→K(G, n) can be homotoped to take basepoint to basepoint, and every homotopy between basepoint-preserving maps can be homotoped to be basepoint-preserving since the target space K(G, n) is simply-connected. When n = 1 it is still true that [X, K(G, n)] = hX, K(G, n)i for abelian G according to an exercise for §4.A. For n = 0 it is elementary that H 0(X; G) = [X, K(G, 0)] and H 0(X; G) = hX, K(G, 0)i. e It is possible to give a direct proof of the theorem, constructing maps and ho- motopies cell by cell. This provides geometric insight into why the result is true, but unfortunately the technical details of this proof are rather tedious. So we shall take a diﬀerent approach, one that has the advantage of placing the result in its natural context via general machinery that turns out to be quite useful in other situations as well. The two main steps will be the following assertions. (1) The functors hn(X) = hX, K(G, n)i deﬁne a reduced cohomology theory on the category of basepointed CW complexes. (2) If a reduced cohomology theory h∗ deﬁned on CW complexes has coeﬃcient groups hn(S 0) which are zero for n ≠ 0, then there are
natural isomorphisms hn(X) ≈ H n(X; h0(S 0)) for all CW complexes X and all n. e Towards proving (1) we will study a more general question: When does a sequence of spaces Kn deﬁne a cohomology theory by setting hn(X) = hX, Kni? Note that this will be a reduced cohomology theory since hX, Kni is trivial when X is a point. The ﬁrst question to address is putting a group structure on the set hX, Ki. This requires that either X or K have some special structure. When X = S n we have hS n, Ki = πn(K), which has a group structure when n > 0. The deﬁnition of this group structure works more generally whenever S n is replaced by a suspension SX, with the sum of maps f, g : SX→K deﬁned as the composition SX→SX ∨ SX→K where the ﬁrst map collapses an ‘equatorial’ X ⊂ SX to a point and the second map consists of f and g on the two summands. However, for this to make sense we must be talking about basepoint-preserving maps, and there is a problem with where to choose the basepoint in SX. If x0 is a basepoint of X, the basepoint of SX should be somewhere along the segment {x0}× I ⊂ SX, most likely either an endpoint or the Connections with Cohomology Section 4.3 395 midpoint, but no single choice of such a basepoint gives a well-deﬁned sum. The sum would be well-deﬁned if we restricted attention to maps sending the whole segment {x0}× I to the basepoint. This is equivalent to considering basepoint-preserving maps X→K where X = SX/({x0}× I) and the image of {x0}× I in X is taken to be If X is a CW complex with x0 a 0 cell, the quotient map SX→ the basepoint. Σ Σ is a homotopy equivalence since it collapses a contractible subcomplex of SX to a point, so we can identify hSX, Ki with h X is called the reduced X
, Ki. The space X Σ Σ suspension of X when we want to distinguish it from the ordinary suspension SX. Σ Σ It is easy to check that h X, Ki is a group with respect to the sum deﬁned above, inverses being obtained by reﬂecting the I coordinate in the suspension. However, what we would really like to have is a group structure on hX, Ki arising from a special Σ structure on K rather than on X. This can be obtained using the following basic adjoint relation: h X, Ki = hX, Ki where K is the space of loops in K at its chosen basepoint Ω Ω K. and the constant loop is taken as the basepoint of Σ K, called the loopspace of K, is topologized as a subspace of the space KI The space of all maps I→K, where KI is given the compact-open topology; see the Appendix for the deﬁnition and basic properties of this topology. The adjoint relation h X, Ki = X→K are exactly the same as hX, basepoint-preserving maps X→ K, the correspondence being given by associating to Ω X→K the family of loops obtained by restricting f to the images of the segments f : Ki holds because basepoint-preserving maps Ω Ω Σ Σ Ω {x}× I in X. Σ Σ Taking X = S n in the adjoint relation, we see that πn+1(K) = πn( K) for all n ≥ 0. Thus passing from a space to its loopspace has the eﬀect of shifting homotopy K(G, n) is a K(G, n − 1). This groups down a dimension. In particular we see that Ω fact will turn out to be important in what follows. Note that the association X ֏ X→ f : X→Y induces a map induces a homotopy f : Ω X is a functor: A basepoint-preserving map Y by composition with f. A homotopy f ≃ g Ω It is a theorem of [Milnor 1959] that the loopspace of a CW complex has the f ≃ Ω g, so it follows formally that
X ≃ Y implies Ω Ω X ≃ Y. homotopy type of a CW complex. This may be a bit surprising since loopspaces are Ω Ω Ω Ω usually quite large spaces, though of course CW complexes can be quite large too, in terms of the number of cells. What often happens in practice is that if a CW complex X has only ﬁnitely many cells in each dimension, then X is homotopy equivalent to a CW complex with the same property. We will see explicitly how this happens for X = S n in §4.J. Ω Composition of loops deﬁnes a map K, and this gives a sum operKi by setting (f + g)(x) = f (x) g(x), the composition of the loops ation in hX, K × K→ f (x) and g(x). Under the adjoint relation this is the same as the sum in h X, Ki Ω Ω Ω deﬁned previously. If we take the composition of loops as the sum operation then it Ω Σ 396 Chapter 4 Homotopy Theory is perhaps somewhat easier to see that hX, which shows that π1(K) is a group can be applied. Ω Ki is a group since the same reasoning Ω Since cohomology groups are abelian, we would like the group hX, Ki to be ( ( 2K = has a double loopspace K) and inductively an n fold loopspace abelian. This can be achieved by iterating the operation of forming loopspaces. One nK = n−1K). The evident bijection KY × Z ≈ (KY )Z is a homeomorphism for locally compact Hausdorﬀ spaces Y and Z, as shown in Proposition A.16 in the Appendix, Ω nK can be regarded as the space of maps and from this it follows by induction that In→K sending ∂In to the basepoint. Taking n = 2, we see that the argument that Ω 2Ki is an abelian group. Iterating π2(K) is abelian shows more generally that hX, nKi, so this is an abelian group for all the adjoint relation gives h nX, Ki
= hX Thus for a sequence of spaces Kn to deﬁne a cohomology theory hn(X) = hX, Kni we have been led to the assumption that each Kn should be a loopspace and in fact a double loopspace. Actually we do not need Kn to be literally a loopspace since it would suﬃce for it to be homotopy equivalent to a loopspace, as hX, Kni depends only on the homotopy type of Kn. In fact it would suﬃce to have just a weak homotopy equivalence Kn→ Ln for some space Ln since this would induce a bijection Lni by Proposition 4.22. In the special case that Kn = K(G, n) for all hX, Kni = hX, K(G, n + 1) n, we can take Ln = Kn+1 = K(G, n + 1) by the earlier observation that is a K(G, n). Thus if we take the K(G, n) ’s to be CW complexes, the map Kn→ Kn+1 is just a CW approximation K(G, n)→ Ω There is another reason to look for weak homotopy equivalences Kn→ Kn+1. Ω For a reduced cohomology theory hn(X) there are natural isomorphisms hn(X) ≈ hn+1( X) coming from the long exact sequence of the pair (CX, X) with CX the cone on X, so if hn(X) = hX, Kni for all n then the isomorphism hn(X) ≈ hn+1( translates into a bijection hX, Kni ≈ h thing would be for this to come from a weak equivalence Kn→ lences of this form would give also weak equivalences Kn→ we would automatically obtain an abelian group structure on hX, Kni ≈ hX, X) Kn+1i and the most natural Kn+1. Weak equiva2Kn+2 and so 2Kn+2i. These observations lead to the following deﬁnition. An W spectrum is a sequence Ω of CW complexes K1, K2, ··· together with weak homotopy equivalences Kn→ Kn+1 for all n. By using the theorem of Mil
nor mentioned above it would be possible to X, Kn+1i = hX, Kn+1→ Ω K(G, n + 1 replace ‘weak homotopy equivalence’ by ‘homotopy equivalence’ in this deﬁnition. However it does not noticeably simplify matters to do this, except perhaps psycho- logically. Notice that if we discard a ﬁnite number of spaces Kn from the beginning of spectrum K1, K2, ···, then these omitted terms can be reconstructed from the an remaining Kn ’s since each Kn determines Kn−1 as a CW approximation to Kn. So it is not important that the sequence start with K1. By the same token, this allows us Ω Ω Ω Connections with Cohomology Section 4.3 397 to extend the sequence of Kn ’s to all negative values of n. This is signiﬁcant because a general cohomology theory hn(X) need not vanish for negative n. Theorem 4.58. If {Kn} is an spectrum, then the functors X ֏ hn(X) = hX, Kni, n ∈ Z, deﬁne a reduced cohomology theory on the category of basepointed CW com- plexes and basepoint-preserving maps. Ω Rather amazingly, the converse is also true: Every reduced cohomology theory on CW complexes arises from an spectrum in this way. This is the Brown repre- sentability theorem which will be proved in §4.E. Ω Ω A space Kn in an spectrum is sometimes called an inﬁnite loopspace since there are weak homotopy equivalences Kn→ kKn+k for all k. A number of important spaces in algebraic topology turn out to be inﬁnite loopspaces. Besides Eilenberg– Ω MacLane spaces, two other examples are the inﬁnite-dimensional orthogonal and unitary groups O and U, for which there are weak homotopy equivalences O→ 8O and U→ 2U by a strong form of the Bott periodicity theorem, as we will show in [VBKT]. spectra, hence periodic cohomology theories known as So O and U give periodic Ω real and complex
K–theory. For a more in-depth introduction to the theory of inﬁnite loopspaces, the book [Adams 1978] can be much recommended. Ω Ω Proof: Two of the three axioms for a cohomology theory, the homotopy axiom and the wedge sum axiom, are quite easy to check. For the homotopy axiom, a basepointpreserving map f : X→Y induces f ∗ : hY, Kni→hX, Kni by composition, sending a f-----→ Y →Kn. Clearly f ∗ depends only on the basepoint-preserving map Y →Kn to X homotopy class of f, and it is obvious that f ∗ is a homomorphism if we replace Kn Kn+1 and use the composition of loops to deﬁne the group structure. The wedge by α Xα→Kn W sum axiom holds since in the realm of basepoint-preserving maps, a map is the same as a collection of maps Xα→Kn. Ω The bulk of the proof involves associating a long exact sequence to each CW pair (X, A). As a ﬁrst step we build the following diagram: The ﬁrst row is obtained from the inclusion A ֓ X by iterating the rule, ‘attach a cone on the preceding subspace’, as shown in the pictures below. The three downward arrows in the diagram (1) are quotient maps collapsing the most recently attached cone to a point. Since cones are contractible, these downward maps 398 Chapter 4 Homotopy Theory are homotopy equivalences. The second and third of them have homotopy inverses In the lower row of the evident inclusion maps, indicated by the upward arrows. the diagram the maps are the obvious ones, except for the map X/A→SA which is the composition of a homotopy inverse of the quotient map X ∪ CA→X/A followed by the maps X ∪ CA→(X ∪ CA) ∪ CX→SA. Thus the square containing this map commutes up to homotopy. It is easy to check that the same is true of the right-hand square as well. The whole construction can now be repeated with SA ֓ SX in place of A ֓ X, then with double
suspensions, and so on. The resulting inﬁnite sequence can be written in either of the following two forms: A→X→X ∪ CA→SA→SX→S(X ∪ CA)→S 2A→S 2X→ ··· A→X→X/A→SA→SX→SX/SA→S 2A→S 2X→ ··· In the ﬁrst version we use the obvious equality SX ∪ CSA = S(X ∪ CA). The ﬁrst version has the advantage that the map X ∪CA→SA is easily described and canonical, whereas in the second version the corresponding map X/A→SA is only deﬁned up to homotopy since it depends on choosing a homotopy inverse to the quotient map X ∪ CA→X/A. The second version does have the advantage of conciseness, however. When basepoints are important it is generally more convenient to use reduced cones and reduced suspensions, obtained from ordinary cones and suspensions by collapsing the segment {x0}× I where x0 is the basepoint. The image point of this segment in the reduced cone or suspension then serves as a natural basepoint in the quotient. Assuming x0 is a 0 cell, these collapses of {x0}× I are homotopy equivalences. Using reduced cones and suspensions in the preceding construction yields a sequence (2) A ֓ X→X/A→ A ֓ X→ (X/A)→ 2A ֓ 2X→ ··· where we identify X/ A with Σ (X/A), and all the later maps in the sequence are sus- Σ Σ Σ Σ pensions of the ﬁrst three maps. This sequence, or its unreduced version, is called the Σ Σ Σ coﬁbration sequence or Puppe sequence of the pair (X, A). It has an evident naturality property, namely, a map (X, A)→(Y, B) induces a map between the coﬁbration sequences of these two pairs, with homotopy-commutative squares: Taking basepoint-preserving homotopy classes of maps from the spaces in (2) to a ﬁxed space K gives a sequence (3
) hA, Ki← hX, Ki← hX/A, Ki← h A, Ki← h X, Ki← ··· whose maps are deﬁned by composition with those in (2). For example, the map hX, Ki→hA, Ki sends a map X→K to A→X→K. The sets in (3) are groups starting Σ Σ Connections with Cohomology Section 4.3 399 Σ with h A, Ki, and abelian groups from h 2A, Ki onward. It is easy to see that the maps between these groups are homomorphisms since the maps in (2) are suspensions from A→ ‘zero’ elements, the constant maps. Σ X onward. In general the ﬁrst three terms of (3) are only sets with distinguished A key observation is that the sequence (3) is exact. To see this, note ﬁrst that the Σ Σ diagram (1) shows that, up to homotopy equivalence, each term in (2) is obtained from its two predecessors by the same procedure of forming a mapping cone, so it suﬃces to show that hA, Ki← hX, Ki← hX ∪ CA, Ki is exact. This is easy: A map f : X→K goes to zero in hA, Ki iﬀ its restriction to A is nullhomotopic, ﬁxing the basepoint, and this is equivalent to f extending to a map X ∪ CA→K. If we have a weak homotopy equivalence K→ K′ for some space K′, then the sequence (3) can be continued three steps to the left via the commutative diagram Ω Thus if we have a sequence of spaces Kn together with weak homotopy equivalences Kn→ Kn+1, we can extend the sequence (3) to the left indeﬁnitely, producing a long exact sequence Ω ··· ← hA, Kni← hX, Kni← hX/A, Kni← hA, Kn−1i← hX, Kn−1i← ··· All the terms here are abelian groups and the maps homomorphisms. This long exact sequence is natural with respect to maps (X
, A)→(Y, B) since coﬁbration sequences ⊔⊓ are natural. There is no essential diﬀerence between cohomology theories on basepointed CW complexes and cohomology theories on nonbasepointed CW complexes. Given a h∗, one gets an unreduced theory by setting reduced basepointed cohomology theory hn(X/A), where X/∅ = X+, the union of X with a disjoint basepoint. hn(X, A) = e This is a nonbasepointed theory since an arbitrary map X→Y induces a basepointpreserving map X+→Y+. Furthermore, a nonbasepointed unreduced theory h∗ gives hn(point )→hn(X), a nonbasepointed reduced theory by setting where the map is induced by the constant map X→point. One could also give an argument using suspension, which is always an isomorphism for reduced theories, hn(X) = Coker e e and which takes one from the nonbasepointed to the basepointed category. Theorem 4.59. If h∗ is an unreduced cohomology theory on the category of CW pairs and hn(point ) = 0 for n ≠ 0, then there are natural isomorphisms hn(X, A) ≈ H n for all CW pairs (X, A) and all n. The corresponding statement for homology theories is also true. X, A; h0(point) 400 Chapter 4 Homotopy Theory Proof: The case of homology is slightly simpler, so let us consider this ﬁrst. For CW complexes, relative homology groups reduce to absolute groups, so it suﬃces to deal with the latter. For a CW complex X the long exact sequences of h∗ homology groups for the pairs (X n, X n−1) give rise to a cellular chain complex ··· --------→ hn+1(X n+1, X n) dn+1 -----------------→ hn(X n, X n−1) dn-------------→ hn−1(X n−1, X n−2) --------→ ··· just as for ordinary homology. The hypothesis that hn(point ) = 0 for n ≠ 0 implies that this chain complex has
homology groups hn(X) by the same argument as for ordinary homology. The main thing to verify now is that this cellular chain complex is isomorphic to the cellular chain complex in ordinary homology with coeﬃcients in the group G = h0(point ). Certainly the cellular chain groups in the two cases are isomorphic, being direct sums of copies of G with one copy for each cell, so we have only to check that the cellular boundary maps are the same. Σ to 0 chains since one can always pass from X to isomorphism in any homology theory, and the double suspension It is not really necessary to treat the cellular boundary map d1 from 1 chains X, suspension being a natural 2X has no 1 cells. The calculation of cellular boundary maps dn for n > 1 in terms of degrees of certain maps between spheres works equally well for the homology theory h∗, where ‘degree’ now means degree with respect to the h∗ theory, so what is needed is the fact that a map S n→S n of degree m in the usual sense induces multiplication by m on hn(S n) ≈ G. This is obviously true for degrees 0 and 1, represented by a constant map and the identity map. Since πn(S n) ≈ Z, every map S n→S n is homotopic to some multiple of the identity, so the general case will follow if we know that degree in the h∗ theory is additive with respect to the sum operation in πn(S n). This is a special case of the following more general assertion: Σ Lemma 4.60. If a functor h from basepointed CW complexes to abelian groups satisﬁes the homotopy and wedge axioms, then for any two basepoint-preserving maps X→K, we have (f + g)∗ = f∗ + g∗ if h is covariant and (f + g)∗ = f ∗ + g∗ f, g : if h is contravariant. Σ Σ X X X ∨ c-----→ f ∨g ------------------→ K where c is the Proof: The map f + g is the composition quotient map collapsing an equatorial copy of X. In the covariant case consider the diagram at the right, where i1 and i2 are the
inclusions X. Let X -→ q1, q2 : Σ maps restricting to the identity on the Σ X ֓ X be the quotient summand indicated by the subscript and collapsing the other summand to a point. Then q1∗ ⊕ q2∗ is an inverse to i1∗ ⊕ i2∗ since qj ik is the identity map for j = k and the constant map for An element x in the left-hand group h( X) in the diagram is sent by the compo- sition (q1∗ ⊕ q2∗)c∗ to the element (x, x) in the lower group h( Σ X)⊕ h( X) since Σ Σ Connections with Cohomology Section 4.3 401 q1c and q2c are homotopic to the identity. The composition (f ∨g)∗(i1∗ ⊕i2∗) sends (x, 0) to f∗(x) and (0, y) to g∗(y) since (f ∨ g)i1 = f and (f ∨ g)i2 = g. Hence (x, y) is sent to f∗(x) + g∗(y). Combining these facts, we see that the composition across the top of the diagram is x ֏ f∗(x) + g∗(x). But this composition is also (f + g)∗ since f + g = (f ∨ g)c. This ﬁnishes the proof in the covariant case. X)⊕ h( 1 ⊕ i∗ 1 ⊕ q∗ 2 is q∗ The contravariant case is similar, using the corresponding diagram with arrows reversed. The inverse of i∗ 2 by the same reasoning. An element u in the right-hand group h(K) maps to the element (f ∗(u), g∗(u)) in the lower group X) since (f ∨ g)i1 = f and (f ∨ g)i2 = g. An element (x, 0) in the h( lower group in the diagram maps to the element x in the left-hand group since q1c is homotopic to
the identity, and similarly (0, y) maps to y. Hence (x, y) maps to x + y in the left-hand group. We conclude that u ∈ h(K) maps by the composition across the top of the diagram to f ∗(u) + g∗(u) in h( (f + g)∗ by deﬁnition. X). But this composition is ⊔⊓ Σ Σ Σ Returning to the proof of the theorem, we see that the cellular chain complexes for h∗(X) and H∗(X; G) are isomorphic, so we obtain isomorphisms hn(X) ≈ Hn(X; G) for all n. To verify that these isomorphisms are natural with respect to maps f : X→Y we may ﬁrst deform such a map f to be cellular. Then f takes each pair (X n, X n−1) to the pair (Y n, Y n−1), hence f induces a chain map of cellular chain complexes in the h∗ theory, as well as for H∗(−; G). To compute these chain maps we may pass to the quotient maps X n/X n−1→Y n/Y n−1. These are maps of the form α→ β S n β, α→S n so the induced maps f∗ on hn are determined by their component maps f∗ : S n β. W This is exactly the same situation as with the cellular boundary maps before, where we saw that the degree of a map S n→S n determines the induced map on hn. We conclude that the cellular chain map induced by f in the h∗ theory agrees exactly with the cellular chain map for H∗(−; G). This implies that the isomorphism between the two theories is natural. α S n W The situation for cohomology is quite similar, but there is one point in the ar- gument where a few more words are needed. For cohomology theories the cellular cochain groups are the direct product, rather than the direct sum, of copies of the coeﬃcient group G = h0(point ), with one copy per cell. This means that when there are inﬁnitely many cells in a given dimension, it is not automatically true that
the cellular coboundary maps are uniquely determined by how they map factors of one direct product to factors of the other direct product. To be precise, consider the cellular coboundary map dn : hn(X n, X n−1)→hn+1(X n+1, X n). Decomposing the latter group as a product of copies of G for the (n + 1) cells, we see that dn is determined by the maps hn(X n/X n−1)→hn(S n α ) associated to the attaching maps ϕα of the cells en+1. The thing to observe is that since ϕα has compact image, meeting only ﬁnitely α many n cells, this map hn(X n/X n−1)→hn(S n α ) is ﬁnitely supported in the sense that 402 Chapter 4 Homotopy Theory there is a splitting of the domain into a product of ﬁnitely many factors and a product of the remaining possibly inﬁnite number of factors, such that the map is zero on the latter product. Finitely supported maps have the good property that they are determined by their restrictions to the G factors of hn(X n/X n−1). From this we deduce, using the lemma, that the cellular coboundary maps in the h∗ theory agree with those in ordinary cohomology with G coeﬃcients. This extra argument is also needed to prove naturality of the isomorphisms hn(X) ≈ H n(X; G). This completes the proof of Theorem 4.59. ⊔⊓ Proof of Theorem 4.57: The functors hn(X) = hX, K(G, n)i deﬁne a reduced cohomology theory, and the coeﬃcient groups hn(S i) = πi(K(G, n)) are the same as H n(S i; G), so Theorem 4.59, translated into reduced cohomology, gives natural isomorphisms T : hX, K(G, n)i→ e H n(X; G) for all CW complexes X. It remains to see that T ([f ]) = f ∗(α) for some α ∈ H n(K(
G, n); G), independent of f. This is purely formal: Take α = T (11) for 11 the identity map of K(G, n), and then naturality gives T ([f ]) = T (f ∗(11)) = f ∗T (11) = f ∗(α), where the ﬁrst f ∗ refers to induced homomorphisms for the functor hn, which means composition ⊔⊓ with f. e e The fundamental class α = T (11) can be made more explicit if we choose for K(G, n) a CW complex K with (n − 1) skeleton a point. Denoting hX, K(G, n)i by hn(X), then we have hn(K) ≈ hn(Kn+1) ≈ Ker d : hn(Kn)→hn+1(Kn+1, Kn) The map d is the cellular coboundary in h∗ cohomology since we have hn(Kn) = hn(Kn, Kn−1) because Kn−1 is a point and h∗ is a reduced theory. The isomorphism of hn(K) with Ker d is given by restriction of maps K→K to Kn, so the element 11 ∈ hn(K) deﬁning the fundamental class T (11) corresponds, under the isomorphism hn(K) ≈ Ker d, to the inclusion Kn֓K viewed as an element of hn(Kn). As a cellular cocycle this element assigns to each n cell of K the element of the coeﬃcient group G = πn(K) given by the inclusion of the closure of this cell into K. This means that the fundamental class α ∈ H n(K; G) is represented by the cellular cocycle assigning to each n cell the element of πn(K) given by a characteristic map for the cell. By naturality of T it follows that for a cellular map f : X→K, the corresponding element of H n(X; G) is represented by the cellular cocycle sending each n cell of X to the element of G = πn(K) represented by the composition of f with a characteristic map for the cell. The natural isomorphism H n(X; G) ≈
hX, K(G, n)i leads to a basic principle which reappears many places in algebraic topology, the idea that the occurrence or nonoccurrence of a certain phenomenon is governed by what happens in a single spe- cial case, the universal example. To illustrate, let us prove the following special fact: Connections with Cohomology Section 4.3 403 The map H 1(X; Z)→H 2(X; Z), α ֏ α2, is identically zero for all spaces X. By taking a CW approximation to X we are reduced to the case that X is a CW complex. Then every element of H 1(X; Z) has the form f ∗(α) for some f : X→K(Z, 1), with α a fundamental class in H 1(K(Z, 1); Z), further reducing us to verifying the result for this single α, the ‘universal example’. And for this universal α it is evident that α2 = 0 since S 1 is a K(Z, 1) and H 2(S 1; Z) = 0. Does this fact generalize? It certainly does not hold if we replace the coeﬃcient ring Z by Z2 since H ∗(RP∞; Z2) = Z2[x]. Indeed, the example of RP∞ shows more generally that the fundamental class α ∈ H n(K(Z2, n); Z2) generates a polynomial subalgebra Z2[α] ⊂ H ∗(K(Z2, n); Z2) for each n ≥ 1, since there is a map f : RP∞→K(Z2, n) with f ∗(α) = xn and all the powers of xn are nonzero, hence also all the powers of α. By the same reasoning, the example of CP∞ shows that the fundamental class α ∈ H 2n(K(Z, 2n); Z) generates a polynomial subalgebra Z[α] in H ∗(K(Z, 2n); Z). As we shall see in [SSAT], H ∗(K(Z, 2n); Z)/torsion is exactly this polynomial algebra Z[α]. A little more subtle is the question of identifying the subalgebra of H �
�(K(Z, n); Z) generated by the fundamental class α for odd n ≥ 3. By the commutativity property of cup products we know that α2 is either zero or of order two. To see that α2 is nonzero it suﬃces to ﬁnd a single space X with an element γ ∈ H n(X; Z) such that γ2 ≠ 0. The ﬁrst place to look might be RP∞, but its cohomology with Z coeﬃcients is concentrated in even dimensions. Instead, consider X = RP∞× RP∞. This has Z2 cohomology Z2[x, y] and Example 3E.5 shows that its Z cohomology is the Z2[x2, y 2] submodule generated by 1 and x2y + xy 2, except in dimension zero of course, where 1 generates a Z rather than a Z2. In particular we can take γ = x2k(x2y + xy 2) for any k ≥ 0, and then all powers γm are nonzero since we are inside the polynomial ring Z2[x, y]. It follows that the subalgebra of H ∗(K(Z, n); Z) generated by α is Z[α]/(2α2) for odd n ≥ 3. These examples lead one to wonder just how complicated the cohomology of K(G, n) ’s is. The general construction of a K(G, n) is not very helpful in answering this question. Consider the case G = Z for example. Here one would start with S n and attach (n + 2) cells to kill πn+1(S n). Since πn+1(S n) happens to be cyclic, only one (n + 2) cell is needed. To continue, one would have to compute generators for πn+2 of the resulting space S n ∪ en+2, use these to attach (n + 3) cells, then compute the resulting πn+3, and so on for each successive dimension. When n = 2 this procedure happens to work out very neatly, and the resulting K(Z, 2) is CP∞ with its usual CW structure having one cell in each even dimension, according to an exercise at the end of the
section. However, for larger n it quickly becomes impractical to make this procedure explicit since homotopy groups are so hard to compute. One can get some idea of the diﬃculties of the next case n = 3 by considering the homology groups of K(Z, 3). Using techniques in [SSAT], the groups Hi(K(Z, 3); Z) for 0 ≤ i ≤ 12 can be 404 Chapter 4 Homotopy Theory computed to be Z, 0, 0, Z, 0, Z2, 0, Z3, Z2, Z2, Z3, Z10, Z2 To get this sequence of homology groups would require quite a few cells, and the situation only gets worse in higher dimensions, where the homology groups are not always cyclic. Indeed, one might guess that computing the homology groups of K(Z, n) ’s would be of the same order of diﬃculty as computing the homotopy groups of spheres, but by some miracle this is not the case. The calculations are indeed complicated, but they were completely done by Serre and Cartan in the 1950s, not just for K(Z, n) ’s, but for all K(G, n) ’s with G ﬁnitely generated abelian. For example, H ∗(K(Z, 3); Z2) is the polynomial algebra Z2[x3, x5, x9, x17, x33, ···] with generators of dimensions 2i + 1, indicated by the subscripts. And in general, for G ﬁnitely generated abelian, H ∗(K(G, n); Zp) is a polynomial algebra on generators of speciﬁed dimensions if p is 2, while for p an odd prime one gets the tensor product of a polynomial ring on generators of speciﬁed even dimensions and an exterior algebra on generators of speciﬁed odd dimensions. With Z coeﬃcients the description of the cohomology is not nearly so neat, however. We will study these questions in some detail in [SSAT]. There is a good reason for being interested in the cohomology of K(G, n) ’s, arising from the equivalence H n(X; G) ≈ hX, K(
G, n)i. Taking Z coeﬃcients for simplicity, an element of Hm(K(Z, n); Z) corresponds to a map θ : K(Z, n)→K(Z, m). We can compose θ with any map f : X→K(Z, n) to get a map θf : X→K(Z, m). Letting f vary and keeping θ ﬁxed, this gives a function H n(X; Z)→Hm(X; Z), depending only on θ. This is the idea of cohomology operations, which we study in more detail in §4.L. The equivalence H n(X; G) ≈ hX, K(G, n)i also leads to a new viewpoint toward cup products. Taking G to be a ring R and setting Kn = K(R, n), then if we are given maps f : X→Km and g : Y →Kn, we can deﬁne the cross product of the corresponding cohomology classes by the composition X × Y f × g -----------------→ Km × Kn ------→ Km ∧ Kn µ------------→ Km+n where the middle map is the quotient map and µ can be deﬁned in the following way. The space Km ∧ Kn is (m + n − 1) connected, so by the Hurewicz theorem and the K¨unneth formula for reduced homology we have isomorphisms πm+n(Km ∧ Kn) ≈ Hm+n(Km ∧ Kn) ≈ Hm(Km) ⊗ Hn(Kn) ≈ R ⊗ R. By Lemmas 4.7 and 4.31 there is then a map µ : Km ∧ Kn→Km+n inducing the multiplication map R ⊗ R→R on πm+n. Or we could use the isomorphism Hm+n(Km ∧ Kn; R) ≈ Hom(Hm+n(Km ∧ Kn), R) and let µ be the map corresponding to the cohomology class given by the multiplication homomorphism R ⊗ R→R. Connections with Cohomology Section 4.3 405 The case R = Z is particularly simple. We can
take Sm as the (m + 1) skeleton of Km, and similarly for Kn, so Km ∧ Kn has Sm ∧ S n as its (m + n + 1) skeleton and we can obtain µ by extending the inclusion Sm ∧ S n = Sm+n ֓ Km+n. It is not hard to prove the basic properties of cup product using this deﬁnition, and in particular the commutativity property becomes somewhat more transparent from this viewpoint. For example, when R = Z, commutativity just comes down to the fact that the map Sm ∧ S n→S n ∧ Sm switching the factors has degree (−1)mn when regarded as a map of Sm+n. Fibrations Recall from §4.2 that a ﬁbration is a map p : E→B having the homotopy lifting property with respect to all spaces. In a ﬁber bundle all the ﬁbers are homeomorphic by deﬁnition, but this need not be true for ﬁbrations. An example is the linear projec- tion of a 2 simplex onto one of its edges, which is a ﬁbration according to an exercise at the end of the section. The following result gives some evidence that ﬁbrations should be thought of as a homotopy-theoretic analog of ﬁber bundles: Proposition 4.61. For a ﬁbration p : E→B, the ﬁbers Fb = p−1(b) over each path component of B are all homotopy equivalent. Proof: A path γ : I→B gives rise to a homotopy gt : Fγ(0)→B with gt(Fγ(0)) = γ(t). The inclusion Fγ(0) ֓ E provides a lift g0, so by the homotopy lifting property we g1 gives have a homotopy e a map Lγ : Fγ(0)→Fγ(1). The association γ ֏ Lγ has the following basic properties: (a) If γ ≃ γ′ rel ∂I, then Lγ ≃ Lγ′. In particular the homotopy class of Lγ is inde- gt(Fγ(0)) ⊂ Fγ(t)
for all t. In particular, gt : Fγ(0)→E with e e e pendent of the choice of the lifting gt of gt. (b) For a composition of paths γγ′, Lγγ′ is homotopic to the composition Lγ′ Lγ. e From these statements it follows that Lγ is a homotopy equivalence with homotopy inverse Lγ, where γ is the inverse path of γ. Before proving (a), note that a ﬁbration has the homotopy lifting property for pairs (X × I, X × ∂I) since the pairs (I × I, I × {0}∪∂I × I) and (I × I, I × {0}) are homeo- morphic, hence the same is true after taking products with X. To prove (a), let γ(s, t) be a homotopy from γ(t) to γ′(t), (s, t) ∈ I × I. This determines a family gst : Fγ(0)→B with gst(Fγ(0)) = γ(s, t). Let g1,t be lifts gs,0 be the inclusion Fγ(0) ֓ E for all s. Using the deﬁning Lγ and Lγ′, and let homotopy lifting property for the pair (Fγ(0) × I, Fγ(0) × ∂I), we can extend these lifts to lifts gst for (s, t) ∈ I × I. Restricting to t = 1 then gives a homotopy Lγ ≃ Lγ′. g0,t and e e e Property (b) holds since for lifts e deﬁning Lγγ′ by taking g′ g2t for 0 ≤ t ≤ 1/2 and e gt and e t deﬁning Lγ and Lγ′ we obtain a lift g′ ⊔⊓ 2t−1Lγ for 1/2 ≤ t ≤ 1. One may ask whether ﬁbrations satisfy a homotopy analog of the local triviality property of ﬁber bundles. Observe ﬁrst that for a ﬁbration p : E→B, the restriction e e 406 Chapter
4 Homotopy Theory p : p−1(A)→A is a ﬁbration for any subspace A ⊂ B. So we can ask whether every point of B has a neighborhood U for which the ﬁbration p−1(U)→U is equivalent in some homotopy-theoretic sense to a projection U × F→U. The natural notion of equivalence for ﬁbrations is deﬁned in the following way. Given ﬁbrations p1 : E1→B and p2 : E2→B, a map f : E1→E2 is called ﬁber-preserving if p1 = p2f, or in other 2 (b) for all b ∈ B. A ﬁber-preserving map f : E1→E2 is a words, f (p−1 ﬁber homotopy equivalence if there is a ﬁber-preserving map g : E2→E1 such that both compositions f g and gf are homotopic to the identity through ﬁber-preserving 1 (b)) ⊂ p−1 maps. A ﬁber homotopy equivalence can be thought of as a family of homotopy equivalences between corresponding ﬁbers of E1 and E2. An interesting fact is that a ﬁber-preserving map that is a homotopy equivalence is a ﬁber homotopy equivalence; this is an exercise for §4.H. We will show that a ﬁbration p : E→B is locally ﬁber-homotopically trivial in the sense described above if B is locally contractible. In order to do this we ﬁrst digress to introduce another basic concept. Given a ﬁbration p : E→B and a map f : A→B, there is a pullback or induced ﬁbration f ∗(E)→A obtained by setting f ∗(E) = {(a, e) ∈ A× E \|\| f (a) = p(e)}, with the projections of f ∗(E) onto A and E giving a commutative diagram as shown at the right. The homotopy lifting property holds for f ∗(E)→A since a
homotopy gt : X→A gives the ﬁrst gt : X→f ∗(E), the second coordinate being coordinate of a lift a lifting to E of the composed homotopy f gt. e Proposition 4.62. Given a ﬁbration p : E→B and a homotopy ft : A→B, the pullback ﬁbrations f ∗ 1 (E)→A are ﬁber homotopy equivalent. 0 (E)→A and f ∗ Proof: Let F : A× I→B be the homotopy ft. The ﬁbration F ∗(E)→A× I contains 0 (E) and f ∗ f ∗ 1 (E) over A× {0} and A× {1}. So it suﬃces to prove the following: For a ﬁbration p : E→B × I, the restricted ﬁbrations Es = p−1(B × {s})→B are all ﬁber homotopy equivalent for s ∈ [0, 1]. To prove this assertion the idea is to imitate the construction of the homotopy equivalences Lγ in the proof of Proposition 4.61. A path γ : [0, 1]→I gives rise to a ﬁber-preserving map Lγ : Eγ(0)→Eγ(1) by lifting the homotopy gt : Eγ(0)→B × I, gt(x) = (p(x), γ(t)), starting with the inclusion Eγ(0) ֓ E. As before, one shows the two basic properties (a) and (b), noting that in (a) the homotopy Lγ ≃ Lγ′ is ﬁberpreserving since it is obtained by lifting a homotopy ht : Eγ(0) × [0, 1]→B × I of the form ht(x, u) = (p(x), −). From (a) and (b) it follows that Lγ is a ﬁber homotopy ⊔⊓ equivalence with inverse Lγ. Corollary 4.63. A ﬁbration E→B over a contractible base B is ﬁber
homotopy equivalent to a product ﬁbration B × F→B. Connections with Cohomology Section 4.3 407 Proof: The pullback of E by the identity map B→B is E itself, while the pullback by a constant map B→B is a product B × F. ⊔⊓ Thus we see that if B is locally contractible then any ﬁbration over B is locally ﬁber homotopy equivalent to a product ﬁbration. Pathspace Constructions There is a simple but extremely useful way to turn arbitrary mappings into ﬁbrations. Given a map f : A→B, let Ef be the space of pairs (a, γ) where a ∈ A and γ : I→B is a path in B with γ(0) = f (a). We topologize Ef as a subspace of A× BI, where BI is the space of mappings I→B with the compact-open topology; see the Appendix for the deﬁnition and basic properties of this topology, in particular Proposition A.14 which we will be using shortly. Proposition 4.64. The map p : Ef →B, p(a, γ) = γ(1), is a ﬁbration. Proof: Continuity of p follows from (a) of Proposition A.14 in the Appendix which says that the evaluation map BI × I→B, (γ, s) ֏ γ(s), is continuous. To verify the ﬁbration property, let a homotopy gt : X→B and a lift g0 : X→Ef g0(x) = (h(x), γx) for h : X→A and γx : I→B. Deﬁne a lift of g0 be given. Write gt : X→Ef by gt(x) = (h(x), γx g[0,t](x)), the second coordinate being the path γx followed by the path traced out by gs(x) for 0 ≤ s ≤ t. This composition of paths is e deﬁned since g0(x) = p gt is a continuous homotopy we regard it as a map X × I→Ef ⊂
A× BI and then apply (b) of Proposition A.14 which in the current context asserts that continuity of a map X × I→A× BI is equivalent to continuity of the associated map X × I × I→A× B. ⊔⊓ g0(x) = γx(1). To check that e e e e e We can regard A as the subspace of Ef consisting of pairs (a, γ) with γ the constant path at f (a), and Ef deformation retracts onto this subspace by restricting all the paths γ to shorter and shorter initial segments. The map p : Ef →B restricts to f on the subspace A, so we have factored an arbitrary map f : A→B as the composition A ֓ Ef →B of a homotopy equivalence and a ﬁbration. We can also think of this construction as extending f to a ﬁbration Ef →B by enlarging its domain to a homotopy equivalent space. The ﬁber Ff of Ef →B is called the homotopy ﬁber of f. It consists of all pairs (a, γ) with a ∈ A and γ a path in B from f (a) to a basepoint b0 ∈ B. If f : A→B is the inclusion of a subspace, then Ef is the space of paths in B starting at points of A. In this case a map (Ii+1, ∂Ii+1, J i)→(B, A, x0) is the same as a map (Ii, ∂Ii)→(Ff, γ0) where γ0 is the constant path at x0 and Ff is the ﬁber of Ef over x0. This means that πi+1(B, A, x0) can be identiﬁed with πi(Ff, γ0), hence the long exact sequences of homotopy groups of the pair (B, A) and of the ﬁbration Ef →B can be identiﬁed. Ω 408 Chapter 4 Homotopy Theory An important special case is when f is the inclusion of the basepoint b0 into B. Then Ef is the space
P B of paths in B starting at b0, and p : P B→B sends each path to its endpoint. The ﬁber p−1(b0) is the loopspace B consisting of all loops in B based at b0. Since P B is contractible by progressively truncating paths, the long exact sequence of homotopy groups for the path ﬁbration P B→B yields another proof that πn(X, x0) ≈ πn−1( X, x0) for all n. Ω As we mentioned in the discussion of loopspaces earlier in this section, it is a theorem of [Milnor 1959] that the loopspace of a CW complex is homotopy equivalent to a CW complex. Milnor’s theorem is actually quite a bit more general than this, and implies in particular that the homotopy ﬁber of an arbitrary map between CW complexes has the homotopy type of a CW complex. One can usually avoid quoting these results by using CW approximations, though it is reassuring to know they are available if needed, or if one does not want to bother with CW approximations. If the ﬁbration construction f ֏ Ef is applied to a map p : E→B that is already a ﬁbration, one might expect the resulting ﬁbration Ep→B to be closely related to the original ﬁbration E→B. This is indeed the case: Proposition 4.65. If p : E→B is a ﬁbration, then the inclusion E ֓ Ep is a ﬁber homotopy equivalence. In particular, the homotopy ﬁbers of p are homotopy equivalent to the actual ﬁbers. g0 : Ep→E, g0(e, γ) = e. The lifting Proof: We apply the homotopy lifting property to the homotopy gt : Ep→B, gt(e, γ) = gt : Ep→E is then the ﬁrst γ(t), with initial lift coordinate of a homotopy ht : Ep→Ep whose second coordinate is the restriction of the paths γ to the interval [t, 1]. Since the endpoints of the paths γ are unchanged, ht is ﬁber-preserving
. We have h0 = 11, h1(Ep) ⊂ E, and ht(E) ⊂ E for all t. If we let i denote the inclusion E ֓ Ep, then ih1 ≃ 11 via ht and h1i ≃ 11 via ht \|\| E, so i is a ⊔⊓ ﬁber homotopy equivalence. e e e We have seen that loopspaces occur as ﬁbers of ﬁbrations P B→B with con- tractible total space P B. Here is something of a converse: Proposition 4.66. If F→E→B is a ﬁbration or ﬁber bundle with E contractible, then there is a weak homotopy equivalence F→ B. Ω Proof: If we compose a contraction of E with the projection p : E→B then we have for each point x ∈ E a path γx in B from p(x) to a basepoint b0 = p(x0), where x0 is the point to which E contracts. This yields a map E→P B, x ֏ γx, whose composition with the ﬁbration P B→B is p. By restriction this gives a map F→ B where F = p−1(b0), and the long exact sequence of homotopy groups for F→E→B maps to the long Ω exact sequence for implies that the map F→ B→P B→B. Since E and P B are contractible, the ﬁve-lemma ⊔⊓ B is a weak homotopy equivalence. Ω Ω Connections with Cohomology Section 4.3 409 Gn(R∞), U(n) ≃ Examples arising from ﬁber bundles constructed earlier in the chapter are O(n) ≃ In particular, taking n = 1 HP∞. Note that in all in the latter two examples, we have S 1 ≃ Ω Ω these examples it is a topological group that is homotopy equivalent to a loopspace. Gn(H∞). CP∞ and S 3 ≃ Ω Gn(C∞), and Sp(n) ≃ In [Milnor 1956] this is shown to
hold in general: For each topological group G there is a ﬁber bundle G→EG→BG with EG contractible, hence by the proposition there is a weak equivalence G ≃ BG. There is also a converse statement: The loopspace Ω Ω of a CW complex is homotopy equivalent to a topological group. Ω Ω The relationship between X and X has been much studied, particularly the case that X has the homotopy type of a ﬁnite CW complex, which is of special interest because of the examples of the classical Lie groups such as O(n), U(n), and Sp(n). Ω See [Kane 1988] for an introduction to this subject. It is interesting to see what happens when the process of forming homotopy ﬁbers is iterated. Given a ﬁbration p : E→B with ﬁber F = p−1(b0), we know that the inclusion of F into the homotopy ﬁber Fp is a homotopy equivalence. Recall that Fp consists of pairs (e, γ) with e ∈ E and γ a path in B from p(e) to b0. The inclusion F ֓ E extends to a map i : Fp→E, i(e, γ) = e, and this map is obviously a ﬁbration. In fact it is the pullback via p of the path ﬁbration P B→B. This allows us to iterate, taking the homotopy ﬁber Fi with its map to Fp, and so on, as in the ﬁrst row of the following diagram: The actual ﬁber of i over a point e0 ∈ p−1(b0) consists of pairs (e0, γ) with γ a B ֓ Fi loop in B at the basepoint b0, so this ﬁber is just B→F is is a homotopy equivalence. In the second row of the diagram the map Ω Ω B ֓ Fi→Fp→F where the last map is a homotopy inverse to the the composition inclusion F ֓ Fp, so the square in the diagram containing these maps commutes up to homotopy.
The homotopy ﬁber Fi consists of pairs (γ, η) where η is a path in E ending at e0 and γ is a path in B from p(η(0)) to b0. A homotopy inverse to the B sending (γ, η) to the loop obtained by inclusion B, and the inclusion Ω Ω B ֓ Fi is the retraction Fi→ composing the inverse path of pη with γ. Ω Ω These constructions can now be iterated to produce a ﬁbration sequence, also known as a Puppe sequence, ··· → 2B→ F→ E→ B→F→E→B Here any two consecutive maps form a ﬁbration, up to homotopy equivalence, and Ω Ω Ω Ω all the maps to the left of B are obtained by applying the functor to the later maps. The long exact sequence of homotopy groups for any ﬁbration in the sequence coincides with the long exact sequence for F→E→B, as the reader can check. Ω Ω 410 Chapter 4 Homotopy Theory Postnikov Towers A Postnikov tower for a path-connected space X is a commutative diagram as at the right, such that: (1) The map X→Xn induces an isomorphism on πi for i ≤ n. (2) πi(Xn) = 0 for i > n. As we saw in Example 4.16, every connected CW complex X has a Postnikov tower, and this is unique up to homotopy equivalence by Corollary 4.19. If we convert the map Xn→Xn−1 into a ﬁbration, its ﬁber Fn is a K(πnX, n), as is apparent from a brief inspection of the long exact sequence of homotopy groups for the ﬁbration: πi+1(Xn)→πi+1(Xn−1)→πi(Fn)→πi(Xn)→πi(Xn−1) We can replace each map Xn→Xn−1 by a ﬁbration X ′ with X2→X1 and working upward. For the inductive step we convert the
composition Xn→Xn−1 ֓ X ′ n−1 ﬁtting into the commutative diagram at the right. Thus we obtain a n−1 into a ﬁbration X ′ n→X ′ n→X ′ n−1 in succession, starting Postnikov tower satisfying also the condition (3) The map Xn→Xn−1 is a ﬁbration with ﬁber a K(πnX, n). To the extent that ﬁbrations can be regarded as twisted products, up to homotopy equivalence, the spaces Xn in a Postnikov tower for X can be thought of as twisted products of Eilenberg-MacLane spaces K(πnX, n). For many purposes, a CW complex X can be replaced by one of the stages Xn in a Postnikov tower for X, for example if one is interested in homotopy or homology groups in only a ﬁnite range of dimensions. However, to determine the full homotopy type of X from its Postnikov tower, some sort of limit process is needed. Let us investigate this question is somewhat greater generality. Given a sequence of maps ··· →X2→X1, deﬁne their inverse limit lim ←-- Xn to be n Xn consisting of sequences of points xn ∈ Xn with the subspace of the product xn mapping to xn−1 under the map Xn→Xn−1. The corresponding algebraic notion ←-- Gn of a sequence of group homomorphisms ··· →G2→G1, is the inverse limit lim n Gn consisting of sequences of elements gn ∈ Gn with which is the subgroup of gn mapping to gn−1 under the homomorphism Gn→Gn−1. Q Q ←-- Xn)→ lim Proposition 4.67. For an arbitrary sequence of ﬁbrations ··· →X2→X1 the natural map λ : πi(lim ←-- πi(Xn) is surjective, and λ is injective if the maps πi+1(Xn)→πi+1(Xn−1) are surjective for n suﬃciently large. Proof: Represent an element of lim←-- πi(Xn
) by maps fn : (S i, s0)→(Xn, xn). Since the projection pn : Xn→Xn−1 takes [fn] to [fn−1], by applying the homotopy lifting Connections with Cohomology Section 4.3 411 property for the pair (S i, s0) we can homotope fn, ﬁxing s0, so that pnfn = fn−1. Doing this inductively for n = 2, 3, ···, we get pnfn = fn−1 for all n simultaneously, which gives surjectivity of λ. For injectivity, note ﬁrst that inverse limits are unaﬀected by throwing away a ﬁnite number of terms at the end of the sequence of spaces or groups, so we may assume the maps πi+1(Xn)→πi+1(Xn−1) are surjective for all n. Given a map f : S i→ lim ←-- Xn, suppose we have nullhomotopies Fn : Di+1→Xn of the coordinate functions fn : S i→Xn of f. We have pnFn = Fn−1 on S i, so pnFn and Fn−1 are the restrictions to the two hemispheres of S i+1 of a map gn−1 : S i+1→Xn−1. If the map πi+1(Xn)→πi+1(Xn−1) is surjective, we can rechoose Fn so that the new gn−1 is nullhomotopic, that is, so that pnFn ≃ Fn−1 rel S i. Applying the homotopy lifting property for (Di+1, S i), we can make pnFn = Fn−1. Doing this inductively for n = 2, 3, ···, we see that f : S i→ lim ⊔⊓ ←-- Xn is nullhomotopic and λ is injective. In fact, Ker λ is naturally isomorphic to lim ←-- One might wish to have a description of the kernel of λ in the case of an arbitrary sequence of ﬁbrations ··· →X2→X1, though for our present purposes this question 1
πi+1(Xn), where lim is not relevant. ←-is the functor deﬁned in §3.F. Namely, if f : S i→ lim ←-- Xn determines an element of Ker λ, then the sequence of maps gn : S i+1→Xn constructed above gives an element n πi+1(Xn), well-deﬁned up to the choice of the nullhomotopies Fn. Any new of choice of Fn is obtained by adding a map Gn : S i+1→Xn to Fn. The eﬀect of this is to 1πi+1(Xn) is the quotient of change gn to gn +Gn and gn−1 to gn−1 −pnGn. Since lim ←-1πi+1(Xn). Thus n πi+1(Xn) under exactly these identiﬁcations, we get Ker λ ≈ lim←-- Q 1 for each i > 0 there is a natural exact sequence Q 0 -→ lim ←-- 1πi+1(Xn) -→ πi(lim ←-- Xn) -→ lim ←-- πi(Xn) -→ 0 1 term vanishes if the maps πi+1(Xn)→πi+1(Xn−1) The proposition says that the lim ←-- are surjective for suﬃciently large n. Corollary 4.68. For the Postnikov tower of a connected CW complex X the natural map X→ lim ←-- Xn is a weak homotopy equivalence, so X is a CW approximation to lim ←-- Xn. Proof: The composition πi(X) -→ πi(lim πi(X)→πi(Xn) is an isomorphism for large n. ←-- πi(Xn) is an isomorphism since ⊔⊓ ←-- Xn) λ-----→ lim Having seen how to decompose a space X into the terms in its Postnikov tower, we consider now the inverse process of building a Postnikov tower, starting with X1 as a K(π, 1) and inductively constructing Xn from Xn−1. It
would be very nice if the ﬁbration K(π, n)→Xn→Xn−1 could be extended another term to the right, to form a ﬁbration sequence K(π, n)→Xn→Xn−1→K(π, n + 1) 412 Chapter 4 Homotopy Theory for this would say that Xn is the homotopy ﬁber of a map Xn−1→K(π, n + 1), and homotopy classes of such maps are in one-to-one correspondence with elements of H n+1(Xn−1; π ) by Theorem 4.57. Since the homotopy ﬁber of Xn−1→K(π, n + 1) is the same as the pullback of the path ﬁbration P K(π, n + 1)→K(π, n + 1), its homotopy type depends only on the homotopy class of the map Xn−1→K(π, n + 1), by Proposition 4.62. Note that the last term K(π, n + 1) in the ﬁbration sequence above cannot be anything else but a K(π, n+1) since its loopspace must be homotopy equivalent to the ﬁrst term in the sequence, a K(π, n). In general, a ﬁbration F→E→B is called principal if there is a commutative diagram where the second row is a ﬁbration sequence and the vertical maps are weak ho- motopy equivalences. Thus if all the ﬁbrations in a Postnikov tower for X happen to be principal, we have a diagram as at the right, where each Xn+1 is, up to weak homotopy equivalence, the homotopy ﬁber of the map kn : Xn→K(πn+1X, n + 2). The map kn is equivalent to a class in H n+2 Xn; πn+1X called the nth k invariant of X. These classes specify how to construct X inductively from Eilenberg–MacLane spaces. For example, if all the kn ’s are zero, X is just the product of the spaces K(πnX, n),
and in the general case X is some sort of twisted product of K(πnX, n) ’s. To actually build a space from its k invariants is usually too unwieldy a procedure to be carried out in practice, but as a theoretical tool this procedure can be quite useful. The next result tells us when this tool is available: Theorem 4.69. A connected CW complex X has a Postnikov tower of principal ﬁbrations iﬀ π1(X) acts trivially on πn(X) for all n > 1. Notice that in the deﬁnition of a principal ﬁbration, the map F→ B′ automatically exists and is a homotopy weak equivalence once one has the right-hand square of the commutative diagram with its vertical maps weak homotopy equivalences. Thus the question of whether a ﬁbration is principal can be rephrased in the following way: Given a map A→X, which one can always replace by an equivalent ﬁbration if one likes, does there exist a ﬁbration F→E→B and a commutative square as at the right, with the vertical maps weak homotopy equivalences? By replacing A and X with CW approximations and converting the resulting map A→X into an inclusion via a mapping cylinder, the question becomes whether a CW pair (X, A) is equivalent to a ﬁbration pair (E, F ), that Ω Connections with Cohomology Section 4.3 413 is, whether there is a ﬁbration F→E→B and a map (X, A)→(E, F ) for which both X→E and A→F are weak homotopy equivalences. In general the answer will rarely be yes, since the homotopy ﬁber of A ֓ X would have to have the weak homotopy type of a loopspace, which is a rather severe restriction. However, in the situation of Postnikov towers, the homotopy ﬁber is a K(π, n) with π abelian since n ≥ 2, so it is a loopspace. But there is another requirement: The action of π1(A) on πn(X, A) must be trivial for all n ≥ 1. This is equivalent to
the action of π1(F ) on πn(E, F ) being trivial, which is always the case in a ﬁbration since under the isomorphism p∗ : πn(E, F )→πn(B, x0) an element γα−α, with γ ∈ π1(F ) and α ∈ πn(E, F ), maps to p∗(γ)p∗(α) − p∗(α) which is zero since p∗(γ) lies in the trivial group π1(x0). The relative group πn(X, A) is always isomorphic to πn−1 of the homotopy ﬁber of the inclusion A ֓ X, so in the case at hand when the homotopy ﬁber is a K(π, n), the only nontrivial relative homotopy group is πn+1(X, A) ≈ π. In this case the necessary condition of trivial action is also suﬃcient: Lemma 4.70. Let (X, A) be a CW pair with both X and A connected, such that the homotopy ﬁber of the inclusion A ֓ X is a K(π, n), n ≥ 1. Then there exists a ﬁbration F→E→B and a map (X, A)→(E, F ) inducing weak homotopy equivalences X→E and A→F iﬀ the action of π1(A) on πn+1(X, A) is trivial. Proof: It remains only to prove the ‘if’ implication. As we noted just before the statement of the lemma, the groups πi(X, A) are zero except for πn+1(X, A) ≈ π. If the action of π1(A) on πn+1(X, A) is trivial, the relative Hurewicz theorem gives an isomorphism πn+1(X, A) ≈ Hn+1(X, A). Since (X, A) is n connected, we may assume A contains the n skeleton of X, so X/A is n connected and the absolute H
urewicz theorem gives πn+1(X/A) ≈ Hn+1(X/A). Hence the quotient map X→X/A induces an isomorphism πn+1(X, A) ≈ πn+1(X/A) since the analogous statement for homology is certainly true. Since πn+1(X/A) ≈ π, we can build a K(π, n + 1) from X/A by attaching cells of dimension n + 3 and greater. This leads to the commutative diagram at the right, where the vertical maps are inclusions and the lower row is obtained by converting the map k into a ﬁbration. The map A→Fk is a weak homotopy equivalence by the ﬁve-lemma applied to the map between the long exact sequences of homotopy groups for the pairs (X, A) and (Ek, Fk), since the only nontrivial relative groups are πn+1, both of which map isomorphically to ⊔⊓ πn+1(K(π, n + 1)). Proof of 4.69: In view of the lemma, all that needs to be done is identify the action of π1(X) on πn(X) with the action of π1(Xn) on πn+1(Xn−1, Xn) for n ≥ 2, thinking of the map Xn→Xn−1 as an inclusion. From the exact sequence 0 = πn+1(Xn−1) -→ πn+1(Xn−1, Xn) ∂-----→ πn(Xn) -→ πn(Xn−1) = 0 414 Chapter 4 Homotopy Theory we have an isomorphism πn+1(Xn−1, Xn) ≈ πn(Xn) respecting the action of π1(Xn). And the map X→Xn induces isomorphisms on π1 and πn, so we are done. ⊔⊓ Let us consider now a natural generalization of Postnikov towers, in which one starts with a map f : X→Y between path-connected spaces rather than just a single space X.
A Moore–Postnikov tower for f is a commutative diagram as shown at the right, with each composition X→Zn→Y homotopic to f, and such that: (1) The map X→Zn induces an isomorphism on πi for i < n and a surjection for i = n. (2) The map Zn→Y induces an isomorphism on πi for i > n and an injection for i = n. (3) The map Zn+1→Zn is a ﬁbration with ﬁber a K(πnF, n) where F is the homo- topy ﬁber of f. A Moore–Postnikov tower specializes to a Postnikov tower by taking Y to be a point and then setting Xn = Zn+1, discarding the space Z1 which has trivial homotopy groups. Theorem 4.71. Every map f : X→Y between connected CW complexes has a Moore– Postnikov tower, which is unique up to homotopy equivalence. A Moore–Postnikov tower of principal ﬁbrations exists iﬀ π1(X) acts trivially on πn(Mf, X) for all n > 1, where Mf is the mapping cylinder of f. Proof: The existence and uniqueness of a diagram satisfying (1) and (2) and commutative at least up to homotopy follows from Propositions 4.17 and 4.18 applied to the pair (Mf, X) with Mf the mapping cylinder of f. Having such a diagram, we proceed as in the earlier case of Postnikov towers, replacing each map Zn→Zn−1 by a homotopy equivalent ﬁbration, starting with Z2→Z1 and working upward. We can then apply the homotopy lifting property to make all the triangles in the left half of the tower strictly commutative. After these steps the triangles in the right half of the diagram commute up to homotopy, and to make them strictly commute we can just replace each map to Y by the composition through Z1. To see that the ﬁbers of the maps Zn+1→Zn are Eilenberg–MacLane spaces as in condition (3), consider two successive levels of the tower. We may arrange that the maps X→Zn+1
→Zn→Y are inclusions by taking mapping cylinders, ﬁrst of X→Zn+1, then of the new Zn+1→Zn, and then of the new Zn→Y. From the left-hand triangle we see that Zn+1→Zn induces an isomorphism on πi for i < n and a surjection for i = n, hence πi(Zn, Zn+1) = 0 for i < n + 1. Similarly, the other triangle gives πi(Zn, Zn+1) = 0 for i > n + 1. To show that πn+1(Zn, Zn+1) ≈ πn+1(Y, X) we use the following diagram: Connections with Cohomology Section 4.3 415 The upper-right vertical map is injective and the lower-left vertical map is surjective, so the ﬁve-lemma implies that the two middle vertical maps are isomorphisms. Since the homotopy ﬁber of an inclusion A ֓ B has πi equal to πi+1(B, A), we see that condition (3) is satisﬁed. The statement about a tower of principal ﬁbrations can be obtained as an appli- cation of Lemma 4.70. As we saw in the previous paragraph, there are isomorphisms πn+1(Y, X) ≈ πn+1(Zn, Zn+1), and these respect the action of π1(X) ≈ π1(Zn+1), so ⊔⊓ Lemma 4.70 gives the result. Besides the case that Y is a point, which yields Postnikov towers, another in- teresting special case of Moore–Postnikov towers is when X is a point. In this case the space Zn is an n connected covering of Y, as in Example 4.20. The n connected covering of Y can also be obtained as the homotopy ﬁber of the nth stage Y →Yn of a Postnikov tower for Y. The tower of n connected coverings of Y can be realized by principal ﬁbrations by taking Zn to be the homotopy ﬁber of the map
Zn−1→K(πnY, n) that is the ﬁrst nontrivial stage in a Postnikov tower for Zn−1. A generalization of the preceding theory allowing nontrivial actions of π1 can be found in [Robinson 1972]. Obstruction Theory It is very common in algebraic topology to encounter situations where one would like to extend or lift a given map. Obvious examples are the homotopy extension and homotopy lifting properties. In their simplest forms, extension and lifting questions can often be phrased in one of the following two ways: The Extension Problem. Given a CW pair (W, A) and a map A→X, does this extend to a map W→X? The Lifting Problem. Given a ﬁbration X→Y and a map W→Y, is there a lift W→X? In order for the lifting problem to include things like the homotopy lifting property, it should be generalized to a relative form: 416 Chapter 4 Homotopy Theory The Relative Lifting Problem. Given a CW pair (W, A), a ﬁbration X→Y, and a map W→Y, does there exist a lift W→X extending a given lift on A? Besides reducing to the absolute lifting problem when A = ∅, this includes the exten- sion problem by taking Y to be a point. Of course, one could broaden these questions by dropping the requirements that (W, A) be a CW pair and that the map X→Y be a ﬁbration. However, these conditions are often satisﬁed in cases of interest, and they make the task of ﬁnding solutions much easier. The term ‘obstruction theory’ refers to a procedure for deﬁning a sequence of cohomology classes that are the obstructions to ﬁnding a solution to the extension, lifting, or relative lifting problem. In the most favorable cases these obstructions lie in cohomology groups that are all zero, so the problem has a solution. But even when the obstructions are nonzero it can be very useful to have the problem expressed in cohomological terms. There are two ways of developing obstruction theory, which produce essentially the same result in the end. In the more elementary approach one tries to construct the extension or lifting one cell of W at a time, proceeding inductively over
skeleta of W. This approach has an appealing directness, but the technical details of working at the level of cochains are perhaps a little tedious. Instead of pursuing this direct line we shall follow the second approach, which is slightly more sophisticated but has the advantage that the theory becomes an almost trivial application of Postnikov towers for the extension problem, or Moore–Postnikov towers for the lifting problem. The cellular viewpoint is explained in [VBKT], where it appears in the study of characteristic classes of vector bundles. Let us consider the extension problem ﬁrst, where we wish to extend a map A→X to the larger complex W. Suppose that X has a Postnikov tower of principal ﬁbrations. Then we have a commutative diagram as shown below, where we have enlarged the tower by adjoining the space X0, which is just a point, at the bottom. The map X1→X0 is then a ﬁbration, and to say it is principal says that X1, which in any case is a K(π1X, 1), is the loopspace of K(π1X, 2), hence π1(X) must be abelian. Conversely, if π1(X) is abelian and acts trivially on all the higher homotopy groups of X, then there is an extended Postnikov tower of principal ﬁbrations as shown. Our strategy will be to try to lift the constant map W→X0 to maps W→Xn for n = 1, 2, ··· in succession, extending the given maps A→Xn. If we are able to ﬁnd all these lifts W→Xn, there will then be no diﬃculty in constructing the desired extension W→X. Connections with Cohomology Section 4.3 417 For the inductive step we have a commutative diagram as at the right. Since Xn is the pullback, its points are pairs consisting of a point in Xn−1 and a path from its image in K to the basepoint. A lift W→Xn therefore amounts to a nullhomotopy of the composition W→Xn−1→K. We already have such a lift deﬁned on A, hence a nullhomotopy of A→K, and we want a nullhomotopy of W→K extending this nullhomotopy on A
. The map W→K together with the nullhomotopy on A gives a map W ∪ CA→K, where CA is the cone on A. Since K is a K(πnX, n + 1), the map W ∪ CA→K determines an obstruction class ωn ∈ H n+1(W ∪ CA; πnX) ≈ H n+1(W, A; πnX). Proposition 4.72. A lift W→Xn extending the given A→Xn exists iﬀ ωn = 0. Proof: We need to show that the map W ∪ CA→K extends to a map CW→K iﬀ ωn = 0, or in other words, iﬀ W ∪ CA→K is homotopic to a constant map. Suppose that gt : W ∪ CA→K is such a homotopy. The constant map g1 then extends to the constant map g1 : CW→K, so by the homotopy extension property for the pair (CW, W ∪ CA), applied to the reversed homotopy g1−t, we have a homotopy gt : CW→K extending the previous homotopy gt : W ∪ CA→K. The map g0 : CW→K then extends the given map W ∪ CA→K. Conversely, if we have an extension CW→K, then this is nullhomotopic since the ⊔⊓ cone CW is contractible, and we may restrict such a nullhomotopy to W ∪ CA. ←-- Xn extending the given A→X→ lim ←-- Xn. Since the restriction of W→ lim If we succeed in extending the lifts A→Xn to lifts W→Xn for all n, then we obtain a map W→ lim ←-- Xn. Let M be the mapping cylinder of X→ lim ←-- Xn ⊂ M to A factors through X, this gives a homotopy of this restriction to the map A→X ⊂ M. Extend this to a homotopy of W→M, producing a map (W, A)→(M, X). Since the map X→ lim ←-- Xn is a weak homotopy equivalence, πi(M, X) = 0 for all
i, so by Lemma 4.6, the compression lemma, the map (W, A)→(M, X) can be homotoped to a map W→X extending the given A→X, and we have solved the extension problem. Thus if it happens that at each stage of the inductive process of constructing lifts W→Xn the obstruction ωn ∈ H n+1(W, A; πnX) vanishes, then the extension problem has a solution. In particular, this yields: Corollary 4.73. If X is a connected abelian CW complex and (W, A) is a CW pair such that H n+1(W, A; πnX) = 0 for all n, then every map A→X can be extended to a map W→X. ⊔⊓ This is a considerable improvement on the more elementary result that extensions exist if πn(X) = 0 for all n such that W − A has cells of dimension n + 1, which is Lemma 4.7. 418 Chapter 4 Homotopy Theory We can apply the Hurewicz theorem and obstruction theory to extend the homology version of Whitehead’s theorem to CW complexes with trivial action of π1 on all homotopy groups: Proposition 4.74. If X and Y are connected abelian CW complexes, then a map f : X→Y inducing isomorphisms on all homology groups is a homotopy equivalence. Proof: Taking the mapping cylinder of f reduces us to the case of an inclusion X֓Y of a subcomplex. If we can show that π1(X) acts trivially on πn(Y, X) for all n, then the relative Hurewicz theorem will imply that πn(Y, X) = 0 for all n, so X→Y will be a weak homotopy equivalence. The assumptions guarantee that π1(X)→π1(Y ) is an isomorphism, so we know at least that π1(Y, X) = 0. We can use obstruction theory to extend the identity map X→X to a retraction Y →X. To apply the theory we need π1(X) acting trivially on πn(X), which holds by hypothesis. Since the inclusion X ֓ Y induces isomorphisms
on homology, we have H∗(Y, X) = 0, hence H n+1(Y, X; πn(X)) = 0 for all n by the universal coeﬃcient theorem. So there are no obstructions, and a retraction Y →X exists. This implies that the maps πn(Y )→πn(Y, X) are onto, so trivial action of π1(X) on πn(Y ) implies ⊔⊓ trivial action on πn(Y, X) by naturality of the action. The generalization of the preceding analysis of the extension problem to the rela- tive lifting problem is straightforward. Assuming the ﬁbration p : X→Y in the statement of the relative lifting problem has a Moore–Postnikov tower of principal ﬁbrations, we have the diagram at the right, where F is the ﬁber of the ﬁbration X→Y. The ﬁrst step is to lift the map W→Y to Z1, extending the given lift on A. We may take Z1 to be the covering space of Y corresponding to the subgroup p∗(π1(X)) of π1(Y ), so covering space theory tells us when we can lift W→Y to Z1, and the unique lifting property for covering spaces can be used to see whether a lift can be chosen to agree with the lift on A given by the diagram; this could only be a problem when A has more than one component. Having a lift to Z1, the analysis proceeds exactly as before. One ﬁnds a sequence of obstructions ωn ∈ H n+1(W, A; πnF ), assuming π1F is abelian in the case n = 1. A lift to X exists, extending the given lift on A, if each successive ωn is zero. One can ask the converse question: If a lift exists, must the obstructions ωn all be zero? Since Proposition 4.72 is an if and only if statement, one might expect the answer to be yes, but upon closer inspection the matter becomes less clear. The diﬃculty is that, even if at some stage the obstruction ωn is zero, so a lift to Zn+1 exists, there may
be many choices of such a lift, and diﬀerent choices could lead to diﬀerent ωn+1 ’s, some zero and others nonzero. Examples of such ambiguities are not hard to produce, for both the lifting and the extension problems, and the Connections with Cohomology Section 4.3 419 ambiguities only become worse with each subsequent choice of a lift. So it is only in rather special circumstances that one can say that there are well-deﬁned obstructions. A simple case is when πi(F ) = 0 for i < n, so the Moore– Postnikov factorization begins with Zn In this as in the diagram at the right. case the composition across the bottom of the diagram gives a well-deﬁned primary obstruction ωn ∈ H n+1(W, A; πnF ). e Exercises 1. Show there is a map RP∞→CP∞ = K(Z, 2) which induces the trivial map on H ∗(−; Z). How is this consistent with the universal H∗(−; Z) but a nontrivial map on coeﬃcient theorem? e 2. Show that the group structure on S 1 coming from multiplication in C induces a group structure on hX, S 1i such that the bijection hX, S 1i→H 1(X; Z) of Theorem 4.57 is an isomorphism. 3. Suppose that a CW complex X contains a subcomplex S 1 such that the inclusion S 1 ֓ X induces an injection H1(S 1; Z)→H1(X; Z) with image a direct summand of H1(X; Z). Show that S 1 is a retract of X. 4. Given abelian groups G and H and CW complexes K(G, n) and K(H, n), show that the map hK(G, n), K(H, n)i→Hom(G, H) sending a homotopy class [f ] to the induced homomorphism f∗ : πn(K(G, n))→πn(K(H, n)) is a bijection. 5. Show that [X, S n] ≈ H n(X; Z) if X is an n dimensional CW complex
. K(Z, n) from S n by attaching cells of dimension ≥ n + 2.] 6. Use Exercise 4 to construct a multiplication map µ : K(G, n)× K(G, n)→K(G, n) for any abelian group G, making a CW complex K(G, n) into an H–space whose multipli- [Build a cation is commutative and associative up to homotopy and has a homotopy inverse. Show also that the H–space multiplication µ is unique up to homotopy. 7. Using an H–space multiplication µ on K(G, n), deﬁne an addition in hX, K(G, n)i by [f ] + [g] = [µ(f, g)] and show that under the bijection H n(X; G) ≈ hX, K(G, n)i this addition corresponds to the usual addition in cohomology. 8. Show that a map p : E→B is a ﬁbration iﬀ the map π : EI→Ep, π (γ) = (γ(0), pγ), has a section, that is, a map s : Ep→EI such that π s = 11. 9. Show that a linear projection of a 2 simplex onto one of its edges is a ﬁbration but not a ﬁber bundle. [Use the preceding problem.] 10. Given a ﬁbration F→E→B, use the homotopy lifting property to deﬁne an action of π1(E) on πn(F ), a homomorphism π1(E)→Aut, such that the composiπn(F ) tion π1(F )→π1(E)→Aut is the usual action of π1(F ) on πn(F ). Deduce πn(F ) that if π1(E) = 0, then the action of π1(F ) on πn(F ) is trivial. 420 Chapter 4 Homotopy Theory 11. For a space B, let F(B) be the set of ﬁber homotopy equivalence classes of ﬁbrations E→B. Show that a map f :
B1→B2 induces f ∗ : F(B2)→F(B1) depending only on the homotopy class of f, with f ∗ a bijection if f is a homotopy equivalence. 12. Show that for homotopic maps f, g : A→B the ﬁbrations Ef →B and Eg→B are ﬁber homotopy equivalent. 13. Given a map f : A→B and a homotopy equivalence g : C→A, show that the ﬁbrations Ef →B and Ef g→B are ﬁber homotopy equivalent. [One approach is to use Corollary 0.21 to reduce to the case of deformation retractions.] 14. For a space B, let M(B) denote the set of equivalence classes of maps f : A→B where f1 : A1→B is equivalent to f2 : A2→B if there exists a homotopy equivalence g : A1→A2 such that f1 ≃ f2g. Show the natural map F(B)→M(B) is a bijection. [See Exercises 11 and 13.] 15. If the ﬁbration p : E→B is a homotopy equivalence, show that p is a ﬁber homotopy equivalence of E with the trivial ﬁbration 11 : B→B. 16. Show that a map f : X→Y of connected CW complexes is a homotopy equivalence if it induces an isomorphism on π1 and its homotopy ﬁber Ff has H∗(Ff ; Z) = 0. X is an H–space with multiplication the composition of loops. 17. Show that B→F→E→B induces a long exact se18. Show that a ﬁbration sequence ··· → Ω quence ··· →hX, Bi→hX, F i→hX, Ei→hX, Bi, with groups and group homomorphisms except for the last three terms, abelian groups except for the last six terms. 19. Given a ﬁbration F -→ E B on the homotopy ﬁber Fp and use this to show that exactness at hX, F i in the long
exact sequence in the preceding problem can be improved to the statement that two elements of hX, F i have the same image in hX, Ei iﬀ they are in the same orbit of the induced action of p-----→ B, deﬁne a natural action of Ω Ω Ω e hX, Bi on hX, F i. 20. Show that by applying the loopspace functor to a Postnikov tower for X one Ω obtains a Postnikov tower of principal ﬁbrations for X. 21. Show that in the Postnikov tower of an H–space, all the spaces are H–spaces and the maps are H–maps, commuting with the multiplication, up to homotopy. 22. Show that a principal ﬁbration C -→ E p-----→ B is ﬁber homotopy equivalent to the product C × B iﬀ it has a section, a map s : B→E with ps = 11. Ω Ω 23. Prove the following uniqueness result for the Quillen plus construction: Given a connected CW complex X, if there is an abelian CW complex Y and a map X→Y inducing an isomorphism H∗(X; Z) ≈ H∗(Y ; Z), then such a Y is unique up to homotopy equivalence. [Use Corollary 4.73 with W the mapping cylinder of X→Y.] 24. In the situation of the relative lifting problem, suppose one has two diﬀerent lifts W→X that agree on the subspace A ⊂ W. Show that the obstructions to ﬁnding a homotopy rel A between these two lifts lie in the groups H n(W, A; πnF ). Ω Basepoints and Homotopy Section 4.A 421 In the ﬁrst part of this section we will use the action of π1 on πn to describe the diﬀerence between πn(X, x0) and the set of homotopy classes of maps S n→X without conditions on basepoints. More generally, we will compare the set hZ, Xi of basepoint-preserving homotopy classes of maps (Z, z0)→(X, x0) with
the set [Z, X] of unrestricted homotopy classes of maps Z→X, for Z any CW complex with basepoint z0 a 0 cell. Then the section concludes with an extended example exhibiting some rather subtle nonﬁnite generation phenomena in homotopy and homology groups. We begin by constructing an action of π1(X, x0) on hZ, Xi when Z is a CW complex with basepoint 0 cell z0. Given a loop γ in X based at x0 and a map f0 : (Z, z0)→(X, x0), then by the homotopy extension property there is a homotopy fs : Z→X of f0 such that fs(z0) is the loop γ. We might try to deﬁne an action of π1(X, x0) on hZ, Xi by [γ][f0] = [f1], but this deﬁnition encounters a small problem when we compose loops. For if η is another loop at x0, then by applying the homotopy extension property a second time we get a homotopy of f1 restricting to η on, x0, and the two homotopies together give the relation [γ][f0] in view of our convention that the product γη means ﬁrst γ, then η. This is not quite the relation we want, but the problem is easily corrected by letting the action be an action on the right rather than on the left. Thus we set [f0][γ] = [f1], and then [f0] Let us check that this right action is well-deﬁned. Suppose we start with maps f0, g0 : (Z, z0)→(X, x0) representing the same class in hZ, Xi, together with homotopies fs and gs of f0 and g0 such that fs(z0) and gs(z0) are homotopic loops. These various homotopies deﬁne a map H : Z × I × ∂I ∪ Z × {0}× I ∪ {z0}× I × I -→ X which is fs on Z × I × {0}, gs on Z × I × {1}, the bas
epoint-preserving homotopy from f0 to g0 on Z × {0}× I, and the homotopy from fs(z0) to gs(z0) on {z0}× I × I. We would like to extend H over Z × I × I. The pair (I × I, I × ∂I ∪ {0}× I) is homeo- [f0] = [η] [f0][γ] [γ][η] [γ][η] [η]. = morphic to (I × I, I × {0}), and via this homeomorphism we can view H as a map Z × I × {0} ∪ {z0}× I × I -→ X, that is, a map Z × I→X with a homotopy on the subcomplex {z0}× I. This means the homotopy extension property can be applied to produce an extension of the original H to Z × I × I. Restricting this extended H to Z × {1}× I gives a basepoint-preserving homotopy f1 ≃ g1, which shows that [f0][γ] is well-deﬁned. Note that in this argument we did not have to assume the homotopies fs and gs were constructed by applying the homotopy extension property. Thus we have proved 422 Chapter 4 Homotopy Theory the following result: Proposition 4A.1. There is a right action of π1(X, x0) on hZ, Xi deﬁned by setting [f0][γ] = [f1] whenever there exists a homotopy fs : Z→X from f0 to f1 such that ⊔⊓ fs(z0) is the loop γ, or any loop homotopic to γ. It is easy to convert this right action into a left action, by deﬁning [γ][f0] = [f0][γ]−1. This just amounts to choosing the homotopy fs so that fs(z0) is the inverse path of γ. When Z = S n this action reduces to the usual action of π1(X, x0) on πn(X, x0) since in the original deﬁnition of �
�f in terms of maps (In, ∂In)→(X, x0), a homotopy from γf to f is obtained by restricting γf to smaller and smaller concentric cubes, and on the ‘basepoint’ ∂In this homotopy traces out the loop γ. Proposition 4A.2. If (Z, z0) is a CW pair and X is a path-connected space, then the natural map hZ, Xi→[Z, X] induces a bijection of the orbit set hZ, Xi/π1(X, x0) onto [Z, X]. In particular, this implies that [Z, X] = hZ, Xi if X is simply-connected. Proof: Since X is path-connected, every f : Z→X can be homotoped to take z0 to the basepoint x0, via homotopy extension, so the map hZ, Xi→[Z, X] is onto. If f0 and f1 are basepoint-preserving maps that are homotopic via the homotopy fs : Z→X, then by deﬁnition [f1] = [f0][γ] for the loop γ(s) = fs(z0), so [f0] and [f1] are in the same orbit under the action of π1(X, x0). Conversely, two basepoint-preserving ⊔⊓ maps in the same orbit are obviously homotopic. Example 4A.3. If X is an H–space with identity element x0, then the action of π1(X, x0) on hZ, Xi is trivial since for a map f : (Z, z0)→(X, x0) and a loop γ in X based at x0, the multiplication in X deﬁnes a homotopy fs (z) = f (z)γ(s). This starts and ends with a map homotopic to f, and the loop fs(z0) is homotopic to γ, both these homotopies being basepoint-preserving by the deﬁnition of an H–space. The set of orbits of the π1 action on πn does not generally inherit a group structure from π
n. For example, when n = 1 the orbits are just the conjugacy classes in π1, and these form a group only when π1 is abelian. Basepoints are thus a necessary technical device for producing the group structure in homotopy groups, though as we have shown, they can be ignored in simply-connected spaces. For a set of maps S n→X to generate πn(X) as a module over Z[π1(X)] means that all elements of πn(X) can be represented by sums of these maps along arbitrary paths in X, where we allow reversing orientations to get negatives and repetitions to get arbitrary integer multiples. Examples of ﬁnite CW complexes X for which πn(X) is not ﬁnitely generated as a module over Z[π1(X)] were given in Exercise 38 in §4.2, provided n ≥ 3. Finding such an example for n = 2 seems to be more diﬃcult. The Basepoints and Homotopy Section 4.A 423 rest of this section will be devoted to a somewhat complicated construction which does this, and is interesting for other reasons as well. An Example of Nonﬁnite Generation We will construct a ﬁnite CW complex having πn not ﬁnitely generated as a Z[π1] module, for a given integer n ≥ 2. The complex will be a subcomplex of a K(π, 1) having interesting homological properties: It is an (n + 1) dimensional CW complex with Hn+1 nonﬁnitely generated, but its n skeleton is ﬁnite so Hi is ﬁnitely generated for i ≤ n and π is ﬁnitely presented if n > 1. The ﬁrst such example was found in [Stallings 1963] for n = 2. Our construction will be essentially the n dimensional generalization of this, but described in a more geometric way as in [Bestvina & Brady 1997], which provides a general technique for constructing many examples of this sort. To begin, let X be the product of n copies of S 1 ∨ S 1. Since S 1 ∨ S 1 is the 1 skeleton of the torus T 2 = S 1 × S 1 in its usual CW structure, X can be regarded as a subcomplex of
the 2n dimensional torus T 2n, the product of 2n circles. Deﬁne f : T 2n→S 1 by f (θ1, ···, θ2n) = θ1 + ··· + θ2n where the coordinates θi ∈ S 1 are viewed as angles measured in radians. The space Z = X ∩ f −1(0) will provide the example we are looking for. As we shall see, Z is a ﬁnite CW complex of dimension n − 1, with πn−1(Z) nonﬁnitely generated as a module over π1(Z) if n ≥ 3. We will also see that πi(Z) = 0 for 1 < i < n − 1. The induced homomorphism f∗ : π1(T 2n)→π1(S 1) = Z sends each generator T 2n→T 2n be the covering space corresponding coming from an S 1 factor to 1. Let to the kernel of f∗. This is a normal covering space since it corresponds to a normal T 2n projecting subgroup, and the deck transformation group is Z. The subcomplex of X→X with the same group of deck transformations. to X is a normal covering space Since π1(X) is the product of n free groups on two generators, X is the covering e space of X corresponding to the kernel of the homomorphism π1(X)→Z sending each of the two generators of each free factor to 1. Since X is a K(π, 1), so is X. For example, when n = 1, X is the union of two helices on the inﬁnite cylinder e e e T 2 : e e e The map f lifts to a map f : T 2n→R, and Z lifts homeomorphically to a subspace Z ⊂ X, namely f −1(0) ∩ X. We will show: e e e (∗) X is homotopy equivalent to a space Y obtained from Z by attaching an inﬁnite e e sequence of n cells. e Assuming this is true, it follows that Hn(Y ) is not ﬁnitely generated since in the exact sequence Hn(Z)→Hn(Y )→Hn(Y,
Z)→Hn−1(Z) the ﬁrst term is zero and the last term is ﬁnitely generated, Z being a ﬁnite CW complex of dimension n − 1, 424 Chapter 4 Homotopy Theory while the third term is an inﬁnite sum of Z ’s, one for each n cell of Y. If πn−1(Z) were ﬁnitely generated as a π1(Z) module, then by attaching ﬁnitely many n cells to Z we could make it (n − 1) connected since it is already (n − 2) connected as the (n − 1) skeleton of the K(π, 1) Y. Then by attaching cells of dimension greater than n we could build a K(π, 1) with ﬁnite n skeleton. But this contradicts the fact that Hn(Y ) is not ﬁnitely generated. To begin the veriﬁcation of (∗), consider the torus T m. The standard cell structure on T m lifts to a cubical cell structure on the universal cover Rm, with vertices the integer lattice points Zm. The function f lifts to a linear projection L : Rm→R, L(x1, ···, xm) = x1 + ··· + xm. The planes in L−1(Z) cut the cubes of Rm into convex polyhedra which we call slabs. There are m slabs in each m dimensional cube. The boundary of a slab in L−1[i, i + 1] consists of lateral faces that are slabs for lower-dimensional cubes, together with a lower face in L−1(i) and an upper face in L−1(i + 1). In each cube there are two exceptional slabs whose lower or upper face degenerates to a point. These are the slabs containing the vertices of the cube where L has its maximum and minimum values. A slab defor- mation retracts onto the union of its lower and lateral faces, provided that the slab has an upper face that is not just a point. Slabs of the latter type are m simplices, and we will refer to them as cones in what follows. These are the slabs containing the vertex of a cube on which L takes its maximal value. The lateral faces of
a cone are also cones, of lower dimension. The slabs, together with all their lower-dimensional faces, give a CW structure on Rm with the planes of L−1(Z) as subcomplexes. These structures are preserved by the deck transformations of the cover Rm→T m so there is an induced CW structure in the quotient T m, with f −1(0) as a subcomplex. If X is any subcomplex of T m in its original cubical cell structure, then the slab CW structure on T m restricts to a CW structure on X. In particular, we obtain a CW structure on Z = X ∩ f −1(0). Likewise we get a lifted CW structure on the cover f −1[i, j]. The deformation retractions of noncone slabs X ⊂ X[i, i + 1] X[i, i + 1]. These cones are X[i + 1], so Ci onto their lateral and lower faces give rise to a deformation retraction of e onto e e X[i] ∪ Ci where Ci consists of all the cones in attached along their lower faces, and they all have the same vertex in T m. Let X[i, j] = X ∩ e e e e is itself a cone in the usual sense, attached to e X[i] along its base. e For the particular X we are interested in, we claim that each Ci is an n disk attached along its boundary sphere. When n = 1 this is evident from the earlier e picture of X as the union of two helices on a cylinder. For larger n we argue by induction. Passing from n to n + 1 replaces X by two copies of X × S 1 intersecting in X, one copy for each of the additional S 1 factors of T 2n+2. Replacing X by X × S 1 changes Ci to its join with a point in the base of the new Ci. The two copies of this e Basepoints and Homotopy Section 4.A 425 join then yield the suspension of Ci attached along the suspension of the base. The same argument shows that X[−i − 1, −i] deformation retracts onto X[−i] e with an n cell attached. We build the space Y and a homotopy equivalence g : Y → X by an inductive procedure, starting with Y0 = Z. Assuming that Yi and a
homotopy e equivalence gi : Yi→ X[−i, i] have already been deﬁned, we form Yi+1 by attaching two n cells by the maps obtained from the attaching maps of the two n cells in X[−i, i] by composing with a homotopy inverse to gi. This allows X[−i − 1, i + 1]. Taking the X[−i − 1, i + 1] − gi to be extended to a homotopy equivalence gi+1 : Yi+1→ e union over i gives g : Y → that it induces isomorphisms on all homotopy groups, using the standard compact- X. One can check this is a homotopy equivalence by seeing e e e e e ness argument. This ﬁnishes the veriﬁcation of (∗). It is interesting to see what the complex Z looks like in the case n = 3, when Z is 2 dimensional and has π2 nonﬁnitely generated over Z[π1(Z)]. In this case X is the product of three S 1 ∨ S 1 ’s, so X is the union of the eight 3 tori obtained by choosing one of the two S 1 summands in each S 1 ∨ S 1 factor. We denote these 3 tori S 1 ±. Viewing each of these 3 tori as the cube in the previous ﬁgure with opposite faces identiﬁed, we see that Z is the union of the eight 2 tori formed ± × S 1 ± × S 1 by the two sloping triangles in each cube. Two of these 2 tori intersect along a circle when the corresponding 3 tori of X intersect along a 2 torus. This happens when the triples of ± ’s for the two 3 tori diﬀer in exactly one entry. The pattern of intersection of the eight 2 tori of Z can thus be described combinatorially via the 1 skeleton of the cube, with vertices (±1, ±1, ±1). There is a torus of Z for each vertex of the cube, and two tori intersect along a circle when the corresponding vertices of the cube are the endpoints of an edge of the cube. All eight tori contain the single 0 cell of Z. To obtain a model of Z itself, consider a
regular octahedron inscribed in the cube with vertices (±1, ±1, ±1). If we identify each pair of oppo- site edges of the octahedron, each pair of opposite triangular faces becomes a torus. However, there are only four pairs of opposite faces, so we get only four tori this way, not eight. To correct this problem, regard each triangular face of the octa- hedron as two copies of the same triangle, distinguished from each other by a choice of normal direction, an arrow attached to the triangle pointing either inside the octahedron or outside it, that is, either to- ward the nearest vertex of the surrounding cube or toward the opposite vertex of the cube. Then each pair of opposite triangles of the octahedron having normal vectors pointing toward the same vertex of the cube determines a torus, when opposite edges are identiﬁed as before. Each edge of the original octahedron is also replaced by two edges oriented either toward the interior or exterior of the octahedron. The vertices of the octahedron may be left unduplicated since they will all be identiﬁed to a single point anyway. With this scheme, the two tori corresponding to the vertices at the ends 426 Chapter 4 Homotopy Theory of an edge of the cube then intersect along a circle, as they should, and other pairs of tori intersect only at the 0 cell of Z. This model of Z has the advantage of displaying the symmetry group of the cube, a group of order 48, as a symmetry group of Z, corresponding to the symmetries of X permuting the three S 1 ∨ S 1 factors and the two S 1 ’s of each S 1 ∨ S 1. Undoubtedly Z would be very pretty to look at if we lived in a space with enough dimensions to see all of it at one glance. It might be interesting to see an explicit set of maps S 2→Z generating π2(Z) as a Z[π1] module. One might also ask whether there are simpler examples of these nonﬁnite generation phenomena. Exercises 1. Show directly that if X is a topological group with identity element x0, then any two maps f, g : (Z, z0)→(X, x0) which are homotopic are homotopic through basepointpreserving maps
. 2. Show that under the map hX, Y i→Hom, [f ] ֏ f∗, the acπn(X, x0), πn(Y, y0) tion of π1(Y, y0) on hX, Y i corresponds to composing with the action on πn(Y, y0), that is, (γf )∗ = βγf∗. Deduce a bijection of [X, K(π, 1)] with the set of orbits of Hom(π1(X), π ) under composition with inner automorphisms of π. In particular, if π is abelian then [X, K(π, 1)] = hX, K(π, 1)i = Hom(π1(X), π ). 3. For a space X let Aut(X) denote the group of homotopy classes of homotopy equivalences X→X. Show that for a CW complex K(π, 1), Aut phic to the group of outer automorphisms of π, that is, automorphisms modulo inner is isomor- K(π, 1) automorphisms. 4. With the notation of the preceding problem, show that Aut( n S k) ≈ GLn(Z) for n S k denotes the wedge sum of n copies of S k and GLn(Z) is the k > 1, where W group of n× n matrices with entries in Z having an inverse matrix of the same form. W [ GLn(Z) is the automorphism group of Zn ≈ πk( 5. This problem involves the spaces constructed in the latter part of this section. n S k) ≈ Hk( W n S k).] W (a) Compute the homology groups of the complex Z in the case n = 3, when Z is 2 dimensional. e (b) Letting Xn−1→ X, show that inductively from Xn denote the n dimensional complex e ating deck transformation Xn can be obtained Xn−1 as the union of two copies of the mapping torus of the generXn−1 in these two mapping Xn−1. Xn→S 1 ∨ S 1 with ﬁber tori identiﬁed. Thus there is a ﬁ
ber bundle (c) Use part (b) to ﬁnd a presentation for π1( Xn), and show this presentation reduces e to a ﬁnite presentation if n > 2 and a presentation with a ﬁnite number of generators e if n = 2. In the latter case, deduce that π1( X2) has no ﬁnite presentation from the fact that H2( X2) is not ﬁnitely generated. Xn−1, with copies of e e e e e e e e The Hopf Invariant Section 4.B 427 In §2.2 we used homology to distinguish diﬀerent homotopy classes of maps S n→S n via the notion of degree. We will show here that cup product can be used to do something similar for maps S 2n−1→S n. Originally this was done by Hopf using more geometric constructions, before the invention of cohomology and cup products. In general, given a map f : Sm→S n with m ≥ n, we can form a CW complex Cf by attaching a cell em+1 to S n via f. The homotopy type of Cf depends only on the homotopy class of f, by Proposition 0.18. Thus for maps f, g : Sm→S n, any invariant of homotopy type that distinguishes Cf from Cg will show that f is not homotopic to g. For example, if m = n and f has degree d, then from the cellular chain complex of Cf we see that Hn(Cf ) ≈ Z\|d\|, so the homology of Cf detects the degree of f, up to sign. When m > n, however, the homology of Cf consists of Z ’s in dimensions 0, n, and m + 1, independent of f. The same is true of cohomology groups, but cup products have a chance of being nontrivial in H ∗(Cf ) when m = 2n − 1. In this case, if we choose generators α ∈ H n(Cf ) and β ∈ H 2n(Cf ), then the multiplicative structure of H ∗(Cf ) is determined by a relation α2 = H(f )β for an integer H(f ) called the Hopf invariant of f. The sign of
H(f ) depends on the choice of the gener- ator β, but this can be speciﬁed by requiring β to correspond to a ﬁxed generator of H 2n(D2n, ∂D2n) under the map H 2n(Cf ) ≈ H 2n(Cf, S n)→H 2n(D2n, ∂D2n) induced by the characteristic map of the cell e2n, which is determined by f. We can then change the sign of H(f ) by composing f with a reﬂection of S 2n−1, of degree −1. If f ≃ g, then under the homotopy equivalence Cf ≃ Cg the chosen generators β for H 2n(Cf ) and H 2n(Cg) correspond, so H(f ) depends only on the homotopy class of f. If f is a constant map then Cf = S n ∨ S 2n and H(f ) = 0 since Cf retracts onto S n. Also, H(f ) is always zero for odd n since in this case α2 = −α2 by the commutativity property of cup product, hence α2 = 0. Three basic examples of maps with nonzero Hopf invariant are the maps deﬁning the three Hopf bundles in Examples 4.45, 4.46, and 4.47. The ﬁrst of these Hopf maps is the attaching map f : S 3→S 2 for the 4 cell of CP 2. This has H(f ) = 1 since H ∗(CP2; Z) ≈ Z[α]/(α3) by Theorem 3.19. Similarly, HP2 gives rise to a map S 7→S 4 of Hopf invariant 1. In the case of the octonionic projective plane OP2, which is built from the map S 15→S 8 deﬁned in Example 4.47, we can deduce that H ∗(OP2; Z) ≈ Z[α]/(α3) either from Poincar´e duality as in Example 3.40 or from Exercise 5 for §4.D. It is a fundamental theorem of [Adams 1960] that a map f : S 2n−1→S n of Hopf invariant 1 exists only when
n = 2, 4, 8. This has a number of very interesting con- sequences, for example: 428 Chapter 4 Homotopy Theory Rn is a division algebra only for n = 1, 2, 4, 8. S n is an H–space only for n = 0, 1, 3, 7. S n has n linearly independent tangent vector ﬁelds only for n = 0, 1, 3, 7. The only ﬁber bundles S p→S q→S r occur when (p, q, r ) = (0, 1, 1), (1, 3, 2), (3, 7, 4), and (7, 15, 8). The ﬁrst and third assertions were in fact proved shortly before Adams’ theorem in [Kervaire 1958] and [Milnor 1958] as applications of a theorem of Bott that π2nU(n) ≈ Zn!. A full discussion of all this, and a proof of Adams’ theorem, is given in [VBKT]. Though maps of Hopf invariant 1 are rare, there are maps S 2n−1→S n of Hopf invariant 2 for all even n. Namely, consider the space J2(S n) constructed in §3.2. This has a CW structure with three cells, of dimensions 0, n, and 2n, so J2(S n) has the form Cf for some f : S 2n−1→S n. We showed that if n is even, the square of a generator of H n(J2(S n); Z) is twice a generator of H 2n(J2(S n); Z), so H(f ) = ±2. From this example we can get maps of any even Hopf invariant when n is even via the following fact. Proposition 4B.1. The Hopf invariant H : π2n−1(S n)→Z is a homomorphism. Proof: For f, g : S 2n−1→S n, let us compare Cf +g with the space Cf ∨g obtained from S n by attaching two 2n cells via f and g. There is a natural quotient map q : Cf +g→Cf ∨g collapsing the equatorial disk of the 2n cell of Cf +g to a point. The
induced cellular chain map q∗ sends e2n f +g to e2n g. In cohomology this implies that q∗(βf ) = q∗(βg) = βf +g where βf, βg, and βf +g are the cohomology classes dual to the 2n cells. Letting αf +g and αf ∨g be the cohomology classes corresponding to the n cells, we have q∗(αf ∨g) = αf +g since q is a homeomorphism on the n cells. By restricting to the subspaces Cf and Cg of Cf ∨g we see that α2 f ∨g = H(f )βf + H(g)βg. Thus α2 ⊔⊓ f ∨g) = H(f )q∗(βf ) + H(g)q∗(βg) = f +g = q∗(α2 H(f ) + H(g) f +e2n βf +g. Corollary 4B.2. π2n−1(S n) contains a Z direct summand when n is even. Proof: Either H or H/2 is a surjective homomorphism π2n−1(S n)→Z. ⊔⊓ Exercises 1. Show that the Hopf invariant of a composition S 2n−1 f-----→ S 2n−1 g-----→ S n is given by H(gf ) = (deg f )H(g), and for a composition S 2n−1 f-----→ S n g-----→ S n the Hopf invariant satisﬁes H(gf ) = (deg g)2H(f ). 2. Show that if S k -→ Sm p-----→ S n is a ﬁber bundle, then m = 2n − 1, k = n − 1, and, when n > 1, H(p) = ±1. [Show that Cp is a manifold and apply Poincar´e duality.] Minimal Cell Structures Section 4.C 429 We can apply the homology version of Whitehead’s theorem, Corollary 4.33, to show that a simply-connected CW complex with ﬁnitely
generated homology groups is always homotopy equivalent to a CW complex having the minimum number of cells consistent with its homology, namely, one n cell for each Z summand of Hn and a pair of cells of dimension n and n + 1 for each Zk summand of Hn. Proposition 4C.1. Given a simply-connected CW complex X and a decomposition of each of its homology groups Hn(X) as a direct sum of cyclic groups with speciﬁed generators, then there is a CW complex Z and a cellular homotopy equivalence f : Z→X such that each cell of Z is either : (a) a ‘generator’ n cell en α, which is a cycle in cellular homology mapped by f to a cellular cycle representing the speciﬁed generator α of one of the cyclic summands of Hn(X) ; or (b) a ‘relator’ (n + 1) cell en+1 α, with cellular boundary equal to a multiple of the generator n cell en α, in the case that α has ﬁnite order. In the nonsimply-connected case this result can easily be false, counterexamples being provided by acyclic spaces and the space X = (S 1 ∨ S n) ∪ en+1 constructed in Example 4.35, which has the same homology as S 1 but which must have cells of dimension greater than 1 in order to have πn nontrivial. Proof: We build Z inductively over skeleta, starting with Z 1 a point since X is simplyconnected. For the inductive step, suppose we have constructed f : Z n→X inducing an isomorphism on Hi for i < n and a surjection on Hn. For the mapping cylinder Mf we then have Hi(Mf, Z n) = 0 for i ≤ n and Hn+1(Mf, Z n) ≈ πn+1(Mf, Z n) by the Hurewicz theorem. To construct Z n+1 we use the following diagram: By induction we know the map Hn(Z n)→Hn(Mf ) ≈ Hn(X) exactly, namely, Z n has generator n cells, which are cellular cycles mapping to the given generators of Hn(X), along with relator n cells that
do not contribute to Hn(Z n). Thus Hn(Z n) is free with basis the generator n cells, and the kernel of Hn(Z n)→Hn(X) is free with basis given by certain multiples of some of the generator n cells. Choose ‘relator’ elements ρi in Hn+1(Mf, Z n) mapping to this basis for the kernel, and let the ‘generator’ elements γi ∈ Hn+1(Mf, Z n) be the images of the chosen generators of Hn+1(Mf ) ≈ Hn+1(X). Via the Hurewicz isomorphism Hn+1(Mf, Z n) ≈ πn+1(Mf, Z n), the homology classes ρi and γi are represented by maps ri, gi : (Dn+1, S n)→(Mf, Z n). We form 430 Chapter 4 Homotopy Theory Z n+1 from Z n by attaching (n + 1) cells via the restrictions of the maps ri and gi to S n. The maps ri and gi themselves then give an extension of the inclusion Z n ֓ Mf to a map Z n+1→Mf, whose composition with the retraction Mf →X is the extended map f : Z n+1→X. This gives us the lower row of the preceding diagram, with commutative squares. By construction, the subgroup of Hn+1(Z n+1, Z n) generated by the relator (n + 1) cells maps injectively to Hn(Z n), with image the kernel of Hn(Z n)→Hn(X), so f∗ : Hn(Z n+1)→Hn(X) is an isomorphism. The elements of Hn+1(Z n+1, Z n) represented by the generator (n+1) cells map to the γi ’s, hence map to zero in Hn(Z n), so by exactness of the second row these generator (n + 1) cells are cellular cycles representing elements of Hn+1(Z n+1) mapped by f∗ to the given generators of Hn+1(X).
In particular, f∗ : Hn+1(Z n+1)→Hn+1(X) is surjective, and the induction step is ﬁnished. Doing this for all n, we produce a CW complex Z and a map f : Z→X with the ⊔⊓ desired properties. Example 4C.2. Suppose X is a simply-connected CW complex such that for some n ≥ 2, the only nonzero reduced homology groups of X are Hn(X), which is ﬁnitely generated, and Hn+1(X), which is ﬁnitely generated and free. Then the proposition says that X is homotopy equivalent to a CW complex Z obtained from a wedge sum of n spheres by attaching (n + 1) cells. The attaching maps of these cells are determined up to homotopy by the cellular boundary map Hn+1(Z n+1, Z n)→Hn(Z n) since πn(Z n) ≈ Hn(Z n). So the attaching maps are either trivial, in the case of generator (n + 1) cells, or they represent some multiple of an inclusion of one of the wedge summands, in the case of a relator (n + 1) cell. Hence Z is the wedge sum of spheres S n and S n+1 together with Moore spaces M(Zm, n) of the form S n ∪ en+1. In particular, the homotopy type of X is uniquely determined by its homology groups. Proposition 4C.3. Let X be a simply-connected space homotopy equivalent to a CW complex, such that the only nontrivial reduced homology groups of X are H2(X) ≈ Zm and H4(X) ≈ Z. Then the homotopy type of X is uniquely determined by the cup product ring H ∗(X; Z). In particular, this applies to any simply-connected closed 4 manifold. j of m 2 spheres S 2 Proof: By the previous proposition we may assume X is a complex Xϕ obtained from j by attaching a cell e4 via a map ϕ : S 3 wedge sum j ) is free with basis the Hopf maps ηj : S 3→S 2 j S 2 As shown in Example 4.52, π3( j and W W
the Whitehead products [ij, ik], j < k, where ij is the inclusion S 2 j. Since W a homotopy of ϕ does not change the homotopy type of Xϕ, we may assume ϕ is j<k ajk[ij, ik]. We need to see how the coeﬃcients a linear combination aj and ajk determine the cup product H 2(X; Z)× H 2(X; Z)→H 4(X; Z). j S 2 W j ajηj + j ֓ P P This cup product can be represented by an m× m symmetric matrix (bjk) where the cup product of the cohomology classes dual to the j th and k th 2-cells is bjk Cohomology of Fiber Bundles Section 4.D 431 j × S 2 times the class dual to the 4 cell. We claim that bjk = ajk for j < k and bjj = aj. If ϕ is one of the generators ηi or [ij, ik] this is clear, since if ϕ = ηj then Xϕ is the wedge sum of CP2 with m − 1 2 spheres, while if ϕ = [ij, ik] then Xϕ is the wedge sum of S 2 k with m − 2 2 spheres. The claim is also true when ϕ is −ηj or −[ij, ik] since changing the sign of ϕ amounts to composing ϕ with a reﬂection of S 3, and this changes the generator of H 4(Xϕ; Z) to its negative. The general case now follows by induction from the assertion that the matrix (bjk) for Xϕ+ψ is the sum of the corresponding matrices for Xϕ and Xψ. This assertion can be proved as j S 2 follows. By attaching two 4 cells to j by ϕ and ψ we obtain a complex Xϕ,ψ which we can view as Xϕ ∪ Xψ. There is a quotient map q : Xϕ+ψ -→ Xϕ,ψ that is a W homeomorphism on the 2 skeleton and collapses the closure of an equatorial 3 disk in the 4 cell of Xϕ+ψ to a point. The
induced map q∗ : H 4(Xϕ,ψ)→H 4(Xϕ+ψ) sends each of the two generators corresponding to the 4 cells of Xϕ,ψ to a generator, and the assertion follows. Now suppose Xϕ and Xψ have isomorphic cup product rings. This means bases for H ∗(Xϕ; Z) and H ∗(Xψ; Z) can be chosen so that the matrices specifying the cup product H 2× H 2→H 4 with respect to these bases are the same. The preceding proposition says that any choice of basis can be realized as the dual basis to a cell structure on a CW complex homotopy equivalent to the given complex. Therefore we may assume the matrices (bjk) for Xϕ and Xψ are the same. By what we have shown in the preceding paragraph, this means ϕ and ψ are homotopic, hence Xϕ and Xψ are homotopy equivalent. For the statement about simply-connected closed 4 manifolds, Corollaries A.8 and A.9 and Proposition A.11 in the Appendix say that such a manifold M has the ho- motopy type of a CW complex with ﬁnitely generated homology groups. Then Poincar´e duality and the universal coeﬃcient theorem imply that the only nontrivial homology groups Hi(M) are Z for i = 0, 4 and Zm for i = 2, for some m ≥ 0. ⊔⊓ This result and the example preceding it are special cases of a homotopy classi- ﬁcation by Whitehead of simply-connected CW complexes with positive-dimensional cells in three adjacent dimensions n, n + 1, and n + 2 ; see [Baues 1996] for a full treatment of this. While the homotopy groups of the three spaces in a ﬁber bundle ﬁt into a long exact sequence, the relation between their homology or cohomology groups is much more complicated. The K¨unneth formula shows that there are some subtleties even for a product bundle, and for general bundles the machinery of spectral sequences, Φ 432 Chapter 4 Homotopy Theory developed in [SSAT], is required. In this section we will describe a few special sorts of ﬁber bundles where more elementary techniques su
ﬃce. As applications we calculate the cohomology rings of some important spaces closely related to Lie groups. In particular we ﬁnd a number of spaces with exterior and polynomial cohomology rings. The Leray–Hirsch Theorem This theorem will be the basis for all the other results in this section. It gives hypotheses suﬃcient to guarantee that a ﬁber bundle has cohomology very much like that of a product bundle. p-----→ B be a ﬁber bundle such that, for some commutative Theorem 4D.1. Let F -→ E coeﬃcient ring R : (a) H n(F ; R) is a ﬁnitely generated free R module for each n. (b) There exist classes cj ∈ H kj (E; R) whose restrictions i∗(cj) form a basis for H ∗(F ; R) in each ﬁber F, where i : F→E is the inclusion. : H ∗(B; R) ⊗R H ∗(F ; R)→H ∗(E; R), Then the map ij bi ⊗ i∗(cj)֏ ij p∗(bi)`cj, is an isomorphism of R modules. P P The conclusion can be restated as saying that H ∗(E; R) is a free H ∗(B; R) module with basis {cj}, where we view H ∗(E; R) as a module over the ring H ∗(B; R) by deﬁning scalar multiplication by bc = p∗(b) ` c for b ∈ H ∗(B; R) and c ∈ H ∗(E; R). In the case of a product E = B × F with H ∗(F ; R) free over R, we can pull back a basis for H ∗(F ; R) via the projection E→F to obtain the classes cj. Thus the Leray–Hirsch theorem generalizes the version of the K¨unneth formula involving cup products, Theorem 3.15, at least as far as the additive structure and the module structure over H ∗(B; R) are concerned. However, the Leray–H
irsch theorem does not assert that the isomorphism H ∗(E; R) ≈ H ∗(B; R) ⊗R H ∗(F ; R) is a ring isomorphism, and in fact this need not be true, for example for the Klein bottle viewed as a bundle with ﬁber and base S 1, where the Leray–Hirsch theorem applies with Z2 coeﬃcients. An example of a bundle where the classes cj do not exist is the Hopf bundle S 1→S 3→S 2, since H ∗(S 3) 6≈ H ∗(S 2) ⊗ H ∗(S 1). Proof: We ﬁrst prove the result for ﬁnite-dimensional CW complexes B by induction on their dimension. The case that B is 0 dimensional is trivial. For the induction step, suppose B has dimension n, and let B′ ⊂ B be the subspace obtained by deleting a α of B. Let E′ = p−1(B′). Then we have a point xα from the interior of each n cell en commutative diagram, with coeﬃcients in R understood: The map on the left is deﬁned exactly as in the absolute case, using the relative cup product H ∗(E, E′) ⊗R H ∗(E)→H ∗(E, E′). The ﬁrst row of the diagram is exact since Φ Cohomology of Fiber Bundles Section 4.D 433 tensoring with a free module preserves exactness. The second row is of course exact also. The commutativity of the diagram follows from the evident naturality of in the case of the two squares shown. For the other square involving coboundary maps, if we start with an element b ⊗ i∗(cj) ∈ H ∗(B′) ⊗R H ∗(F ) and map this horizontally we get δb ⊗ i∗(cj) which maps vertically to p∗(δb)` cj, whereas if we ﬁrst map vertically we get p∗(b)` cj which maps horizontally to δ(p∗(b)` cj) = δp∗
(b)` cj = p∗(δb)` cj since δcj = 0. Φ The space B′ deformation retracts onto the skeleton Bn−1, and the following lemma implies that the inclusion p−1(Bn−1) ֓ E′ is a weak homotopy equivalence, hence induces an isomorphism on all cohomology groups: Lemma 4D.2. Given a ﬁber bundle p : E→B and a subspace A ⊂ B such that (B, A) is k connected, then is also k connected. E, p−1(A) Proof: For a map g : (Di, ∂Di)→ with i ≤ k, there is by hypothesis a homotopy ft : (Di, ∂Di)→(B, A) of f0 = pg to a map f1 with image in A. The homotopy lifting property then gives a homotopy gt : (Di, ∂Di)→ of g to a map with image in p−1(A). E, p−1(A) E, p−1(A) ⊔⊓ The theorem for ﬁnite-dimensional B will now follow by induction on n and the ﬁve-lemma once we show that the left-hand in the diagram is an isomorphism. Φ p−1(U), p−1(U ′) By the ﬁber bundle property there are open disk neighborhoods Uα ⊂ en α of the points xα such that the bundle is a product over each Uα. Let U = α Uα and let U ′ = U ∩ B′. By excision we have H ∗(B, B′) ≈ H ∗(U, U ′), and H ∗(E, E′) ≈ H ∗. This gives a reduction to the problem of showing that the : H ∗(U, U ′) ⊗R H ∗(F )→H ∗(U × F, U ′ × F ) is an isomorphism. For this we can map either appeal to the relative K¨unneth formula in Theorem 3.18 or we can argue again by induction, applying the ﬁve-lemma to the
diagram with (B, B′) replaced by (U, U ′), induction implying that the theorem holds for U and U ′ since they deformation retract onto complexes of dimensions 0 and n − 1, respectively, and by the lemma we S Φ can restrict to the bundles over these complexes. Next there is the case that B is an inﬁnite-dimensional CW complex. Since (B, Bn). Hence in is n connected, the lemma implies that the same is true of E, p−1(Bn) the commutative diagram at the right the horizontal maps are isomorphisms below dimension n. Then the fact that the right- hand is an isomorphism, as we have al- ready shown, implies that the left-hand is an isomorphism below dimension n. Φ Since n is arbitrary, this gives the theorem for all CW complexes B. To extend to the case of arbitrary base spaces B we need the notion of a pull- back bundle which is used quite frequently in bundle theory. Given a ﬁber bundle Φ 434 Chapter 4 Homotopy Theory p : E→B and a map f : A→B, let f ∗(E) = {(a, e) ∈ A× E a commutative diagram as at the right, where the two maps from f ∗(E) are (a, e) ֏ a and (a, e) ֏ e. It is a simple exercise to verify that the projection f ∗(E)→A is a ﬁber bundle with the same ﬁber as E→B, since a local trivialization of E→B over U ⊂ B gives rise to a local trivialization of f ∗(E)→A over f −1(U). f (a) = p(e)}, so there is If f : A→B is a CW approximation to an arbitrary base space B, then f ∗(E)→E induces an isomorphism on homotopy groups by the ﬁve-lemma applied to the long exact sequences of homotopy groups for the two bundles E→B and f ∗(E)→A with ﬁber F. Hence f ∗(E)→E is also an isomorphism on cohomology. The classes cj pull back to classes in H �
�(f ∗(E); R) which still restrict to a basis in each ﬁber, and so the ⊔⊓ naturality of reduces the theorem for E→B to the case of f ∗(E)→A. Φ Corollary 4D.3. (a) H ∗(U(n); Z) ≈ generators xi of odd dimension i. (b) H ∗(SU(n); Z) ≈ (c) H ∗(Sp(n); Z) ≈ Z[x3, x5, ···, x2n−1]. Z[x3, x7, ···, x4n−1]. Λ Λ Z[x1, x3, ···, x2n−1], the exterior algebra on These are ring isomorphisms, and the proof will involve bundles where the iso- Λ morphism in the Leray–Hirsch theorem happens to be an isomorphism of rings. U(n), U(n−1) Proof: For (a), assume inductively that the result holds for U(n−1). From the bundle U(n−1)→U(n)→S 2n−1 we see that the pair is (2n − 2) connected, so H i(U(n); Z)→H i(U(n − 1); Z) is an isomorphism for i ≤ 2n − 3 and the classes x1, ···, x2n−3 ∈ H ∗(U(n − 1); Z) given by induction are the restrictions of classes c1, ···, c2n−3 ∈ H ∗(U(n); Z). The products of distinct xi ’s form an additive basis for H ∗(U(n − 1); Z) ≈ Z[x1, ···, x2n−3], and these products are restrictions of the corresponding products of ci ’s, so the Leray–Hirsch theorem applies to give an additive basis for H ∗(U(n); Z) consisting of all products of distinct elements x1 = c1, ···, x2n−3 = c2n−3 and a new generator x2n−1 coming from H 2n−1(S 2n
−1; Z). By commutativity of cup product this is the exterior algebra Z[x1, ···, x2n−1]. Λ The same proof works for Sp(n) using the bundle Sp(n − 1)→Sp(n)→S 4n−1. In the case of SU(n) one uses the bundle SU(n − 1)→SU(n)→S 2n−1. Since SU(1) is the trivial group, the bundle SU(1)→SU(2)→S 3 shows that SU(2) = S 3, so the ⊔⊓ ﬁrst generator is x3. Λ It is illuminating to look more closely at how the homology and cohomology of O(n), U(n), and Sp(n) are related to their bundle structures. For U(n) one has the sequence of bundles Cohomology of Fiber Bundles Section 4.D 435 If all these were product bundles, U(n) would be homeomorphic to the product S 1 × S 3 × ··· × S 2n−1. cohomology of U(n) are the same as for this product of spheres, including the cup In actuality the bundles are nontrivial, but the homology and product structure. For Sp(n) the situation is quite similar, with the corresponding product of spheres S 3 × S 7 × ··· × S 4n−1. For O(n) the corresponding sequence of bundles is The calculations in §3.D show that H∗(O(n); Z2) ≈ H∗(S 0 × S 1 × ··· × S n−1; Z2), but with Z coeﬃcients this no longer holds. Instead, consider the coarser sequence of bundles where the last bundle O(2k)→S 2k−1 is omitted if n = 2k − 1. As we remarked at the end of §3.D in the case of SO(n), the integral homology and cohomology groups of O(n) are the same as if these bundles were products, but the cup product structure for O(n) with Z2 coeﬃcients is not the same as in this product. Cohomology of Grassmannians Here is an important application of the Leray–Hirsch theorem,
generalizing the calculation of the cohomology rings of projective spaces: Theorem 4D.4. If Gn(C∞) is the Grassmann manifold of n dimensional vector subspaces of C∞, then H ∗(Gn(C∞); Z) is a polynomial ring Z[c1, ···, cn] on generators ci of dimension 2i. Similarly, H ∗(Gn(R∞); Z2) is a polynomial ring Z2[w1, ···, wn] on generators wi of dimension i, and H ∗(Gn(H∞); Z) ≈ Z[q1, ···, qn] with qi of dimension 4i. The plan of the proof is to apply the Leray–Hirsch theorem to a ﬁber bundle p-----→ Gn(C∞) where E has the same cohomology ring as the product of n copies F -→ E of CP∞, a polynomial ring Z[x1, ···, xn] with each xi 2 dimensional. The induced map p∗ : H ∗(Gn(C∞); Z)→H ∗(E; Z) will be injective, and we will show that its image consists of the symmetric polynomials in Z[x1, ···, xn], the polynomials invariant under permutations of the variables xi. It is a classical theorem in algebra that the symmetric polynomials themselves form a polynomial ring Z[σ1, ···, σn] where σi is a certain symmetric polynomial of degree i, namely the sum of all products of i distinct xj ’s. This gives the result for Gn(C∞), and the same argument will also apply in the real and quaternionic cases. 436 Chapter 4 Homotopy Theory Proof: Deﬁne an n ﬂag in Ck to be an ordered n tuple of orthogonal 1 dimensional vector subspaces of Ck. Equivalently, an n ﬂag could be deﬁned as a chain of vector subspaces V1 ⊂ ··· ⊂ Vn of Ck where Vi has dimension i. Why either of these objects should be called a �
�ﬂag’ is not exactly clear, but that is the traditional name. The set of all n ﬂags in Ck forms a subspace Fn(Ck) of the product of n copies of CPk−1. There is a natural ﬁber bundle Fn(Cn) ------→ Fn(Ck) p------------→ Gn(Ck) where p sends an n tuple of orthogonal lines to the n plane it spans. The local triviality property can be veriﬁed just as was done for the analogous Stiefel bundle Vn(Cn)→Vn(Ck)→Gn(Ck) in Example 4.53. The case k = ∞ is covered by the same argument, and this case will be the bundle F→E→Gn(C∞) alluded to in the paragraph preceding the proof. The ﬁrst step in the proof is to show that H ∗(Fn(C∞); Z) ≈ Z[x1, ···, xn] where xi is the pullback of a generator of H 2(CP∞; Z) under the map Fn(C∞)→CP∞ projecting an n ﬂag onto its ith line. This can be seen by considering the ﬁber bundle CP∞ ------→ Fn(C∞) p------------→ Fn−1(C∞) where p projects an n ﬂag onto the (n−1) ﬂag obtained by ignoring its last line. The local triviality property can be veriﬁed by the argument in Example 4.54. The Leray– Hirsch theorem applies since the powers of xn restrict to a basis for H ∗(CP∞; Z) in the ﬁbers CP∞, each ﬁber being the space of lines in a vector subspace C∞ of the standard C∞. The elements xi for i < n are the pullbacks via p of elements of H ∗(Fn−1(C∞); Z) deﬁned in the same way. By induction H ∗(Fn−1(C∞); Z) is a polynomial ring on these elements. From the Leray–Hirsch theorem we conclude that the products of powers of the xi �

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