топология / Tom Dieck T. Algebraic topology (EMS, 2008)

508 Chapter 20. Homology and Homotopy

sequence for the triple .Ek ; Ek 1; E 1/, and similarly for B, and applies the C- Five Lemma to the resulting diagram induced by the various morphisms p . For the epimorphism statement we also use part (2) in (20.4.3). The colimit k ! 1 causes no problem for singular homology.

The statement of (20.4.1) is adapted to the method of proof. The hypotheses can be weakened as follows.

(20.4.5) Remark. Let .X; A/ be an .s 1/-connected pair of spaces. Then there exists a weak relative homotopy equivalence .B; A/ ! .X; A/ from a relative CWcomplex .B; A/ with A D Bs 1. We pull back a fibration over q W E ! X along this equivalence and use the fact that weak equivalences induce isomorphisms in singular homology. Then part (1) of (20.4.1) yields that q W Hj .E; q 1.A/I M / ! Hj .X; AI M / is a C- isomorphism for j r C s 1 and a C-epimorphism for i D r C s. Þ

(20.4.6) Remark. Let X be a 1-connected space such that Hj .XI Z/ is finitely generated for i e. Then there exists a weak equivalence B ! X such that B has only a finite number of t-cells for t e. We use this result in the next section. Þ

Problems

1. Suppose that Hj .F I M / D 0 for 0 < j < r. Let B be .s 1/-connected. Then p W Hj .E; F I M / ! Hj .B; I M / is an isomorphism for j r C s 1. We can now insert this isomorphism into the exact homology sequence of the pair .E; F / and obtain an exact sequence

HrCs 1.F I M / ! HrC1 1.EI M / ! HrCs 1.BI M / !

! H1.F I M / ! H1.EI M / ! H1.BI M / ! 0

which is analogous to the exact sequence of homotopy groups. Compare these sequences via the Hurewicz homomorphism .M D Z/.

20.5 Consequences of the Fibration Theorem

We use exact sequences and the fibration theorem to derive a number of results. We consider a fibration p W E ! B and assume that B and F D p 1. / are 0-connected; then E is 0-connected too. We use the notation

Z 2 C.r; M / , Hj .ZI M / 2 C for 0 < j < r 1.

(M an R-module. In the case that M D Z we write C.r/. For r D 1 there is no condition.) Let F , G denote the class of finite, finitely generated abelian groups, respectively. We use homology with coefficients in the R-module M if nothing else is specified.

(20.5.1) Remark. The homomorphism p W Hj .E; F I M / ! Hj .B; I M / is always an isomorphism for j 1 and an epimorphism for j D 2 (see (20.4.1)). The homomorphism p W Hj .EI M / ! Hj .BI M / is an isomorphism for j D 0 and

and a C-epimorphism for i D r C s C 1. Moreover p W Hi .E/ ! Hi .B/ is a C-isomorphism for i < r and a C-epimorphism for i D r.

Proof. The first statement is (20.4.1). For the second statement we use in addition the exact homology sequence of the pair .E; F /.

(20.5.3) Theorem. Let C be saturated.

(1)F; B 2 C.r; M / ) E 2 C.r; M /.

(2)F 2 C.r; M /; E 2 C.r C 1; M / ) B 2 C.r C 1; M /.

(3)B 2 C.r C 1; M /; E 2 C.r; M /, B 1-connected ) F 2 C.r; M /.

Proof. (1) and (2) are consequences of the fibration theorem and the exact homology sequence of the pair .E; F /. (3) is proved by induction on r. For r D 1 there is nothing to prove. For the induction step consider

(1) is a C isomorphism for r 1 C s 1 D r, since s D 2 and F 2 C.r 1; M / by induction. From the hypotheses Hr .BI I M /; Hr 1.EI M / 2 C we conclude

Proof. As for (20.5.3). The 1-connectedness of B is needed in order to apply the cell theorem (see (20.4.6)).

(20.5.5) Corollary. Suppose E is contractible (path fibration over B). Then F ' B, the loop space of B. Let B be simply connected. Let C be saturated. Then B 2 C.r C 1; M / if and only if B 2 C.r; M /. Moreover B 2 G .r C 1/ if and only if B 2 G .r/. Similarly for F instead of G .

(20.5.6) Proposition. Let A be a finitely generated abelian group. Then the Eilenberg–Mac Lane spaces K.A; n/ are contained in G .1/. If A is finite, then

510 Chapter 20. Homology and Homotopy

Proof. We use the path fibration K.A; n 1/ ! P ! K.A; n/ with contractible P and induction with (20.5.3) and (20.5.4). In the case that n D 1, standard constructions yield models for K.A; 1/ with a finite number of cells in each dimension. One uses K.A1; 1/ K.A2; 1/ D K.A1 A2; 1/, K.Z; 1/ D S1, and K.Z=m; 1/ D S1=.Z=m/.

Let X be a .k 1/-connected space (k 2). We attach cells of dimension j k C 2 to X in order to kill the homotopy groups j .X/ for j k C 1. The resulting space in an Eilenberg–Mac Lane space K. ; k/, D k .X/, and the inclusion W X ! K. ; k/ induces an isomorphism k . /. We pull back the path fibration over K. ; k/ and obtain a fibration

K. ; k 1/ ! Y ! X

with k-connected Y , and q W Y ! X induces isomorphisms j .q/, j > k. This follows from the exact homotopy sequence. If 2 C and C is saturated, then q W Hj .Y I M / ! Hj .XI M / is a C-isomorphism for j > 0, by the fibration theorem, since K. ; k 1/ 2 C.1/. Similarly for arbitrary C when X is of finite type.

20.6 Hurewicz andWhiteheadTheorems modulo Serre classes

Let C be a Serre class of abelian groups with the additional property: The groups Hk .K.A; 1// 2 C whenever A 2 C and k > 0. In this section we work with integral singular homology.

(20.6.1) Theorem (Hurewicz Theorem mod C). Suppose X is 1-connected and n 2. Assume that either C is saturated or Hi .X/ is finitely generated for i < n and C is arbitrary. Then the following assertions are equivalent:

(1)….n/ W i .X; / 2 C for 1 < i < n.

(2)H.n/ W Hi .X/ 2 C for 1 < i < n.

If ….n/ or H.n/ holds, the Hurewicz homomorphism hn W n.X; / ! Hn.X/ is a C-isomorphism.

Proof. The proof is by induction on n.

f

(1) Let X ! PX ! X be the path fibration with contractible PX. It provides us with a commutative diagram

The boundary maps @ are isomorphisms, since PX is contractible. By a basic property of fibrations n.f / is an isomorphism.

By the ordinary Hurewicz theorem h2 W 2.X; / ! H2.X/ is an isomorphism. (The method of the following proof can also be used to prove the classical Hurewicz theorem.)

(2) Let n > 2. Assume that the theorem holds for n 1 and that ….n/ holds. We want to show that H.n/ holds and that hn W .X; / ! Hn.X/ is a C-isomorphism. We first consider the special (3) case that 2.X/ D 0 and then reduce the general case (4) to this special.

(3) Thus let 2.X/ D 0. Then i . X/ Š iC1.X/ 2 C for 1 < i < n 1, and X is 1-connected. If C is saturated, then hn 1 W n 1. X/ ! Hn 1. X/ is a C-isomorphism by induction. If Hi .X/ is finitely generated for i < n, then by (20.5.3) Hi . X/ is finitely generated for i < n 1 so that by induction hn 1

C-isomorphism. Also Hn 1.X/ 2 C by induction.

(4)Let now 2.X/ D 2 be arbitrary. By assumption, this group is contained in C. There exists a map W X ! K. 2; 2/ which induces an isomorphism 2. We pull back the path fibration along

X2 PK. 2; 2/

'f

Since 2 2 C, we have Hi .K. 2; 1// 2 C for i > 0, by the general assumption in this section. Note that K D K. 2; 1/ is the fibre of ' and f . The exact homotopy sequence of is used to show that ' W i .X2/ Š 2.X/ for i > 2 and that 1.X2/ Š 0 Š 2.X2/. We can therefore apply the special case (3) to X2. Consider the diagram

In order to show that hn.X/ is a C-isomorphism we show two things:

(i)hn.X2/ is a C-isomorphism.

(ii)' W Hn.X2/ ! Hn.X/ is a C-isomorphism.

Part (i) follows from case (3) if we know that Hi .X2/ is finitely generated for i < n. This follows from (20.5.3) applied to the fibration K ! X2 ! X, since X is 1-connected and since 1.K/ Š 2 Š H2.X/ is finitely generated.

For the proof of (ii) we first observe that the canonical map ˇ W Hi .X2/ ! Hi .X2; K/ is a C-isomorphism for i > 0, since Hi .K/ 2 C for i > 0 by the

512 Chapter 20. Homology and Homotopy

general assumption of this section. The fibration theorem and (20.4.6) show that ' W Hi .X2; K/ ! Hi .X; / is a C-isomorphism for 0 < i n. We now compose with ˇ and obtain (ii).

(5) Now assume that H.k/ holds for 2 k < n. Since Hi .X/ 2 C for 2 i < k C 1 the Hurewicz map i .X/ ! Hi .X/ is a C-isomorphism for i k by H.k/, hence i .X/ 2 C for i k. By the first part of the proof, hkC1 is a C-isomorphism, hence ….k C 1/ holds.

We list some consequences of the Hurewicz theorem. Note that the general

(20.6.3) Theorem. Let C be a saturated Serre class. Let f W X ! Y be a map between 1-connected spaces with 1-connected homotopy fibre F . Then the following are equivalent:

(1)k .f / W k .X/ ! k .Y / is a C-isomorphism for k < n and a C-epimor- phism for k D n.

(2)Hk .f / W Hk .X/ ! Hk .Y / is a C-isomorphism for k < n and a C-epimor- phism for k D n.

Proof. The statement (1) is equivalent to k .F / 2 C for k < n (exact homotopy sequence). Suppose this holds. Then Hk .X; F / ! Hk .Y / is a C isomorphism for k n, by the fibration theorem (20.4.1). From the exact homology sequence of the pair .X; F / we now conclude that (2) holds, since Hj .F / 2 C by the Hurewicz theorem. Here we use that 1.F / D 0.

Suppose (2) holds. We show by induction that Hk .F / 2 C for k < n. Then we apply again the Hurewicz theorem. The induction starts with n D 3. Since Y and F are simply connected, the fibration theorem shows that Hj .X; F / ! Hj .Y; / is a C-isomorphism for j 3. The assumption (2) and the homology sequence

of the pair then show H2.F / 2 C. The general induction step is of the same type.

Problems

1. Let T Q be a subring. Let X be a simply connected space such that Hn.XI T / Š Hn.SnI T /. Then there exists a map Sn ! X which induces an isomorphism in T -homo- logy.

2.The finite CW-complex X D S1 _ S2 has 1.X/ Š Z. The group 2.X/ is free abelian with a countably infinite number of generators. (Study the universal covering of X.)

3.Use the fibration theorem and deduce for each 0-connected space a natural exact sequence

2.X/ ! H2.X/ ! H2.K. 1.X/; 1/I Z/ ! 0.

20.7 Cohomology of Eilenberg–Mac Lane Spaces

We compute the cohomology ring H .K.Z; n/I Q/. Let f W Sn ! K.Z; n/ D K.n/ induce an isomorphism n.f / of the n-th homotopy groups. Then also f W H n.K.Z; n/I Q/ ! H n.SnI Q/ is an isomorphism and n is defined such that f . n/ 2 H n.SnI Q/ is a generator.

(20.7.1) Theorem. If n 2 is even, then H .K.Z; n/I Q/ Š QŒ n (polynomial ring). If n is odd, then f W H .K.Z; n/I Q/ Š H .SnI Q/.

Proof. We work with rational cohomology. Since K.Z; 1/ ' S1 and K.Z; 2/ ' CP 1 we know already the cohomology ring for these spaces with coefficients in Z and this implies (20.7.1) in these cases. We prove the theorem by induction on n, and for this purpose we analyze the path-fibration K.Z; n 1/ ! P ! K.Z; n/ with contractible P . There are two cases for the induction step, depending on the parity of n.

2k 1 ) 2k. We have a relative fibration p W .E; E0/ ! K.n/ with E D

K.n/I ; p.w/ D w.1/; P D E0 D p 1. /. The map p W E ! K.n/ is a homotopy equivalence and E0 is contractible. Therefore

.1/ H n.E; E0/ Š H n.E/ Š H n.K.n//;

the latter induced by p. The fibres .F; F 0/ of p have a contractible F and F 0 DK.n/ D K.n 1/ is by induction a rational cohomology .n 1/-sphere (i.e., has the rational cohomology of Sn 1). Hence H k .F; F 0/ Š H k 1.F 0/ Š Q for k D n and Š 0 for k 6Dn. Since K.n/ is simply connected, we have a Thom class tn 2 H n.E; E0/. We can assume that tn is mapped under (1) to n, hence n is the Euler class e associated to tn. The Gysin sequence has the form (n D 2k)

H j .K.n// e H j Cn.K.n// H j Cn.P / :

! ! ! !

Since P is contractible and H j .K.n// D 0 for 0 < j < n, we see induc-

tively that the cup product with the Euler class e is an isomorphism H j .K.n// !

H j C2k .K.n//. Hence H .K.n// Š QŒ n .

2k ) 2k C 1. We reduce the problem to a Wang sequence. Let n D 2k C 1. We consider a pullback

YP

qp

514 Chapter 20. Homology and Homotopy

where f induces an isomorphism n.f /. The Wang sequence for q has the form

We use H .K.n 1// Š QŒ n 1 and the fact that is a derivation. From the definition of Y and the exact sequence of homotopy groups we see

j .Y / D 0; j n; j .Y / Š j .Sn/; j > n:

From the Hurewicz theorem and the universal coefficient theorem we conclude that H j .Y / D 0 for j n. Hence W H 2k .K.n 1// ! H 0.K.n 1//

is an isomorphism. Using the derivation property of we see inductively thatW H 2kr .K.n 1// ! H 2k.r 1/.K.n 1// is an isomorphism, and the Wang

is the homotopy fibre of f , we conclude that f W H .S / ! H .K.n// is an isomorphism (by (20.4.1) say), and similarly for cohomology. This finishes the induction.

20.8 Homotopy Groups of Spheres

Let n > 1 be an odd integer. Let f W Sn ! K.Z; n/ induce an isomorphism n.f / and denote by Y the homotopy fibre of f . In the previous section we have shown that Hj .Y I Q/ D 0 for j > 0. The Hurewicz theorem modulo the class of torsion

groups (D the rational Hurewicz theorem) shows us that the groups j .Y / ˝ Q are zero for j 2 N. From j .Sn/ Š j .Y / for j > n we see that also j .Sn/˝Q D 0

for j > n and odd n. Since we already know that the homotopy groups of spheres are finitely generated we see:

(20.8.1) Theorem. Let n be an odd integer. Then the groups j .Sn/ are finite for

We now investigate the homotopy groups of S2n. Let V D V2.R2nC1/ denote the Stiefel manifold of orthonormal pairs .x; y/ in R2nC1. We have a fibre bundle S2n 1 ! V ! S2n, and V is the unit-sphere bundle of the tangent bundle of S2n. Recall from 14.2.4 the integral homology of V

be a map of degree 1. Then g induces an isomorphism in

Let F be the homotopy fibre of g; it is simply connected. From (20.6.3) we see that g ˝ Q W j .V / ˝ Q ! j .S4n 1/ ˝ Q is always an

isomorphism. We use (20.8.1) and see that the homotopy groups of V are finite, except 4n 1.V / Š Z ˚ E, E finite. Now we go back to the fibration V ! S2n

and its homotopy sequence. It shows:

(20.8.2) Theorem. j .S2n/ is finite for j ¤ 2n; 4n 1 and 4n 1.S2n/ Š Z˚E,

E finite.

The results about the homotopy groups of spheres enable us to prove a refined rational Hurewicz theorem.

(20.8.3) Theorem. Let X be 1-connected. Suppose Hi .XI Z/ is finite for i < k and finitely generated for i 2k 2 (k 2). Then the Hurewicz homomorphism h W r .X/ ! Hr .XI Z/ has finite kernel (cokernel) for r < 2k 1 (r 2k 1).

Proof. The case k D 2 causes no particular problem, since the Hurewicz homomorphism h W mC1.X/ ! HmC1.XI Z/ is surjective for each .m 1/-connected space (m 2). So let k 3. Since X is 1-connected, the Hurewicz theorem shows that r .X/ is finite for r < k and finitely generated for r 2k 2. We writer .X/ D Fr ˚ Tr , Fr free, and Tr finite. We choose basis elements for Fr and representing maps. These representing elements provide us with a map of the form

f W S D Sr.1/ _ _ Sr.t/ ! X

with k r.j / 2k 2. The canonical map

L W

j r .Sr.j // ! r j Sn.j /

is an isomorphism for r 2k 2 and an epimorphism for r D 2k 1. We can now conclude that f W r .S/ ! r .X/ has finite kernel and cokernel for r 2k 2 and finite cokernel for r D 2k 1, since the homotopy groups of spheres are finite in the relevant range. The homotopy fibre F of f has finite homotopy groupsj .F / for j 2k 2. If, moreover, F is 1-connected, then Hj .F I Z/ is finite in the same range, by the Hurewicz theorem. The fibration theorem then yields that f W Hr .S/ ! Hr .X/ is an F -isomorphism (F -epimorphism) for r 2k 2 (r D 2k 1). From our knowledge of the homotopy groups of spheres we see directly that the theorem holds for S. The naturality of the Hurewicz theorem applied to f is now used to see that the desired result also holds for X.

We have used that F is 1-connected. This holds if 2.X/ D 0. Since 2.X/ is finite by assumption (k 3), we can pass to the 2-connected cover q W Xh 2i ! X of X. The map q induces F -isomorphisms in homology and homotopy. Therefore

it suffices to prove the theorem for Xh 2i to which the reasoning above applies.

If one is not interested in finite generation one obtains by a similar reasoning (see also [103]):

We apply the Hurewicz theorem modulo the class of abelian p-torsion groups (p prime) and see that the p-primary component of i .S3/ is zero for 3 < i < 2p and isomorphic to Z=p for i D 2p. In particular an infinite number of homotopy groups n.S3/ is non-zero.

The determination of the homotopy groups of spheres is a difficult problem. You can get an impression by looking into [163]. Individual computations are no longer so interesting; general structural insight is still missing. Since the groups nCk .Sn/

do not change after suspension for k n 2, by the Freudenthal theorem, they are called the stable homotopy groups kS .

We copy a table from [185]; a denotes a cyclic group of order a, and a b is the product of cyclic groups of order a and b, and aj the j -fold product of cyclic groups of order a.

20.8.7 The Hopf invariant. The exceptional case 4n 1.S2n/ is interesting in

many respects. Already Hopf constructed a homomorphism h W 4n 1.S2n/ ! Z, now called the Hopf invariant, and gave a geometric interpretation in the simplicial setting [88]. Let f W S4n 1 ! S2n be a smooth map; and let a, b be two regular

values. The pre-images Ma D f 1.a/ and Mb D f 1.b/ are closed orientable

.2n 1/-manifolds, they have a linking number, and this number is the Hopf

cohomology ring. The graded commutativity of the cup product is used to show that h.f / D 0 for odd k. In the case k D 2n the integer h.f / is the Hopf invariant. Since X.f / depends up to h-equivalence only on the homotopy class of f , the integer h.f / is a homotopy invariant. One shows the elementary properties of this invariant:

(1)h is a homomorphism.

(2)If g W S4n 1 ! S4n 1 has degree d , then h.fg/ D dh.f /.

(3)If k W S2n ! S2n has degree d , then h.kf / D d 2h.f /.