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

(20.1.8) Theorem. Let f W X ! Y be a map between connected CW-complexes

i.e., qF D fp. Suppose F induces an isomorphism of the homology groups. Then f is a homotopy equivalence.

Proof. We choose isomorphisms 1.X/ Š G Š 1.Y / which transform f into the identity of G. We then consider p and q as G-principal bundles with left action

From the assumption and (20.1.5) we see that F is an h-equivalence. The exact homotopy sequence and the Five Lemma show that EG G F induces isomorphisms

that .gx/ D .x/g 1 for x 2 X and g 2 G, see (14.1.4). A map of this type is essentially the same thing as a classifying map of p. Since the fibre of P is contractible, P induces isomorphisms of the homotopy groups, and the same holds then for a section s of P . We see that f D Qı.EG G F /ıs induces isomorphisms of homotopy groups; hence f is a homotopy equivalence.

(20.1.9) Corollary. In the situation of (20.1.8) F is a G-homotopy equivalence.

G-map. A homotopy of hf can be lifted to a G-homotopy of HF . The end of this homotopy is a bundle automorphism.

Let G be a discrete group which acts on the pair .Y; B/. The induced maps of the left translations by group elements yield a left action of G on Hn.Y; B/ via homomorphisms, i.e., Hn.Y; B/ becomes a module over the integral group ring ZG of G.

Suppose X is obtained from A by attaching n-cells .n 3/. Let p W Y ! X be a universal covering and B D p 1.A/. Then Y is obtained from B by attaching n-cells. The group D 1.X/ of deck transformations acts freely on the set of n-cells in Y X B. Hence Hn.Y; B/ is a free Z -module, the basis elements correspond bijectively to the n-cells of X X A. Theorem (20.1.3) now yields:

(20.1.10) Theorem. Let X be a connected CW-complex and let n 3. Thenn.Xn; Xn 1/ is a free Z 1.Xn 1/-module. A basis of this module consists of the characteristic maps of the n-cells. The map h# W n.Xn; Xn 1/ ! Hn.Xn; Xn 1/ is an isomorphism.

500 Chapter 20. Homology and Homotopy

The exact sequence

nC1.XnC1; Xn/ ! n.Xn; Xn 1/ ! n.XnC1; Xn 1/ ! 0

is for n 3 a sequence of Z 1.Xn 1/ Š Z 1.Xn/-modules. Because of the isomorphism n.X; Xn 1/ Š n.XnC1; Xn 1/ the sequence is a presentation of the Z 1.Xn 1/-module n.X; Xn 1/. The induced sequence of the #-groups is also exact. This gives the following theorem for n 3.

(20.1.11) Theorem (Hurewicz). Let .X; A/ be a CW-pair with connected X and A. Let .X; A/ be .n 1/-connected (n 2). Then h# W n#.X; A; / Š Hn.X; A/.

Proof. We now give a purely homological proof of the Hurewicz theorems which also covers the relative case n D 2 for spaces which are not simply connected. The proof is by induction. The induction starts with (9.2.1). We assume the absolute theorem for 1 i n 1 and prove the relative theorem for n. We consider the

Hn.X; A/ for an .n 1/-connected pair .X; A/ with path connected A, see (9.5.4). We have to adapt the homotopy groups to the simplicial setup. We consider elements of n.X; A; / as homotopy classes of maps f W . Œn ; @ Œn ; e0/ !

.X; A; /. For this purpose we fix a homeomorphism ˛ W .D.n/; S.n 1/; / !

. Œn ; @ Œn ; e0/ which sends the generator zQn defined at the beginning to the

reduce the problem to a universal situation. For this purpose we define elements bn 2 n.@ Œn C 1 ; Œn C 1 n 1; e0/,

iD1

where Œvw denotes the affine path class in Œn C 1 from v to w, and we use the transport along this path. (Multiplicative notation in 2, additive notation for n 3. This definition corresponds to the homological boundary operator, but we have to transport the face maps to the base point e0. For b2 we have to pay attention to the order of the factors, since the group is non-abelian.)

Let us write K D Œn C 1 and let W .K; Kn 1; K0/ ! .X; A; / be a basis element of SnnC11.X; A; /. Then

@Œ D Pi . 1/i Œ dinC1 D #Œbn # D #j #Œbn #

where j denotes the inclusion @ Œn C 1 Œn C 1 . Thus it remains to show that j Œbn D 0.

The skeleton Kn 1 is .n 2/-connected (use e.g., the induction hypothesis). Hence, by the inductive assumption, n# 1.Kn 1; e0/ ! Hn 1.Kn 1/ is an isomorphism. The commutativity @h D h@ now shows that @Œbn D 0, since @hŒbn D 0 by the fundamental boundary relation for singular homology. We have the factorization

and @0 is an isomorphism, since K is contractible. This finishes the inductive step for the relative Hurewicz theorem. For n 2 the absolute theorem is a special case of the relative theorem.

An interesting consequence of the homological proof of the Hurewicz theorem is a new proof of the Brouwer–Hopf degree theorem n.Sn/ Š Z.

20.2 Realization of Chain Complexes

The computation of homology groups from the cellular chain complex shows that one needs enough cells to realize the homology groups algebraically as the homology groups of a chain complex. It is interesting to know that in certain cases a converse holds. We work with integral homology.

(20.2.1) Theorem (Cell Theorem). Let Y be a 1-connected CW-complex. Suppose Hj .Y / is finitely generated for j n. Then Y is homotopy-equivalent to a CWcomplex Z with finitely many j -cells for j n.

The proof of this theorem is based on a theorem which says that under suitable hypotheses an algebraic chain complex can be realized as a cellular chain complex. We describe the inductive construction of a realization. We start with the following:

20.2.2 Data and notation.

(1) Y is a CW-complex with i-skeleton Yi .

502Chapter 20. Homology and Homotopy

(2)Zr is an r-dimensional CW-complex.

(3)f W Zr ! Y is a cellular map.

(4)Ci .Z/ D Hi .Zi ; Zi 1I Z/ is the i-th cellular chain group.

(5)f induces a chain map ' W C .Z/ ! C .Y /.

(6)We attach .r C 1/-cells to Zr such that f can be extended to F :

We now start from a diagram in which ArC1 is a free abelian group with basis

.aj j j 2 J /. The horizontal parts should be chain complexes, i.e., dı D 0. Can this diagram be realized geometrically?

(20.2.3) Proposition. A realization exists, if the following holds:

(1) f W Hi .Zr / ! Hi .Y / is bijective for i r 1 and surjective for i D r;

(2)r 2;

(3)Zr and Y are 1-connected.

Proof. Suppose we are given for each j 2 J a diagram

We attach .r C 1/-cells to Zr with attaching maps bj to obtain ZrC1 and use the Bj to extend f to frC1 W ZrC1 ! YrC1. Then ArC1 Š HrC1.ZrC1; Zr / canonically and basis preserving.

We consider f as an inclusion. The assumption (1) is then equivalent to Hi .Y; Zr / D 0 for i r. Since r 2 and 1.Y / D 0 we also have 1.YrC1/ D 0.

Therefore we have the relative Hurewicz isomorphism h W rC1.YrC1; Zr / Š HrC1.YrC1; Zr /, since Hi .YrC1; Zr / Š Hi .Y; Zr / D 0 for i r. The diagram represents an element in rC1.YrC1; Zr /. Let xj D h.ŒBj ; bj / 2 HrC1.YrC1; Zr / be its image under the relative Hurewicz homomorphism. This element can be determined by homological conditions. The correct maps ı and are obtained, if xj has the following properties:

(1)The image of xj under @ W HrC1.YrC1; Zr / ! Hr .Zr / ! Hr .Zr ; Zr 1/ is ı.aj /.

(2) The image of xj under W HrC1.YrC1; Zr / ! HrC1.YrC1; Yr / is .aj /. We show that there exists a unique element xj with these properties. We know HrC1.Y; r; Zr / D 0 and Hj .Yr 1Zr 1/ D 0, j r for reasons of dimension. The exact sequence of the triple .YrC1; Yr ; Zr / shows that is injective. Hence there exists at most one xj with the desired properties. The existence follows if we show that Im.@; / D Ker.'r dr0C1/. This follows by diagram chasing in the next diagram with exact rows

Proof. Since Y is simply connected, we can assume that Y has a single 0-cell and no 1-cells. We construct Z inductively with Z0 D f g and Z1 D f g. We choose a finite number of generators for 2.Y / Š H2.Y / and representing maps S2 ! Y2. They yield a cellular map f2 W Z2 D W S2 ! Y , and the induced map is surjective in H2 and bijective in Hj , j < 2. This starts the inductive construction.

Suppose fr W Zr ! Y is given such that fr W Hi .Zr / ! Hi .Y / is bijective for i r 1 and surjective for i D r. We construct a diagram of type (7) in 20.2.2 as follows. We have Hr .Zr / D Ker.d/ and Hr .Y / D Ker.dr0 /= Im.dr0C1/ and the map .fr / W Hr .Zr / D Ker.d/ ! Hr .Y / is surjective. Let ArC1 be the kernel of .fr / and ı W Hr .Zr / Cr .Zr /. As a subgroup of Cr .Zr / it is free abelian. Since Zr has, by induction, a finite number of r-cells, the group ArC1 is finitely generated. By definition of ArC1, the image of 'r ı is contained in the image of dr0C1. Hence there exists making the diagram commutative. We now apply (20.2.3) in order to attach an .r C 1/-cell for each basis element of ArC1 and to extend fr to fr0C1 W Zr0 C1 ! Y . By construction, .fr0C1/ is now bijective on Hr . If this map is not yet surjective on HrC1 we can achieve this by attaching more

.r C 1/-cells with trivial attaching maps; if HrC1.Y / is finitely generated, we only need a finite number of cells for this purpose. We continue in this manner as long as H .Y / is finitely generated. After that point we do not care about finite generation. The final map f W Z ! Y is a homotopy equivalence by (20.1.5).

504 Chapter 20. Homology and Homotopy

20.3 Serre Classes

Typical qualitative results in algebraic topology are statements of the type that the homotopy or homology groups of a space are (in a certain range) finite or finitely generated or that induced maps have finite or finitely generated kernel and cokernel. A famous result of Serre [170] says that the homotopy groups of spheres are finite, except in the cases already known to Hopf.

Here are three basic ideas of Serre’s approach:

(1)Properties like ‘finite’ or ‘finitely generated’ or ‘rational isomorphism’ have a formal structure. Only this structure matters – and it is axiomatized in the notion of a Serre class of abelian groups or modules.

(2)One has to relate homotopy groups and homology groups, since qualitative results about homology groups are more accessible. The connection is based on the Hurewicz homomorphism.

(3)For inductive proofs one has to relate the homology groups of the basis, fibre, and total space of a (Serre-)fibration. This is the point where Serre uses the method of spectral sequences.

A non-empty class C of modules over a commutative ring R is a Serre class if the

following holds: Let 0 ! A ! B ! C ! 0 be an exact sequence of R-modules.

Then B 2 C if and only if A; C 2 C. We call C saturated if A 2 C implies that

L

arbitrary direct sums j A of copies of A are contained in C. The class consisting of the trivial module alone is saturated. A morphism f W M ! N between R- modules is a C-epimorphism (C-monomorphism) if the cokernel (kernel) of f is in C, and a C-isomorphism if it is a C-epi- and -monomorphism. We use certain facts about these notions, especially the C-Five Lemma. The idea is to neglect modules in C, or, as one says, to work modulo C; so, instead of C-isomorphism, we say isomorphism modulo C. Here are some examples of Serre classes.

(1)The class containing only the trivial group.

(2)The class F of finite abelian groups.

(3)The class G of finitely generated abelian groups.

(4)Let R be a principal ideal domain. The class C consists of the (finitely generated) R-modules. If R is a field, then we are considering the class of (finite-dimensional) vector spaces.

(5)Let R be a principal ideal domain. The class C consists of the (finitely generated) R-torsion modules. A module M is a torsion module, if for each

x 2 M there exists 0 ¤ 2 R such that x D 0.

(6)Let P N be a set of prime numbers. Let ZP Q denote the subring of rational numbers with denominators not divisible by an element of P . If

P D ;, then Z D Q. If P D fpg, then ZP D Z.p/ is the localization of Z at p. If P contains the primes except p, then ZP D ZŒp 1 is the

506Chapter 20. Homology and Homotopy

(2)Let C be arbitrary. Then p is a C-isomorphism for i ˛ and a C-epi- morphism for i D ˛ C 1 where now ˛ D min.e C 1; r C s 1/.

We remark that r 1. For r D 1 we make no further assumptions about F . Since a weak homotopy equivalence induces isomorphisms in singular homo-

logy, we can assume without essential restriction that B is a CW-complex (pull back the fibration along a CW-approximation). We reduce the proof of the theorem by a Five Lemma argument to the attaching of t-cells. We consider the following

situation. Let

.ˆ; '/ W qa .Dat ; Sat 1/ ! .B; B0/

be an attaching of t-cells. Let p W E ! B be a fibration and set E0 D p 1.B0/. We assume that the fibres are 0-connected and homotopy equivalent. We pull back the fibration along ˆ and obtain two pullback diagrams.

We apply homology (always with coefficients in M ) and obtain the diagram

isomorphism, we attach a single t-cell, to simplify the notation. Let B0 be obtained from B by deleting the center ˆ.0/ of the cell.

(20.4.2) Lemma. Let p W X ! B be a fibration with restrictions p0 W X0 ! B0 and p0 W X0 ! B0. Let q W Y ! Dt be the pullback of p, and similarly q0 W Y 0 ! St 1 and q0 W Y0 ! Dt X 0. Then

‰ W hi .Y; Y 0/ ! hi .X; X0/

is an isomorphism for each homology theory.

Proof. We have a commutative diagram

Since B0 is a deformation retract and p a fibration, also X0 is a deformation retract of X0. Therefore (2) is an isomorphism by homotopy invariance. The map (4) is an isomorphism by excision. For similar reasons (1) and (3) are isomorphisms. Finally (5) is induced by a homeomorphism.

By the homotopy theorem for fibrations, a fibration over Dt is fibre-homotopy equivalent to a product projection Dt F ! Dt . We use such equivalences and a suspension isomorphism Hi .Dt F; St 1 F / Š Hi t .F / and obtain altogether a commutative diagram (Pa a point and Fa the fibre over Pa):

Hi .E; E0/ p Hi .B; B0/

(20.4.3) Note. The considerations so far show that the bottom map has the following properties:

(1)Isomorphism for i t (since fibres are 0-connected).

(2)Epimorphism always.

(3)Suppose Hi .Fa/ 2 C for 0 < i < r. Then each particular map Hi t .Fa/ !

Hi t .Pa/ is a C-isomorphism for 0 < i t < r. Thus the total map is a C-isomorphism if either C is saturated or if we attach a finite number of cells. Þ

We apply these considerations to a fibration p W E ! B over a relative CWcomplex .B; A/ as in the statement of the theorem. In this situation the previous considerations yield:

Let C be saturated. Then p W Hi .Et ; Et 1/ ! Hi .Bt ; Bt 1/ is a C-iso- morphism for each t 0, if i < r C s. We only have to consider t s. By (20.4.3), we have a C-isomorphism in the cases i t and i > t > i r. These conditions hold for each t, if s > i r. If, in addition, there are only finitely many t-cells for t e, then we have a C-isomorphism for arbitrary C and each t, if i min.e C 1; r C s 1/, by the same argument.

We finish the proof of theorem (20.4.1) with:

(20.4.4) Lemma. Let .B; A/ be a relative CW-complex with t-skeleton Bt . Let p W E ! B be a fibration and Et D p 1.Bt /. Suppose p W Hi .Et ; Et 1I M / ! Hi .Bt ; Bt 1I M / is a C-isomorphism for each t 0 and each 0 i ˛, then

p W Hi .E; E 1I M / ! Hi .B; B 1I M / is a C-isomorphism for 0 i ˛ and a C-epimorphism for i D ˛ C 1.

Proof. We show by induction on k 0 that p W Hi .Ek ; E 1/ ! Hi .Bk ; B 1/ is a C-isomorphism (k 0). For the induction step one uses the exact homology