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

298 Chapter 11. Homological Algebra

(11.9.7) Proposition. Let f W C ! D be a chain map between complexes of free abelian groups. Suppose that for each field F the map f ˝ F induces isomorphisms of homology groups. Then f is a chain equivalence.

Proof. Let C.f / denote the mapping cone of f . The hypothesis implies that H .C.f / ˝ F / D 0. We use the universal coefficient sequence. It implies that Tor.H .C.f //; Z=p/ D 0 for each prime p. Hence H .C.f // is torsion-free.

11.10 The Künneth Formula

Let C and D be chain complexes of R-modules over a principal ideal domain R. We have the tensor product chain complex C ˝R D and the associated homomorphism

˛ W Hi .C / ˝R Hj .D/ ! HiCj .C ˝R D/; Œx ˝ Œy 7!Œx ˝ y :

We use the notation for TorR. The next theorem and its proof generalizes the universal coefficient formula (11.9.1).

(11.10.1) Theorem (Künneth Formula). Suppose C consists of free R-modules. Then there exists an exact sequence

If also D is a free complex, then the sequence splits.

Proof. We consider the graded modules Z.C / and B.C / of cycles and boundaries as chain complexes with trivial boundary. Since Z.C / is free, we have the equalities (canonical isomorphisms)

.Z.C / ˝ Z.D//n D Ker.1 ˝ @ W .Z.C / ˝ D/n ! .Z.C / ˝ D/n 1/

and

.Z.C / ˝ B.D//n D Im.1 ˝ @ W .Z.C / ˝ D/nC1 ! .Z.C / ˝ D/n/;

and they imply H.Z.C / ˝ D/ Š Z.C / ˝ H.D/ (homology commutes with the tensor product by a free module). In a similar manner we obtain an isomorphism H.B.C /˝D/ Š B.C /˝H.D/. We form the tensor product of the free resolution of chain complexes

i

0 ! B.C / ! Z.C / ! H.C / ! 0

Chapter 12

Cellular Homology

In this chapter we finally show that ordinary homology theory is determined on the category of cell complexes by the axioms of Eilenberg and Steenrod. From the axioms one constructs the cellular chain complex of a CW-complex. This chain complex depends on the skeletal filtration, and the boundary operators of the chain complex are determined by the so-called incidence numbers; these are mapping degrees derived from the attaching maps. The main theorem then says that the algebraic homology groups of the cellular chain complex are isomorphic to the homology groups of the homology theory (if it satisfies the dimension axiom). From this fact one obtains immediately qualitative results and explicit computations of homology groups. Thus if X has k.n/ n-cells, then Hn.XI Z/ is a subquotient of the free abelian group of rank k.n/. A finite cell complex has finitely generated homology groups. We deduce that the combinatorial Euler characteristic is a homotopy invariant that can be computed from the homology groups.

In the case of a simplicial complex we show that singular homology is isomorphic to the classical combinatorial simplicial homology. In this context, simplicial homology is a special case of cellular homology.

12.1 Cellular Chain Complexes

Let h be an additive homology theory. Let X be obtained from A by attaching

302 Chapter 12. Cellular Homology

(12.1.3) Corollary. Let W hk .Dn; Sn 1/ ! hkC1.DnC1; Sn/ be a suspension

isomorphism. Then m.e; f / ı is the multiplication by d.e; f /, provided the relation @00 ı D n ı p holds.

We now write the isomorphism (12.1.1) in a different form. We use an iterated suspension isomorphism n W hk n ! hk .Dn; Sn 1/ in each summand.

(property (W3) of a Whitehead complex). From the preceding discussion we obtain:

(12.1.4) Proposition. Suppose and are chosen such that the relation (12.1.3) holds. Then the diagram

The composition of the boundary operators (belonging to the appropriate triples)

is zero, because the part hm.Xn/ ! hm.Xn; Xn 1/ ! hm 1.Xn 1/ of the exact sequence of the pair .Xn; Xn 1/ is “contained” in this composition. We set hn;k .X/ D hnCk .Xn; Xn 1/. Thus the groups .hn;k .X/ j n 2 Z/ together with the boundary operators just considered form a chain complex h ;k .X/.

(12.1.5) Proposition. The product M.n 1/M.n/ of two adjacent incidence matrices is zero. The cellular chain groups Cn.X/ together with the homomorphisms M.n/ W C.n/ ! C.n 1/ form a chain complex C .X/. This chain complex is called the cellular chain complex of X.

Proof. The relation M.n 1/M.n/ D 0 follows from (12.1.4) applied to the chain complex H ;0.X/ obtained from singular homology with coefficients in Z.

The cellular chain complex has its algebraically defined homology groups. In the next section we prove that in the case of an ordinary homology theory the algebraic homology groups of the cellular chain complex are naturally isomorphic to the homology groups of the theory. We should point out that the algebraic homology groups of the chain complexes h ;k .X/ only depend on the space and the coefficients of the homology theory, so are essentially independent of the theory. Nevertheless, they can be used to obtain further information about general homology theories – this is the topic of the so-called spectral sequences [130].

The definition of incidence numbers uses characteristic maps and a homotopy equivalence . These data are not part of the structure of a CW-complex so that the incidence numbers are not completely determined by the CW-complex. The choice of a characteristic map determines, as one says, an orientation of the cell. If ˆ; ‰ W .Dn; Sn 1/ ! .Xn; Xn 1/ are two characteristic maps of a cell e, then

‰ 1ˆ W Dn=Sn 1 ! Xn=Xn X e Dn=Sn 1

is a homeomorphism and hence has degree ˙1. One concludes that the incidence numbers are defined up to sign by the CW-complex.

(12.1.6) Proposition. A cellular map f W X ! Y induces a chain map with components f W hm.Xn; Xn 1/ ! hm.Y n; Y n 1/. Homotopic cellular maps induce chain homotopic maps.

.X I /n D Xn @I [ Xn 1 I . We define a chain homotopy as the composition

tween additive homology theories such that induces isomorphisms of the coefficient groups W kn.P / Š ln.P /, n 2 Z, P a point. Then is an isomorphism k .X/ ! l .X/ for each CW-complex X.

Proof. Since is compatible with the suspension isomorphism we see from (12.1.1) that W k .Xn; Xn 1/ Š l .Xn; Xn 1/. Now one uses the exact homology sequences and the Five Lemma to prove by induction on n that is an isomorphism for n-dimensional complexes. For the general case one uses (10.8.1).

304 Chapter 12. Cellular Homology

Problems

deletes the last coordinate, has degree 1.

2. Prove M.n 1/M.n/ D 0 without using homology by homotopy theoretic methods.

12.2 Cellular Homology equals Homology

Let H . / D H . I G/ be an ordinary additive homology theory with coefficients in G (not necessarily singular homology). The cellular chain complex C .X/ D C .XI G/ of a CW-complex X with respect to this theory has its algebraically defined homology groups. It is a remarkable and important fact that these algebraic homology groups are naturally isomorphic to the homology groups of the space X. This result says that the homology groups are computable from the combinatorial data (the incidence matrices) of the cellular complex.

(12.2.1) Theorem. The n-th homology group of the cellular chain complex C .X/ is naturally isomorphic to Hn.X/.

Proof. We show that the isomorphism is induced by the correspondence

Hn.Xn; Xn 1/ Hn.Xn/ ! Hn.X/:

We divide the proof into several steps.

(1)A basic input is Hk .Xn; Xn 1/ D 0 for k 6D0; this follows from our determination of the cellular chain groups in (12.1.1) and the dimension axiom.

(2)Hk .Xn/ D 0 for k > n. Proof by induction on n. The result is clear for X0 by the dimension axiom. Let k > n C 1. We have the exact sequence Hk .Xn/ ! Hk .XnC1/ ! Hk .XnC1; Xn/. The first group is zero by induction,

the third by (1).

(3) Since Hn 1.Xn 2/ D 0, the map Hn 1.Xn 1/ ! Hn 1.Xn 1; Xn 2/ is injective. Hence the cycle group Zn of the cellular chain complex is the kernel of

@W Hn.Xn; Xn 1/ ! Hn 1.Xn 1/.

(4)The exact sequence 0 ! Hn.Xn/ ! Hn.Xn; Xn 1/ ! Hn 1.Xn 1/

induces an isomorphism .b/ W Hn.Xn/ Š Zn.

(5) Hk .X; Xn/ D 0 for k n. One shows by induction on t that the groups Hk .XnCt ; Xn/ are zero for t 0 and k n. We know that for an additive theory the canonical map colimt Hk .XnCt ; Xn/ ! Hk .X; Xn/ is an isomorphism (see (10.8.1) and (10.8.4)). For singular homology one can also use that a singular chain has compact support and that a compact subset of X is contained in some skeleton Xm.

306 Chapter 12. Cellular Homology

(12.2.4) Example. The sphere Sn has a CW-decomposition with two i-cells in each dimension i, 0 i n. The attaching diagram is

by the antipodal map and on the left column by permutation of the summands. The equivariant chain groups are therefore isomorphic to the group ring ZG D Z 1 ˚ Z t. In order to determine the equivariant boundary operator we use the fact that we know already the homology of this chain complex. If we add the homology groups in dimension 0 and n we obtain an exact sequence

"

0 ! Z."n/ ! ZG ! ZG ! ! ZG ! Z ! 0:

The map " sends 1; t to 1. The kernel is generated by 1 t. From the geometry we see that the first boundary operator sends the generator 1 2 C1 represented by a suitably oriented 1-cell to ˙.1 t/. We orient the cell such that the plus-sign holds. Then d1 is the multiplication by 1 t. The kernel of d1 is thus generated by 1 C t. We can again orient the 2-cells such that d2 is multiplication by 1 C t. If we continue in this manner, we see that dk D 1 t for k odd, and dk D 1Ct for k even. The homology module Hn.Sn/ D Z."n/ carries the t-action "n D . 1/nC1, the degree of the antipodal map. One can, of course, determine the boundary operator

by a computation of degrees. We leave this as an exercise. Similar results hold for S1. Þ

Let X and Y be CW-complexes. The product inherits a cell decomposition. The cross product induces an isomorphism

LkClDn Hk .Xk ; Xk 1/ ˝ Hl .Y l ; Y l 1/ ! Hn..X Y /n; .X Y /n 1/:

With a careful choice of cell orientations these isomorphisms combine to an isomorphism C .X/ ˝ C .Y / Š C .X Y / of cellular chain complexes.

12.3 Simplicial Complexes

We describe the classical combinatorial definition of homology groups of polyhedra. These groups are isomorphic to the singular groups for this class of spaces. The combinatorial homology groups of a finite polyhedron are finitely generated abelian groups and they are zero above the dimension of the polyhedron. This finite generation is not at all clear from the definition of the singular groups.

Recall that a simplicial complex K D .E; S/ consists of a set E of vertices and a set S of finite subsets of E. A set s 2 S with q C 1 elements is called a q-simplex

of K. We require the following axioms:

(1)A one-point subset of E is a simplex in S.

(2)s 2 S and ; 6Dt s imply t 2 S.

An ordering of a p-simplex is a bijection f0; 1; : : : ; pg ! s. An ordering of K is a partial order on E which induces a total ordering on each simplex. We write s D h v0; : : : ; vp i, if the vertices of s satisfy v0 < v1 < < vp in the given partial ordering. Let Cp.K/ denote the free abelian group with basis the set of p-simplices. Its elements are called the simplicial p-chains of K. Now fix an ordering of K and define a boundary operator

b

The boundary relation @@ holds (we set Cp.K/ D 0 for p 1). We denote the p-th homology group of this chain complex by Hp.K/. This is the classical combinatorial homology group.

A simplicial complex K has a geometric realization jKj. An ordered simplex s D h v0; : : : ; vp i has an associated singular simplex

ˆs W p ! jKj; .t0; : : : ; tp/ 7!P tj vj :

We extend s 7!ˆs by linearity to a homomorphism p W Cp.K/ ! Sp.jKj/. The boundary operators are arranged so that D . p/ is a chain map.

(12.3.1) Theorem. induces isomorphisms Rp W Hp.K/ Š Hp.jKj/.

Proof. We write X D jKj. Let S.p/ be the set of p-simplices. The characteristic maps ˆs W . p; @ p/ ! .Xp; Xp 1/ yield an isomorphism (12.1.1),

Here es is the open simplex which belongs to s. Let s.i/ denote the i-th face of

p and xs.i/ 2 Hp 1.Xp 1; Xp 2/ the corresponding basis element. We claim

@xs D PiD0. 1/i xs.i/. It is clear for geometric reasons that the expression of @xs

p