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

One of the first applications of the homology axioms is the computation H1.S1/ Š Z. Granted the formal result that 1.S1/ is abelian, we obtain yet another proof for 1.S1/ Š Z.

9.3 Homotopy

We prove in this section the homotopy invariance of the singular homology groups. We begin with a special case.

9.3.1 Cone construction. Let X be a contractible space. Define a chain map

" D ."n/ W S .X/ ! S .X/ by "n D 0 for n 6D0 and by "0 P n D P n 0 where 0 W 0 ! fx0g. We associate to each homotopy h W X I ! X from the identity to the constant map with value x0 a chain homotopy s D .sn/ from " to the identity. The homomorphisms s W Sn 1.X/ ! Sn.X/ are obtained from a cone construction. Let

q W n 1 I ! n; .. 0; : : : ; n 1/; t/ 7!.t; .1 t/ 0; : : : ; .1 t/ n 1/:

Given W n 1 ! X, there exists a unique simplex s. / D s W n !X such that h ı . id/ D s. / ı q, since q is a quotient map. For the faces of s we verify

.s /di D s. di 1/, for i > 0, and .s /d0 D . From these data we compute for n > 1,

and @.s / D 0 for a 0-simplex . These relations imply @s C s@ D id ".

transformations between functors; the naturality says that for each continuous map f W X ! Y the commutation relation .f id/ t .X/ D t .Y / f holds.

(9.3.3) Theorem. There exists a natural chain homotopy s from 0 to 1.

For the convenience of the reader we also rewrite the foregoing abstract proof in explicit terms. See also Problem 2 for an explicit chain homotopy and its geometric meaning.

Proof. We have to show: There exist morphisms snX W Sn.X/ ! SnC1.X I / such that

.Kn/ @snX C snX 1@ D 1n.X/ 0n.X/

(chain homotopy), and such that for continuous X ! Y the relations

.Nn/ .f id/# ı snX D snY ı f#

hold (naturality). We construct the sn inductively.

n D 0. In this case, s0 sends the 0-simplex W 0 ! fxg X to the 1-simplex s0 W 1 ! X I , .t0; t1/ 7!.x; t1/. Then the computation

@.s0 / D .s0 /d0 .s0 /d1 D 10 00

shows that .K0/ holds, and also .N0/ is a direct consequence of the definitions. Now suppose that the sk for k < n are given, and that they satisfy .Kk / and .Nk /.

The identity of n is a singular n-simplex; let n 2 Sn. n/ be the corresponding element. The chain to be constructed sn n should satisfy

@.sn n/ D 1n. n/ 0n. n/ sn 1@. n/:

The right-hand side is a cycle in Sn. n I /, as the next computation shows.

@. 1n. n/ 0n. n/ sn 1@. n//

D1n 1.@ n/ 0n 1.@ n/ @sn 1.@ n/

D1n 1.@ n/ 0n 1.@ n/ . 1n 1.@ n/ 0n 1.@ n/ sn 2@@ n/ D 0:

We have used the relation .Kn 1/ for @sn 1@. n/ and that the t are chain maps. Since n I is contractible, there exists, by (9.3.2), an a 2 SnC1. n I / with the property @a D 1n. n/ 0n. n/ sn 1@. n/. We choose an a with this property and define sn. n/ D a and in general sn. / D . id/#a for W n ! X; the required

230 Chapter 9. Singular Homology

naturality .Nn/ forces us to do so. We now verify .Kn/ and .Nn/. We compute

@sn. / D @. id/#a D . id/#@a

D. id/#. 1n n 0n n sn 1@ n/

D1n # n 0n # n sn 1 #@ n

D1n 0n sn 1@ :

We have used: . id/# is a chain map; choice of a; naturality of 1, 0, and

.Nn 1/; # n D ; # is a chain map. Thus we have shown .Kn/. The equalities

.f id/#sn. / D .f id/#. id/#a D .f id/#a D sn.f / D snf#

also a chain homotopy sn W Sn.X; A/ ! SnC1.X I; A I /. The computation

@.f# ı sn/ C .f# ı sn 1/@ D f#@sn C f#sn 1@ D f#. 1 0/ D f#1 f#0 proves the f#sn to be a chain homotopy from f#0 to f#1. Altogether we see:

(9.3.4) Theorem. Homotopic maps induce homotopic chain maps and hence the

X, which coincide on S0.X/. Then there exists a natural chain homotopy from a to a1. This is a consequence of (11.5.1) for F D G and the models n as in the proof of (9.3.3). Þ

Problems

2. One can prove the homotopy invariance by constructing an explicit chain homotopy.

homotopy, if one takes orientations into account. This idea works; one has to decomposen I into simplices, and it suffices to do this algebraically.

In the prism n I let 0; 1; : : : ; n denote the vertices of the base and 00; 10; : : : ; n0 those of the top. In the notation for affine singular simplices introduced later, show that an explicit formula for a D sn n is

(This is a special case of the Eilenberg–Mac Lane shuffle morphism to be discussed in the section on homology products.)

9.4 Barycentric Subdivision. Excision

The basic property of homology is the excision theorem (9.4.7). It is this theorem which allows for effective computations. Its proof is based on subdivision of standard simplices. We have to work out the algebraic form of this subdivision first.

Let D Rn be convex, and let v0; : : : ; vp be elements in D. The affine singular simplex W p ! D, Pi i ei 7! Pi i vi will be denoted D Œv0; : : : ; vp . With

this notation

b

v 2 D we have the contracting homotopy D I ! D, .x; t/ 7!.1 t/x C tv. If we apply the cone construction 9.3.1 to Œv0; : : : ; vp we obtain Œv; v0; : : : ; vp . We denote the chain homotopy associated to the contraction by Sp.D/ ! SpC1.D/, c 7!v c. We have for c 2 Sp.D/:

Bp.X/ D Bp W Sp.X/ ! Sp.X/

to be the homomorphism which sends W p ! X to Bp. / D #Bp. p/, where Bp. p/ is defined inductively as

(9.4.1) Proposition. The Bp constitute a natural chain map which is naturally homotopic to the identity.

Proof. The equalities

f#B D f# #B. p/ D .f /#B. p/ D B.f / D Bf#

prove the naturality. We verify by induction over p that we have a chain map. Let p D 1. Then @B. 1/ D @. ˇ1 B.@ 1// D @ 1 D B@. 1/. For p > 1 we compute

@B p D @. pˇ B.@ p// D B@ p pˇ @B@ p D B@ p pˇ B@@ p D B@ p:

We have used: Definition; (1); inductive assumption; @@ D 0. We now use this special case and the naturality

B@ D B@ # p D B #@ p D #B@ p D #@B p D @ #B p D @B ;

Let U be a family of subsets of X such that their interiors cover X. We call a singular simplex U-small, if its image is contained in some member of U. The subgroup spanned by the U-small simplices is a subcomplex SU.X/ of S .X/ with homology groups denoted by HnU.X/.

(9.4.2) Lemma. The diameter d.v0; : : : ; vp/ of the affine simplex Œv0; : : : ; vp with respect to the Euclidean norm is the maximum of the kvi vj k.

Proof. Let x; y 2 Œv0; : : : ; vp and x D P j vj . Then, because of P j D 1, kx yk D k P j .vj y/k P j kvj yk maxj kvj yk:

This shows in particular ky vi k maxj kvj vi k; we insert this in the above and obtain kx yk maxi;j kvi vj k; hence the diameter is at most as stated.

On the other hand, this value is clearly attained as the distance between two points.

We prove the claim by induction over p. The assertion is obvious for p D 0, a point has diameter zero. By induction hypothesis, the simplices in the chain

(9.4.4) Lemma. Let W p ! X be a singular simplex. Then there exists a k 2 N such that each simplex in the chain Bk has an image contained in a member of U. (Here Bk is the k-fold iteration of B.)

Proof. We consider the open covering . 1.U ı//; U 2 U of p. Let " > 0 be a Lebesgue number of this covering. The simplices of Bk arise by an application ofto the simplices in Bk p. From (9.4.3) we see that the diameter of these simplices

is at most . pCp 1 /k d.e0; : : : ; ep/. If k is large enough, this number is smaller than ".

(9.4.5) Theorem. The inclusion of chain complexes SU.X/ S .X/ induces an isomorphism H U.X/ ! H .X/.

Proof. Let a 2 SnU.X/ be a cycle which represents a homology class in the kernel. Thus a D @b with some b 2 SnC1.X/. By (9.4.4), there exists k such that Bk .b/ 2 SnUC1.X/ (apply (9.4.4) to the finite number of simplices in the linear combination of b). By (9.4.1), there exists a natural chain homotopy Tk between Bk and the identity. Therefore

Bk .b/ b D Tk .@b/ C @Tk .b/ D Tk .a/ C @Tk .b/;

and we conclude

@Bk .b/ @b D @Tk .a/; a D @b D @.Bk .b/ Tk .a//:

From the naturality of Tk and the inclusion a 2 SnU.X/ we see Tk .a/ 2 SnUC1.X/. Therefore a is a boundary in SU.X/. This shows the injectivity of the map in question.

Let a 2 Sn.X/ be a cycle. By (9.4.4), there exists k such that Bk a 2 SnU.X/.

We know that

Bk a a D Tk .@a/ C @Tk .a/ D @Tk .a/:

Since Bk is a chain map, Bk a is a cycle. From the last equality we see that a

Let now .X; A/ be a pair of spaces. We write U \ A D .U \ A j U 2 U/ and

Each row has its long exact homology sequence. We apply (9.4.5) to .X; U/ and

.A; U \ A/, use the Five Lemma (11.2.7), and obtain:

234 Chapter 9. Singular Homology

(9.4.7) Theorem (Excision Theorem). Let Y D Y1ı [ Y2ı. Then the inclusion induces an isomorphism H .Y2; Y1 \ Y2/ Š H .Y; Y1/. Let B A X and

isomorphism H .X X B; A X B/ Š H .X; A/. Again we can invoke (11.6.3) and conclude that the inclusion actually induces chain equivalences between the chain complexes under consideration.

Proof. The covering U D .Y1; Y2/ satisfies the hypothesis of (9.4.5). By definition, we have SnU.X/ D Sn.Y1/ C Sn.Y2/ and also Sn.Y1 \ Y2/ D Sn.Y1/ \ Sn.Y2/. The inclusion S .Y2/ ! S .Y / induces therefore, by an isomorphism theorem of

elementary algebra,

By (9.4.5) and (11.2.7) we see, firstly, that SU.Y /=S .Y1/ ! S .Y /=S .Y1/ and, altogether, that S .Y2/=S .Y1 \ Y2/ ! S .Y /=S .Y1/ induces an isomorphism in homology. The second statement is equivalent to the first; we use X D Y , A D Y1,

Problems

1.Let D Rm and E Rn be convex and let f W D ! E be the restriction of a linear map. Then f#.v c/ D f .v/ f#.c/.

2.Although not necessary for further investigations, it might be interesting to describe the chain BŒv0; : : : ; vp in detail. We use (3) in the proof of (9.4.3). By (2), formula (3) also

holds for Œv0; : : : ; vp . This yields BŒv0; v1 D Œv01; v1 Œv01; v0 with barycenter v01, and for BŒv0; v1; v2 we obtain in short-hand notation what is illustrated by the next figure.

Œ012; 12; 2 Œ012; 12; 1 Œ012; 02; 2 C Œ012; 02; 0 C Œ012; 01; 1 Œ012; 01; 0 :

9.5 Weak Equivalences and Homology

Although singular homology groups are defined for arbitrary topological spaces, they only capture combinatorial information. The theory is determined by its values on cell complexes. Technically this uses two facts: (1) a weak homotopy equivalence induces isomorphisms of homology groups; (2) every topological space is weakly equivalent to a CW-complex. One can use cell complexes to give proofs by induction over the skeleta. Usually the situation for a single cell is quite transparent, and this fact makes the inductive proofs easy to follow and to remember. Once a theorem is known for cell complexes, it can formally be extended to general topological spaces. We now prove this invariance property of singular homology [56], [21].

Let .X; A; / be a pointed pair. Let Œk n be the n-skeleton of the standard

simplicial complex Œk (this is the reason for switching the notation for the standard k-simplex). Let Sk.n;A/.X/ for n 0 denote the subgroup of Sk .X/ spanned by

the singular simplices W Œk ! X with the property

.#/ . Œk n/ A:

The groups .Sk.n;A/.X/ j k 0/ form the Eilenberg subcomplex S.n;A/.X/ of

S .X/.

(9.5.1) Theorem. Let .X; A/ be n-connected. Then the inclusion of the Eilenberg subcomplex ˛ W S.n;A/.X/ ! S .X/ is a chain equivalence.

Proof. We assign to a simplex W Œk ! X a homotopy P . / W Œk I ! X such that

(1)P . /0 D ,

(2)P . /1 satisfies (#),

(3)P . /t D , provided satisfies already (#),

(4)P . / ı .dik id/ D P . ı dik /.

According to (3), the assignment P is defined for simplices which satisfy (#). For the remaining simplices we use an inductive construction.

Suppose k D 0. Then . Œ0 / 2 X is a point. Since .X; A/ is 0-connected, there exists a path from this point to a point in A. We choose a path of this type as P . /.

Suppose P is given for j -simplices, j < k. Then for each k-simplex the homotopy P . ı dik / is already defined, and the P . ı dik / combine to a homotopy @ Œk I ! X. Moreover P . /0 is given. Altogether we obtain

Let k n. Then Œk n D Œk , and similarly for the faces. By the inductive

236 Chapter 9. Singular Homology

There exists a homeomorphism W Œk I ! Œk I which induces homeomorphisms (see (2.3.6))

Œk 0 Š Œk 0 [ @ Œk I; @ Œk 0 Š @ Œk 1;

@ Œk I [ Œk 1 Š Œk 1:

can be extended to a homotopy Q W Œk I ! X which is constant on @ Œk I

where h is the natural chain homotopy between i#0 and i#1, see (9.3.3). The computations

@s. / D @.P . /#h. k // D P . /#@h. k /

DP . /1#. k / P . /0#. k / P . /#h.@ k /

D. / P . /#h.@ k /;

chain equivalence (9.5.1) and the exact homology sequence of .X; A/ now yield:

Let f W X ! Y be a weak homotopy equivalence. We can assume that f is an inclusion (mapping cylinder and homotopy invariance).

(9.5.3) Theorem. A weak homotopy equivalence induces isomorphisms of the singular homology groups.

(9.5.4) Remark. Suppose that .X; A; / is a pointed pair and A is pathwise connected. Then we can define a subcomplex S.X;A; /.X/ of S .X/ where we require

in addition to (#) that . Œk 0/ D f g. Again the inclusion is a chain equivalence.

Þ

9.6 Homology with Coefficients

Let C D .Cn; cn/ be a chain complex of abelian groups and let G be a further abelian group. Then the groups Cn ˝ G and the boundary operators cn ˝ id form again a chain complex (the tensor product is taken over Z). We denote it by C ˝G.

We apply this process to the singular complex S .X; A/ and obtain the complex S .X; A/ ˝ G of singular chains with coefficients in G . Its homology group

in dimension n is denoted Hn.X; AI G/. The cases G D Z; Q; Z=p are often

referred to as integral, rational, mod.p/ homology. Chains in Sn.X; A/ ˝ G can

P

be written as finite formal linear combinations a , a 2 G of singular n- simplices ; this accounts for the name “chain with coefficients”. The sequence 0 ! S .A/ ! S .X/ ! S .X; A/ ! 0 remains exact when tensored with G, i.e., Sn.X; A/ ˝ G Š Sn.X/ ˝ G=Sn.A/ ˝ G. Therefore we still have the exact homology sequence (11.3.2)

@

! Hn.AI G/ ! Hn.XI G/ ! Hn.X; AI G/ ! Hn 1.AI G/ !

and the analogous sequence for triples. The boundary operators @ are again natural transformations. If 0 ! G0 ! G ! G00 ! G is an exact sequence of abelian groups, then the tensor product with S .X; A/ yields again an exact sequence of chain complexes and we obtain from (11.3.2) an exact sequence of the form

! Hn.X; AI G0/ ! Hn.X; AI G/ ! Hn.X; AI G00/ ! Hn 1.X; AI G0/ ! :

The passage from C to C ˝G is compatible with chain maps and chain homotopies. A chain equivalence induces a chain equivalence. This fact yields the homotopy invariance of the homology groups Hn.X; AI G/. The excision theorem still holds. This is a consequence of (9.4.7): Under the hypothesis of the excision theorem, the chain equivalence S .Y1; Y1 \ Y2/ ! S .Y; Y2/ induces a chain equivalence when tensored with G. Hence the functors Hn.X; AI G/ satisfy the axioms of Eilenberg and Steenrod for a homology theory. The dimension axiom holds: We have a canonical isomorphism "P W H0.P / Š G for a point P , which maps the homology class of the chain a to a, where is the unique 0-simplex.

The application of (11.9.1) to topology uses the fact that the singular chain complex consists of free abelian groups. Therefore we obtain: