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

218 Chapter 8. Cell Complexes

In the cases n D 0; 1, the group can be non-abelian. In the case n D 0, we think of K. ; 0/ D with the discrete topology.

(8.8.1) Theorem. Eilenberg–Mac Lane spaces K. ; n/ exist. Proof. Let n 2. There exists an exact sequence

.bj j j 2 J / of F0. Then ˛ is determined by the matrix ˛.ak / D j n.j; k/bj .

is exact. Hence n.X/ Š . Also i .X/ D 0 for i < n. We can now attach cells of dimensions n C 2 to X in order to obtain a K. ; n/, see (8.6.6).

(8.8.2) Examples. The space S1 is a K.Z; 1/. We know 1.S1/ Š Z, and from the exact sequence of the universal covering R ! S1 we know that n.S1/ D 0 for

n 2. The space CP 1 is a model for K.Z; 2/. The space RP 1 is a K.Z=2; 1/.

Þ

The adjunction Œ†X; Y 0 Š ŒX; Y 0 shows that K. ; n C 1/ has the homotopy groups of a K. ; n/. By a theorem of Milnor [132], [67], Y has the homotopy type of a CW-complex if Y is a CW-complex. If one does not want to use this result one has the weaker result that there exists a weak homotopy equivalence K. ; n/ ! K. ; n C 1/.

We now establish further properties of Eilenberg–Mac Lane spaces. We begin by showing that Eilenberg–Mac Lane spaces are H -spaces. Then we construct product pairings K. ; m/ ^ K. ; n/ ! K. ˝ ; m C n/. In this context ˝

denotes the tensor product of the abelian groups and (alias Z-modules) over Z. We call the space K. ; n/ polarized, if we have chosen a fixed isomorphism

˛ W n.K. ; n// ! . If .K. ; n/; ˛/ and .K. ; n/; ˇ/ are polarized complexes, the product K. ; n/ K. ; n/ will be polarized by

n.K. ; n/ K. ; n// Š n.K. ; n// n.K. ; n// ˛ ˇ 1 2:

(8.8.3) Proposition. Having chosen polarizations, we obtain from (8.7.1) an isomorphism ŒK. ; n/; K. ; n/ 0 Š Hom. ; /.

(8.8.4) Theorem. Let be an abelian group. Then an Eilenberg–Mac Lane complex K. ; n/ is a commutative group object in h-TOP.

Proof. Let K D .K. ; n/; ˛/ be a polarized complex with base point a 0-cell e. For an abelian group , the multiplication W ! , .g; h/ 7!gh is a homomorphism. Therefore there exists a map m W K K ! K, unique up to homotopy, which corresponds under (8.8.3) to . Similarly, W ! , g 7!g 1 is a homomorphism and yields a map i W K ! K. Claim: .K; m; i/ is an associative and commutative H -space. The maps m ı .m id/ and m ı .id m/ induce the same homomorphism when n is applied; hence these maps are homotopic. In a similar manner one shows that x 7!m.x; e/ is homotopic to the identity. Since K _ K K K is a cofibration, we can change m by a homotopy such that m.x; e/ D m.e; x/ D x. We write x 7!m.x; i.x// as composition

d id i m

K ! K K ! K K ! K

and apply n; the result is the constant homomorphism. Hence this map is null homotopic. Commutativity is verified in a similar manner by applying n. See also Problem 1.

For the construction of the product pairing we need a general result about products for homotopy groups. We take the smash product of representatives f W I m=@I m ! X, g W I n=@I n ! Y and obtain a well-defined map

m.X/ n.Y / ! mCn.X ^k Y /; .Œf ; Œg / 7!Œf ^ g D Œf ^ Œg :

We call this map the ^-product for homotopy groups. It is natural in the variables X and Y .

(8.8.5) Proposition. The ^-product is bi-additive.

Proof. The additivity in the first variable follows directly from the definition of the addition, if we use C1. We see the additivity in the second variable, if we use the composition laws C1 and CmC1 in the homotopy groups.

(8.8.6) Proposition. Let A be an .m 1/-connected and B an .n 1/-connected CW-complex. Then A ^ B is .m C n 1/-connected and the ^-product

m.A/ ˝ n.B/ ! mCn.A ^k B/

is an isomorphism (m; n 2). If m or n equals 1, then one has to use the abelianized groups.

220 Chapter 8. Cell Complexes

Proof. The assertion about the connectivity was shown in (8.6.3). For A D Sm the

W

assertion holds by the suspension theorem. For A D j Sjm we use the commutative diagram

m W Sjm ˝ n.B/ ^ mCn W Sjm ^k B

(1) is an isomorphism by (8.6.4). (2) is induced by a homeomorphism. (3) is an isomorphism by algebra. (4) is an isomorphism by (8.6.4). (5) is an isomorphism by the suspension theorem. This settles the case of a wedge of m-spheres. Next we let A be the mapping cone of a map f W C ! D where C and D are wedges of m-spheres. Then we have a commutative diagram with exact rows:

The general case now follows from the observation that the inclusion AmC1 ! A induces an isomorphism on m and AmC1 ^k B ! A^k B induces an isomorphism on mCn.

Let .K.G; m/; ˛/, .K.H; n/; ˇ/ and .K.G ˝ H; m C n/; / be polarized Eilen- berg–Mac Lane complexes for abelian groups G and H . A product is a map

m;n W K.G; m/ ^k K.H; n/ ! K.G ˝ H; m C n/

such that the diagram

is commutative. Here we have use the ^-product (8.8.5).

(8.8.7) Theorem. There exists a product. It is unique up to homotopy.

Proof. Let G and H be abelian groups. The first non-trivial homotopy group of K.G; m/ ^k K.H; n/ is mCn and it is isomorphic to G ˝ H , see (8.8.6). By (8.6.6) there exists an inclusion

m;n W K.G; m/ ^k K.H; n/ ! K.G ˝ H; m C n/:

We can choose the polarization so that the diagram above becomes commutative.

mCn;p ı . m;n id/ ' m;nCp ı .id n;p/:

The products are graded commutative in the following sense:

K. ; m C n/ ı m;n ' . 1/mn 0 ı n;m

with the interchange maps 0 W K.m; G/ ^ K.n; H / ! K.n; H / ^ K.m; G/ and

W G ˝ H ! H ˝ G.

Let R be a commutative ring with 1. We think of the multiplication as being a homomorphism W R˝R ! R between abelian groups. From this homomorphism we obtain a unique homotopy class K. / W K.R ˝ R; m/ ! K.R; m/.

We compose K. / with k;l and obtain a product map

mk;l W K.R; k/ ^k K.R; l/ ! K.R; k C l/:

Also these products are associative and graded commutative.

In the associated homotopy groups H k .XI R/ D ŒXC; K.R; k/ 0 we obtain via .f; g/ 7!mk;l .f ^ g/ products

H k .XI R/ ˝ H l .Y I R/ ! H kCl .X Y I R/;

which are also associative and graded commutative. (See also Problem 3.) In a similar manner we can start from an R-module structure R ˝ M ! M on M . Later, when we study singular cohomology, we show that for a CW-complex X the group ŒXC; K.R; k/ 0 is naturally isomorphic to the singular cohomology group H k .XI R/ with coefficients in the ring R. This opens the way to a homotopical study of cohomology. The product (8.8.7) can then be used to construct the so-called cup product in cohomology.

Once singular cohomology theory is constructed one obtains from the representability theorem of Brown Eilenberg–Mac Lane spaces as representing objects.

8.8.8 Eilenberg–Mac Lane spectra. Let A be an abelian group. The Eilenberg– Mac Lane spectrum HA consists of the family .K.A; n/ j n 2 N0/ of Eilenberg– Mac Lane CW-spaces and maps en W †K.A; n/ ! K.A; n C 1/ (an inclusion of

222 Chapter 8. Cell Complexes

subcomplexes; attach cells to †K.A; n/ to obtain a K.A; n C 1/). This spectrum is an -spectrum. We have proved in any case that "n W K.A; n/ ! K.A; n C 1/ is a weak homotopy equivalence. This suffices if one wants to define the cohomology theory only for pointed CW-spaces. Þ

Problems

1. From the natural isomorphism ŒX; K. ; n/ 0 Š ŒX; 2K. ; n C 2/ 0 we see that X 7! ŒX; K. ; n/ 0 is a contravariant functor into the category of abelian groups. Therefore, by category theory, there exists a unique (up to homotopy) structure of a commutative h-group on K. ; n/ inducing the group structures of this functor.

2. Let ˛ 2 m.X/, ˇ 2 n.Y /, and W X ^k Y ! Y ^k X the interchange map. Then

˛ ^ ˇ D . 1/mnˇ ^ ˛.

3.Let M be an R-module. A left translation lr W M ! M , x 7!rx is a homomorphism of the abelian group M and induces therefore a map Lr W K.M; k/ ! K.M; k/. Use these maps to define a natural structure of an R-module on ŒX; K.M; k/ 0.

4.The simply connected surfaces are S2 and R2 [44, p. 87]. If a surface is different from S2 and RP 2, then it is a K. ; 1/.

5.Let E. / ! B. / be a -principal covering with contractible E. /. Then B. / is a K. ; 1/. Spaces of the type B. / will occur later as classifying spaces. There is a bijection ŒK. ; 1/; K. ; 1/ Š Hom. ; /= between homotopy classes and group homomorphisms up to inner automorphisms.

6.Let S1 be the colimit of the unit spheres S.Cn/ S.CnC1/ . This space carries a free action of the cyclic group Z=m S1 by scalar multiplication. Show that S1 with this action is a Z=m-principal covering. The quotient space is a CW-space B.Z=m/ and hence a K.Z=m; 1/.

7.A connected 1-dimensional CW-complex X is a K. ; 1/. Determine from the topology of X.

8.A connected non-closed surface (with or without boundary) is a K. ; 1/.

Chapter 9

Singular Homology

Homology is the most ingenious invention in algebraic topology. Classically, the definition of homology groups was based on the combinatorial data of simplicial complexes. This definition did not yield directly a topological invariant. The definition of homology groups and (dually) cohomology groups has gone through various stages and generalizations.

The construction of the so-called singular homology groups by Eilenberg [56] was one of the definitive settings. This theory is very elegant and almost entirely algebraic. Very little topology is used as an input. And yet the homology groups are defined for arbitrary spaces in an invariant manner. But one has to pay a price: The definition is in no way intuitively plausible. If one does not mind jumping into cold water, then one may well start algebraic topology with singular homology. Also interesting geometric applications are easily obtainable.

In learning about homology, one has to follow three lines of thinking at the same time: (1) The construction. (2) Homological algebra. (3) Axiomatic treatment.

(1)The construction of singular homology groups and the verification of its main properties, now called the axioms of Eilenberg and Steenrod.

(2)A certain amount of algebra, designed for use in homology theory (but also of independent algebraic interest). It deals with diagrams, exact sequences, and chain complexes. Later more advanced topics are needed: Tensor products, linear algebra of chain complexes, derived functors and all that.

(3)The object that one constructs with singular homology is now called a homology theory, defined by the axioms of Eilenberg and Steenrod. Almost all applications of homology are derived from these axioms. The axiomatic treatment has other advantages. Various other homology and cohomology theories are known, either constructed by special input (bordism theories, K-theories, de Rham cohomology) or in a systematic manner via stable homotopy and spectra.

The axioms of a homology or cohomology theory are easily motivated from the view-point of homotopy theory. But we should point out that many results of algebraic topology need the idea of homology: The reduction to combinatorial data via cell complexes, chain complexes, spectral sequences, homological algebra, etc.

Reading this chapter requires a parallel reading of the chapter on homological algebra. Already in the first section we use the terminology of chain complexes and their homology groups and results about exact sequences of homology groups.

ıin W Œn 1 ! Œn be the injective map which misses the value i.

A continuous map W n ! X is called a singular n-simplex in X. The i -th face of is ı din. We denote by Sn.X/ the free abelian group with basis the

set of singular n-simplices in X. (We also set, for formal reasons, Sn.X/ D 0

in the case that n < 0 but disregard mostly this trivial case. If X D ;, we let Sn.X/ D 0.) An element x 2 Sn.X/ is called a singular n-chain. We think of x

as a formal finite linear combination x D P n , n 2 Z. In practice, we skip a summand with n D 0; also we write 1 D . We use without further notice the algebraic fact that a homomorphism from Sn.X/ is determined by its values on

the basis elements W n ! X, and these values can be prescribed arbitrarily. The boundary operator @q is defined for q 1 by

negative of the second sum.

The singular chain groups Sq .X/ and the boundary operators @q form a chain complex, called the singular chain complex S .X/ of X. Its n-th homology group is denoted Hn.X/ D Hn.XI Z/ and called singular homology group of X (with

coefficients in Z). A continuous map f W X ! Y induces a homomorphism f# D Sq .f / W Sq .X/ ! Sq .Y /; 7!f :

The family of the Sq .f / is a chain map S .f / W S .X/ ! S .Y /. Thus we have induced homomorphisms f D Hq .f / W Hq .X/ ! Hq .Y /. In this manner, the Hq become functors from TOP into the category ABEL of abelian groups.

A. Since Sn.;/ D 0, we identify canonically Sn.X/ D Sn.X; ;/. The boundary operator of S .X/ induces a boundary operator @n W Sn.X; A/ ! Sn 1.X; A/ such

that the family of quotient homomorphisms Sn.X/ ! Sn.X; A/ is a chain map. The homology groups Hn.X; A/ D Hn.X; AI Z/ of S .X; A/ are the relative singular homology groups of the pair .X; A/ (with coefficients in Z). A continuous

map f W .X; A/ ! .Y; B/ induces a chain map f W S .X; A/ ! S .Y; B/ and homomorphisms f D Hq .f / W Hq .X; A/ ! Hq .Y; B/. In this way, Hq becomes a functor from TOP.2/ to ABEL.

We apply (11.3.2) to the exact sequence of singular chain complexes

0 ! S .A/ ! S .X/ ! S .X; A/ ! 0

and obtain the associated exact homology sequence:

(9.1.3) Theorem. For each pair .X; A/ the sequence

is exact. The sequence terminates with H0.X/ ! H0.X; A/ ! 0. The undecorated arrows are induced by the inclusions .A; ;/ .X; ;/ and .X; ;/ .X; A/.

Let .X; A; B/ be a triple, i.e., B A X. The inclusion S .A/ ! S .X/ induces by passage to factor groups an inclusion S .A; B/ ! S .X; B/, and its cokernel can be identified with S .X; A/. We apply (11.3.2) to the exact sequence of chain complexes

0 ! S .A; B/ ! S .X; B/ ! S .X; A/ ! 0

and obtain the exact sequence of a triple

The boundary operator @ W Hn.X; B/ ! Hn 1.A; B/ in the exact sequence of a triple is the composition of the boundary operator for .X; A/ followed by the map Hn 1.A/ ! Hn 1.A; B/ induced by the inclusion.

It remains to verify that the connecting morphisms @ constitute a natural transformation, i.e., that for each map between triples f W .X; A; B/ ! .X0; A0; B0/ the

226 Chapter 9. Singular Homology

diagram

is commutative. This is a special case of an analogous fact for morphisms between short exact sequences of chain complexes and their associated connecting morphisms. We leave this as an exercise.

One cannot determine the groups Hq .X; A/ just from its definition (except in a few trivial cases). Note that for open sets in Euclidean spaces the chain groups have an uncountable basis. So it is clear that the setup only serves theoretical purposes. Before we prove the basic properties of the homology functors (the axioms of Eilenberg and Steenrod) we collect a few results which follow directly from the definitions.

9.1.4 Point. Let X D P be a point. There is a unique singular n-simplex, hence Sn.P / Š Z, n 0. The boundary operators @0 and @2iC1 are zero and @2; @4; : : :

connected, and therefore W n ! X has an image in one of the Xj , so we can sort the basis elements of Sn.X/ according to the components Xj . Þ

9.1.6 The groups H0. The group H0.X/ is canonically isomorphic to the free abelian group Z 0.X/ over the set 0.X/ of path components. We identify a singular 0-simplex W 0 ! X with the point . 0/. Then S0.X/ is the free abelian group on the points of X. A singular 1-simplex W 1 ! X is essentially the same thing as a path, only the domain of definition has been changed from I to 1. We associate to the path w W I ! X, t 7! .1 t; t/. Then @0 D w .1/ and @1 D w .0/, hence @ D @0 @1 corresponds to the orientation convention @w D w.1/ w.0/. If two points a; b 2 X are in the same path component, then the zero-simplices a and b are homologous. Hence we obtain a homomorphism from Z 0.X/ into H0.X/, if we assign to the path component of a its homology class. We also have a homomorphism S0.X/ ! Z 0.X/ which sends the singular simplex of a 2 X to the path component of a. This homomorphism sends the image of @ W S1.X/ ! S0.X/ to zero. Hence we obtain an inverse homomorphism

H0.X/ ! Z 0.X/. Þ

9.2 The Fundamental Group

The signs which appear in the definition of the boundary operator have an interpretation in low dimensions. They are a consequence of orientation conventions.

compute @ D c k1 C k0, with a constant c. A constant 1-simplex is a boundary. Hence k0 k1 is a boundary

Œk0 D Œk1 2 C1.X/=B1.X/:

In particular, homotopic loops yield the same element in H1.X/. Thus we obtain a well-defined map h0 W 1.X; x0/ ! H1.X/; by (1), it is a homomorphism. The

fundamental group is in general non-abelian. Therefore we modify h0 algebraically to take this fact into account. Each group G has the associated abelianized factor group Gab D G=ŒG; G ; the commutator group ŒG; G is the normal subgroup

generated by all commutators xyx 1y 1. A homomorphism G ! A to an abelian group A factorizes uniquely over Gab. We apply this definition to h0 and obtain a

homomorphism

h W 1.X; x0/ab ! H1.X/:

(9.2.1) Theorem. Let X be path connected. Then h is an isomorphism.