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

stable Hurewicz homomorphism

W ! Q

hn !n.X/ Hn.X/:

The hn are natural transformations of homology theories (on well-pointed spaces). In order to see this one has to verify that the diagram

is commutative. The commutativity of this diagram is a reason for introducing the sign in the definition D . 1/n l of the suspension isomorphism for spectral homology.

The coefficients of the theory ! . / are the stable homotopy groups of spheres !n.S0/ D colimk nCk .Sk /. These groups are finite for n > 0. Finite abelian groups become zero when tensored with the rational numbers. We thus obtain a natural transformation of homology theories

which are isomorphisms on the coefficients and therefore in general for pointed CW-complexes.

is independent of the chosen representative f of x. We combine these homomorphisms

the multiplication by k. We thus have shown:

Chapter 21

Bordism

We begin with the definition of bordism homology. The geometric idea of homology is perhaps best understood from the view-point of bordism and manifolds. A “singular” cycle is a map from a closed manifold to a space, and the boundary relation is induced by manifolds with boundary. Several of our earlier applications of homology and homotopy can easily be obtained just from the existence of bordism homology, e.g., the Brouwer fixed point theorem, the generalized Jordan separation theorem and the component theorem, and the theorem of Borsuk–Ulam.

Bordism theory began with the fundamental work of Thom [184]. He determined the bordism ring of unoriented manifolds (the coefficient ring of the associated bordism homology theory). This computation was based on a fundamental relation between bordism and homotopy theory, the theorem of Pontrjagin–Thom. In the chapter on smooth manifolds we developed the material which we need for the present proof of this theorem. One application of this theorem is the isomorphism between the geometric bordism theory and a spectral homology theory via the Thom spectrum. From this reduction to homotopy we compute the rational oriented bordism. Hirzebruch used this computation in the proof of his signature theorem. This proof uses almost everything that we developed in this text.

21.1 Bordism Homology

We define the bordism relation and construct the bordism homology theory. Mani-

folds are smooth.

Let X be a topological space. An n-dimensional singular manifold in X is a pair

.B; F / which consists of a compact n-dimensional manifold B and a continuous map F W B ! X. The singular manifold @.B; F / D .@B; F j@B/ is the boundary of .B; F /. If @B D ;, then .B; F / is closed.

A null bordism of the closed singular manifold .M; f / in X is a triple .B; F; '/

which consists of a singular manifold .B; F / in X and a diffeomorphism ' W M ! @B such that .F j@B/ı' D f . If a null bordism exists, then .M; f / is null bordant.

Let .M1; f1/ and .M2; f2/ be singular manifolds in X of the same dimension.

We denote by .M1; f1/ C .M2; f2/ the singular manifold h f1; f2 i W M1 C M2 !

X. We say .M1; f1/ and .M2; f2/ are bordant, if .M1; f1/ C .M2; f2/ is null bordant. A null bordism of .M1; f1/C.M2; f2/ is called bordism between .M1; f1/

and .M2; f2/. The boundary @B of a bordism .B; F; '/ between .M1; f1/ and

.M2; f2/ thus consists of a disjoint sum @1B C @2B, and ' decomposes into two diffeomorphisms 'i W Mi ! @i B.

522 Chapter 21. Bordism

(21.1.1) Proposition. “Bordant” is an equivalence relation.

Proof. Let .M; f / be given. Set B D M I and F D f ı pr W M I ! M ! X. Then @B D M 0 C M 1 is canonically diffeomorphic to M C M and .B; F / is a bordism between .M; f / and .M; f /. The symmetry of the relation is a direct consequence of the definition. Let .B; F; 'i W Mi ! @i B/ be a bordism between

for x 2 @2B. The result D carries a smooth structure, and the canonical maps B !

DC are smooth embeddings (15.10.1). We can factor h F; G i W B C C ! X

We denote by ŒM; f the bordism class of .M; f / and by Nn.X/ the set of bordism classes of n-dimensional closed singular manifolds in X. The set Nn.X/ carries an associative and commutative composition law ŒM; f C ŒN; g D ŒM C N; h f; g i . The reader may check that this is well-defined.

(21.1.2) Proposition. .Nn.X/; C/ is an abelian group. Each element has order at most 2.

Proof. The class of a null bordant manifold serves as neutral element, for example the constant map Sn ! X. (For the purpose at hand it is convenient to think of the empty set as an n-dimensional manifold.) For each .M; f / the sum

In this way Nn. / becomes a functor from TOP to ABEL. Homotopic maps induce the same homomorphism: If F W X I ! Y , f ' g is a homotopy, then .M I; F ı.h id// is a bordism between .M; f h/ and .M; gh/. If X is empty, we consider Nn.X/ as the trivial group.

(21.1.3) Example. A 0-dimensional compact manifold M is a finite discrete set. Hence .M; f / can be viewed as a family .x1; : : : ; xr / of points in X. Points x; y 2 X are bordant if and only if they are contained in the same path component.

(21.1.4) Proposition. Let h W K ! L be a diffeomorphism. Then ŒL; g D ŒK; gh .

Proof. Consider the bordism g ı pr W L I ! X; on the boundary piece L 1 we use the canonical diffeomorphism to L, on the boundary piece we L 0 we compose the canonical diffeomorphism to L with h.

We now make the functors Nn. / part of a homology theory. But this time, for variety, we do not begin with the definition of relative homology groups. The exact homology sequence and the excision axiom are now replaced by a Mayer–Vietoris sequence.

Suppose X is the union of open sets X0 and X1. We construct the boundary operator

We call ˛ in (21.1.5) a separating function. If ˛ is a separating function, then

(21.1.6) Lemma. Let ŒK; f D ŒL; g 2 Nn.X/ and let ˛; ˇ be separating functions for .K; f /; .L; g/. Then ŒK˛; f˛ D ŒLˇ ; gˇ .

Proof. We take advantage of (21.1.4). Let .B; F / be a bordism between .K; f / and .L; g/ with @B D K C L. There exists a smooth function W B ! Œ0; 1 such

that

jK D ˛; jL D ˇ; F 1.X X Xj / 1.j /:

We choose a regular value t for and j@B and obtain a bordism 1.t/ between

(21.1.7) Proposition. Let X be the union of open subspaces X0 and X1. Then the sequence

is exact. Here j.x/ D .j 0.x/; j 1.x// and k.y; z/ D k0y k1z with the inclusions

k

j W X0 \ X1 ! X and k W X ! X. The sequence ends with ! N0.X/ ! 0.

524 Chapter 21. Bordism

Proof. (1) Exactness at Nn 1.X0 \ X1/. Suppose ŒM; f 2 Nn.X/ is given. Then M is decomposed by M˛ into the parts B0 D ˛ 1Œ0; 12 and B1 D ˛ 1Œ 12 ; 1 with common boundary M˛. Since f .B0/ X1, we see via B0 that j 1@ŒM; f is in Nn 1.X1/ the zero element. This shows j ı @ D 0.

Suppose, conversely, that j ŒK; f D 0. Then there exist singular manifolds

.Bi ; Fi / in Xi such that @B0 D K D @B1 and F0jK D f D F1jK. We identify B0 and B1 along K and obtain M ; the maps F0 and F1 can be combined to F W M ! X. There exists a separating function ˛ on M such that M˛ D K: With collars of K

(2)has regular value 12 .

Let .N; f / D . 1. 12 /; F j 1. 12 //. Then . 1Œ0; 12 ; F j 1Œ0; 12 / is a bordism between .N; f / and .M0; f0/ in X0; similarly for .M1; f1/. This shows j ŒN; f D

.x0; x1/.

(3) Exactness at Nn.X/. The relation @ ı k D 0 holds, since we can choose on

.M0; k0f0/ C .M1; k1f1/ a separating function ˛ W M0 C M1 ! Œ0; 1 such that ˛ 1. 12 / is empty.

Conversely, let ˛ be a separating function for .M; f / in X and .B; F / a null bordism of .M˛; f˛/ in X0 \ X1. We decompose M along M˛ into the manifolds B1 D ˛ 1Œ0; 12 and B0 D ˛ 1Œ 12 ; 1 with @B1 D M˛ D @B0. Then we identify B and B0 along M˛ D K and obtain a singular manifold .M0; f0/ D .B0 [K B;

.f jB0/ [K F / in X0, and similarly .M1; f1/ in X1. Once we have shown that in Nn.X/ the equality ŒM0; f0 C ŒM1; f1 D ŒM; f holds, we have verified the exactness. We identify in M0 I CM1 I the parts B 1 in M0 1 and M1 1. The resulting manifold L D .M0 I /[B 1 .M1 I / has the boundary .M0 CM1/CM .

We now define relative bordism groups Nn.X; A/ for pairs .X; A/. Elements of Nn.X; A/ are represented by maps f W .M; @M / ! .X; A/ from a compact n-

manifold M . Again we call .M; f / D .M; @M I f / a singular manifold in .X; A/. The bordism relation is a little more complicated. A bordism between .M0; f0/

and .M1; f1/ is a pair .B; F / with the following properties:

(1) B is a compact .n C 1/-manifold with boundary.

(2)@B is the union of three submanifolds with boundary M0, M1 and M 0, where

@M 0 D @M0 C @M1; Mi \ M 0 D @Mi .

(3)F jMi D fi .

(4)F .M 0/ A.

We call .M0; f0/ and .M1; f1/ bordant, if there exists a bordism between them. Again “bordant” is an equivalence relation. For the proof one uses 15.10.3. The sum in Nn.X; A/ is again induced by disjoint union. Each element in Nn.X; A/ has order at most 2. A continuous map f W .X; A/ ! .Y; B/ induces a homomorphism Nn.f / D f W Nn.X; A/ ! Nn.Y; B/ by composition with f . If f0 and f1 are homotopic as maps between pairs, then Nn.f0/ D Nn.f1/. The assignment ŒM; f 7!Œ@M; f j@M induces a homomorphism (boundary operator)

@ W Nn.X; A/ ! Nn 1.A/. For A D ; the equality Nn.X; ;/ D Nn.X/ holds.

(21.1.8) Lemma. Let M be a closed n-manifold and V M an n-dimensional submanifold with boundary. If f W M ! X is a map which sends M X V into A, then ŒM; f D ŒV; f jV in Nn.X; A/.

Proof. Consider F W M I ! X, .x; t/ 7!f .x/. Then @.M I / D M @I and

V 1 [ M 0 is a submanifold of @.M I / whose complement is mapped under

Proof. (1) Exactness at Nn.A/. The relation i ı @ D 0 is a direct consequence of the definitions. Let .B; F / be a null bordism of f W M ! A in X. Then

@ŒB; F D ŒM; f .

(2)Exactness at Nn.X/. Let ŒM; f 2 Nn.A/ be given. Choose V D ; in (21.1.8). Then ŒM; f D 0 in Nn.X; A/, and this shows j i D 0.

Let j ŒM; f D 0. A null bordism of ŒM; f in .X; A/ is a bordism of .M; f / in X to .K; g/ such that g.K/ A. A bordism of this type shows i ŒK; g D ŒM; f .

(3)Exactness at Nn.X; A/. The relation @ıi D 0 is a direct consequence of the definitions. Let @ŒM; f D 0. Choose a null bordism ŒB; F of .@M; f j@M /. We identify .M; f / and .B; f / along @M and obtain .C; g/. Lemma (21.1.8) shows

A basic property of the relative groups is the excision property. It is possible to give a proof with singular manifolds.

526 Chapter 21. Bordism

The bordism notion can be adapted to manifolds with additional structure. Inter-

esting are oriented manifolds. Let M0 and M1 be closed oriented n-manifolds. An oriented bordism between M0; M1 is a smooth compact oriented .n C 1/-manifold

B with oriented boundary @B together with an orientation preserving diffeomorphism ' W M1 M0 ! @B. Here we have to use the convention about the boundary orientation, and M1 M0 denotes the disjoint sum of the manifolds M1 and M0 where M1 carries the given and M0 the opposite orientation. Again this notion of bordism is an equivalence relation. Singular manifolds are defined as before, and we have bordism groups n.X/ of oriented bordism classes of singular n-manifolds in X. But now elements in the bordism group no longer have order at most 2. For

1 follows from the fact that S is an oriented boundary; the known classification of orientable surfaces as a sphere with handles shows that these surfaces are oriented boundaries. It is a remarkable result that 3.P / D 0: Each oriented closed 3-manifold is an oriented boundary; for a proof of this theorem of Rohlin see [77].

The exact sequences (21.1.7) and (21.1.9) as well as (21.1.10) still hold for the -groups. The definition of the boundary operator @ W n.X; A/ ! n 1.A/ uses the boundary orientation. In order to define the boundary operator of the MV-sequence we have to orient M˛. There exists an open neighbourhood U of M˛ in M and a diffeomorphism ' W V D 1=2 "; 1=2 C "Œ M˛ ! U such that

.˛'/.t; x/ D t. If M is oriented, we have the induced orientation of U , and we orient V such that ' preserves the orientation. We orient M˛ such that V carries the product orientation.

The idea of bordism can be used to acquire an intuitive understanding of homology. A compact n-manifold M has a fundamental class zM 2 Hn.M; @M I F2/ and @zM 2 Hn 1.@M I F2/ is again a fundamental class. Let f W .M; @M / ! .X; A/ be a singular n-manifold. We set .f / D f zM 2 Hn.X; AI F2/. In this manner we obtain a well-defined homomorphism

W Nn.X; A/ ! Hn.X; AI F2/:

The morphisms constitute a natural transformation of homology theories. One of the basic results of bordism theory says that is always surjective. This allows us to view homology classes as being represented by singular manifolds. If, in particular, f is an embedding of manifolds, then we view the image of f as a cycle or a homology class. In bordism theory, the fundamental class of M is M itself, i.e., the identity of M considered as a singular manifold.

The transformation can be improved if we take tangent bundle information into account. Let M be a compact n-manifold and denote by M W M ! BO the classifying map of the stable tangent bundle of M . For a singular manifold f W .M; @M / ! .X; A/ we then have .f; M / W .M; @M / ! .X; A/ BO. Again we take the image of the fundamental class

Œf D .f; M / zM 2 Hn..X; A/ BOI F2/:

We now obtain a natural transformation of homology theories

D .X; A/ W N .X; A/ ! H ..X; A/ BOI F2/;

and in particular for the coefficient ring, the Thom bordism ring N of unoriented manifolds,

W N ! H .BOI F2/:

A fundamental result says that .X; A/ is always injective [28, p. 185].

This transformation is also compatible with the multiplicative structures. The algebra structure of H .X BOI F2/ is induced by the homology product and the H -space structure m W BO BO ! BO which comes from the Whitney sum of bundles. We obtain a natural homomorphism of graded algebras,

H .X BOI F2/ ˝ H .Y BOI F2/ ! H .X BO Y BOI F2/ ! H .X Y BOI F2/I

the first map is the homology product and the second is induced by the permutation of factors and m.

Thom [184] determined the structure of the ring N : It is a graded algebra F2Œu2; u4; u5; : : : with a generator uk in each dimension k which does not have the form k D 2t 1. One can take u2n D ŒRP 2n as generators in even dimensions. The ring H .BOI F2/ is isomorphic to F2Œa1; a2; a3; : : : with a generator ai in dimension i.

Another basic result says that there exists a natural isomorphism N .X/ Š N ˝F2 H .XI F2/ of multiplicative homology theories [160], [28, p. 185]. Thus the homology theory N . / can be reduced to the determination of the coefficient ring N and singular homology with F2-coefficients.

For oriented manifolds the situation is more complicated. One can still define a multiplicative natural transformation of homology theories

W .X; A/ ! H ..X; A/ BSOI Z/

from the fundamental classes of oriented manifolds as above. But this time the transformation is no longer injective and .X; A/ ! H .X; AI Z/ in general not surjective. Also the theory . / cannot be reduced to ordinary homology. But the transformation still carries a lot of information. It induces a natural isomorphism

.X; A/ ˝ Q Š H ..X; A/ BSOI Q/;

and, in particular,

by the stable classifying map of the tangent bundle (see (21.4.2)). The ring ˝ Q is isomorphic to QŒx4; x8; : : : with a generator xn for each n 0.4/. One can take the x4n D ŒCP 2n as polynomial generators (see (21.4.4)).