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

398Chapter 16. Homology of Manifolds

(6)Let K D K1 [ K2 [ K3 [ with compact Ki . Suppose there are pairwise

disjoint open neighbourhoods Ui of Ki . By additivity of homology groups and section groups, J K is the direct sum of the J Ki .

(7)Let A finally be an arbitrary closed subset. Since M is locally compact with countable basis, there exists an exhaustion M D [Ki , K1 K2 by

(16.3.3) Theorem. Suppose A is a closed connected subset of M . Then:

(1)Hn.M; M X AI G/ D 0, if A is not compact.

(2)Hn.M; M X AI G/ Š G, if M is R-orientable along A and A is compact. Moreover Hn.M; M X AI G/ ! Hn.M; M X xI G/ is an isomorphism for each x 2 A.

(3)Hn.M; M X AI Z/ Š 2G D fg 2 G j 2g D 0g, if M is not orientable along A and A is compact.

Proof. (1) Since A is connected, a section in .AI G/ is determined by its value at a single point. If this value is non-zero, then the section is non-zero everywhere. Therefore there do not exist non-zero sections with compact support over a noncompact A, and (16.3.2) shows Hn.M; M X AI G/ D 0.

(2) Let A be compact. Then Hn.M; M X AI G/ Š .AI G/. Again a section is determined by its value at a single point. We have a commutative diagram

If M is orientable along A, then there exists in .A/ an element such that its value at x is a generator. Hence b is an isomorphism and therefore also rxA.

(3) A section in .AI G/ corresponds to a continuous map W Ori.M /jA ! G with t D . If M is not orientable along A, then Ori.M /jA is connected and

Theorem (16.3.3) can be considered as a duality result, since it relates an assertion about A with an assertion about M X A.

(16.3.4) Proposition. Let M be an n-manifold and A M a closed connected subset. Then the torsion subgroup of Hn 1.M; M X AI Z/ is of order 2 if A is compact and M non-orientable along A, and is zero otherwise.

Proof. Let q 2 N and suppose M is orientable along the compact set A. Then

Z=q Š Hn.M; M X AI Z=q/

Š Hn.M; M X AI Z/ ˝ Z=q ˚ Hn 1.M; M X AI Z/ Z=q Š Z=q ˚ Hn 1.M; M X AI Z/ Z=q:

We have used: (16.3.3); universal coefficient theorem; again (16.3.3). This implies that Hn 1.M; M X AI Z/ Z=q D 0. Similarly for non-compact A or q odd

0 Š Hn.M; M X AI Z=q/ Š Hn 1.M; M X AI Z/ Z=q:

Since Tor.G; Z=q/ Š fg 2 G j qg D 0g, this shows that Hn 1.M; M X AI Z/ has no q-torsion in these cases. If A is compact and M non-orientable along A, then

Z=2 Š Hn.M; M X AI Z=4/ Š Hn 1.M; M X AI Z/ Z=4

by (16.3.3) and the universal coefficient formula. Since we know already that the group in question has no odd torsion, we conclude that there exists a single non-zero element of finite order and the order is 2.

Problems

1. Let s be a section over the compact set K. For each x 2 K there exists an open neighbourhood U.x/ and an extension sx of sjU.x/ \ K. Cover K by U.x1/; : : : ; U.xr /. Let

W D fy j sxi .y/ D sxj .x/ if y 2 U.xi / \ U.xj /g and define s.y/; y 2 W as the common value. Show that W is open. Let s; s0 be sections over V which agree on K. Show that they agree in a smaller neighbourhood V1 V of K. (These assertions hold for sections of coverings.)

16.4 Fundamental Class and Degree

The next theorem is a special case of (16.3.3).

(16.4.1) Theorem. Let M be a compact connected n-manifold. Then one of the following assertions holds:

(1)M is orientable, Hn.M / Š Z, and for each x 2 M the restriction Hn.M / ! Hn.M; M X x/ is an isomorphism.

Under the hypothesis of (16.4.1), the orientations of M correspond to the generators of Hn.M /. A generator will be called fundamental class or homological

orientation of the orientable manifold.

We now use fundamental classes in order to define the degree; we proceed as in the special case of Sn. Let M and N be compact oriented n-manifolds. Let

immediately the following properties of the degree:

(1)The degree is a homotopy invariant.

(2)d.f ı g/ D d.f /d.g/.

(3)A homotopy equivalence has degree ˙1.

(4)If M D M1 C M2, then d.f / D d.f jM1/ C d.f jM2/.

(5)If we pass in M or N to the opposite orientation, then the degree changes the sign.

We now come to the computation of the degree in terms of local data of the map. Let M and N be connected and set K D f 1.p/. Let U be an open neighbourhood of K in M . In the commutative diagram

we have f U z.U; K/ D d.f /z.N; p/. Thus the degree only depends on the restriction f U of f to U . One can now extend the earlier investigations of self maps of

Sn to this more general case. The additivity of the degree is proved in exactly the

S

same manner. Let K be finite. Choose U D x2K Ux where the Ux are pair-wise disjoint open neighbourhoods of x. We then have

L

x2K Hn.Ux ; Ux X x/ Š Hn.U; U X K/; Hn.Ux ; Ux X x/ Š Z:

The image z.Ux ; x/ of z.M / is a generator, the local orientation determined by the fundamental class z.M /. The local degree d.f; x/ of f about x is defined by f z.Ux ; x/ D d.f; x/z.N; p/. The additivity yields d.f / D Px2K d.f; x/.

(16.4.2) Remark. Let f be a C 1-map in a neighbourhood of x; this shall mean the following. There exist charts ' W Ux ! Rn centered at x and W V ! Rn centered at p such that f .Ux / V and g D f ' 1 is a C 1-map. We can suppose that the charts preserve the local orientations; this shall mean for ' that

' W Hn.Ux ; Ux X x/ ! Hn.Rn; Rn X 0/ sends z.Ux ; x/ to the standard generator. Such charts are called positive with respect to the given orientations. Suppose now

in addition that the differential of g at x is regular. Then d.f; x/ is the sign of the determinant of the Differential Dg.0/. Þ

(16.4.3) Proposition. Let M be a connected, oriented, closed n-manifold. Then there exists for each k 2 Z a map f W M n ! Sn of degree k.

Proof. If f W M ! Sn has degree a and g W Sn ! Sn degree b, then gf has degree ab. Thus it suffices to realize a degree ˙1. Let ' W Dn ! M be an embedding. Then we have a map f W M ! Dn=Sn 1 which is the inverse of ' on U D '.En/ and sends M X U to the base point. This map has degree ˙1.

Let the manifolds M and N be oriented by the fundamental classes zM 2

Hm.M / and zN 2 Hn.N /. Then the homology product zM zN is a fundamental class for M N , called the product orientation.

Problems

1.Let p W M ! N be a covering of n-manifolds. Then the pullback of Ori.N / ! N along p is Ori.M / ! M .

2.Let p W M ! N be a G-principal covering between connected n manifolds with orientable M . Then N is orientable if and only if G acts by orientation-preserving homeomorphisms.

3.The manifold Ori.M / is always orientable.

4.RP n is orientable if and only if n is odd.

5.Let M be a closed oriented connected n-manifold. Suppose that M carries a CWdecomposition with k-skeleton Mk . The inclusion induces an injective map Hn.M / ! Hn.Mn; Mn 1/. The fundamental class is therefore represented by a cellular chain in Hn.Mn; Mn 1/. If we orient the n-cells in accordance with the local orientations of the manifold, then the fundamental class chain is the sum of the n-cells. This is the classical interpretation of the fundamental class of a triangulated manifold. A similar assertion holds for unoriented manifolds and coefficients in Z=2 and manifolds with boundary (to be considered in the next section).

In a sense, a similar assertion should hold for non-compact manifolds; but the cellular chain would have to be an infinite sum. Therefore a fundamental class has to be defined via an inverse limit.

6.Let M be a closed connected n-manifold. Then Hn.M I Z=2/ Š Z=2 and the restrictions rxM W Hn.M I Z=2/ ! Hn.M; M X xI Z=2/ are isomorphisms.

7.Let f W M ! N be a map between closed connected n-manifold. Then one can define

the degree modulo 2 d2.f / 2 Z=2; it is zero (one) if f W Hn.M I Z=2/ ! Hn.N I Z=2/ is the zero map (an isomorphism). If this degree is non-zero, then f is surjective. If the manifolds are oriented, then d.f / mod 2 D d2.f /.

8.Let G be a compact connected Lie group and let T be a maximal torus of G. The map q W G=T T ! G, .g; t/ 7!gtg 1 has degree jW j. Here W D N T =T is the Weyl group. Since q has non-zero degree, this map is surjective (see [29, IV.1]).

9.Let G be a compact connected Lie group and T a maximal torus of G. The degree of f W G ! G, g 7!gk has degree kr , r D dim T . Let c 2 T be an element such that the powers of c are dense in T , then jf 1.c/j D kr , f 1.c/ T , and c is a regular value of f . [90]

10.Let f W M ! N be a proper map between oriented connected n-manifolds. Define the degree of f .

402 Chapter 16. Homology of Manifolds

16.5 Manifolds with Boundary

Let M be an n-dimensional manifold with boundary. We call z 2 Hn.M; @M / a fundamental class if for each x 2 M X @M the restriction of z is a generator in

Hn.M; M X x/.

(16.5.1) Theorem. Let M be a compact connected n-manifold with non-empty boundary. Then one of the following assertions hold:

Proof. Let W Œ0; 1Œ @M ! U be a collar of M , i.e., a homeomorphism onto an open neighbourhood U of @M such that .0; x/ D x for x 2 @M . For simplicity of notation we identify U with Œ0; 1Œ @M via ; similarly for subsets of U . In this sense @M D 0 @M . We have isomorphisms

Hn.M; @M / Š Hn.M; Œ0; 1Œ @M / Š Hn.M X @M; 0; 1Œ @M / Š .A/:

The first one by h-equivalence; the second one by excision; the third one uses the closed set

A D M X .Œ0; 1Œ @M / M X @M

and (16.3.2). The set A is connected, hence .A/ Š Z or .A/ Š 0. If .A/ Š Z, then M X@M is orientable along A. Instead of A we can argue with the complement of Œ0; "Œ @M . Since each compact subset of M X @M is contained in some such complement, we see that M X @M is orientable along compact subsets, hence orientable (see (16.2.2)). The isomorphism Hn.M X @M; 0; 1Œ @M / Š .A/ says that there exists an element z 2 Hn.M X @M; 0; 1Œ @M / which restricts to a generator of Hn.M X @M; M X @M X x/ for each x 2 A. For the corresponding element z 2 Hn.M; @M / a similar assertion holds for each x 2 M X @M , i.e., z is a fundamental class (move around x within the collar).

It remains to show that @z is a fundamental class. The lower part of the diagram

shows that z yields a fundamental class in Hn.I @M; @I @M /. The upper part shows that this fundamental class corresponds to a fundamental class in Hn 1.@M /, since fundamental classes are characterized by the fact that they are generators (for each component of @M ).

(16.5.2) Example. Suppose the n-manifold M is the boundary of the compact ori- entable .nC1/-manifold B. We have the fundamental classes z.B/ 2 HnC1.B; @B/ and z.M / D @zB 2 Hn.M /. Let f W M ! N be a map which has an extension F W B ! N , then the degree of f (if defined) is zero, d.f / D 0, for we have f z.M / D f @z.B/ D F i @z.B/ D 0, since i @ D 0 as consecutive morphisms

in the exact homology sequence of the pair .B; M /. We call maps f W M ! N orientable bordant if there exists a compact oriented manifold B with oriented bound-

ary @B D M1 M2 and an extension F W B ! N of h f1; f2 i W M1 C M2 ! N . The minus sign in @B D M1 M2 means @z.B/ D z.M1/ z.M2/. Under these assumptions we have d.f1/ D d.f2/. This fact is called the bordism invariance

of the degree; it generalizes the homotopy invariance. Þ

Problems

1. Let M be a compact connected n-manifold with boundary. Then Hn.M; @M I Z=2/ is isomorphic to Z=2; the non-zero element is a Z=2-fundamental class z.M I Z=2/. The restriction to Hn.M; M XxI Z=2/ is for each x 2 M X@M an isomorphism and @z.M I Z=2/ is a Z=2-fundamental class for @M .

2. Show that the degree d2.f / is a bordism invariant.

16.6 Winding and Linking Numbers

Let M be a closed connected oriented n-manifold. Let f W M ! RnC1 and a … Im.f /. The winding number W .f; a/ of f with respect to a is the degree of the

map

for each t, then W .f0; a/ D W .ft ; a/.

(16.6.1) Theorem. Let M be the oriented boundary of the compact smooth oriented manifold B. Let F W B ! RnC1 be smooth with regular value 0 and assume

0 … f .M /. Then

W .f; 0/ D Px2P ".F; x/; P D F 1.0/; f D F j@B

where ".F; x/ 2 f˙1g is the orientation behaviour of the differential Tx F W TB !

T0.RnC1/.

404 Chapter 16. Homology of Manifolds

with k D m C n. Let M N carry the product orientation. The degree of the map fM;N D f W M N ! Sk ; .x; y/ 7!N.x y/

is the linking number L.M; N / of the pair .M; N /. More generally, if the mapsW M m ! RkC1 and W N n ! RkC1 have disjoint images, then the degree of

.x; y/ 7!N. .x/ .y// is the linking number of . ; /.

Problems

1. Let the n-manifold M be the oriented boundary of the smooth connected compact manifold B. Suppose f W M ! Sn has degree zero. Then f can be extended to B.

2.Show L.M; N / D . 1/mCnC1L.N; M /.

3.Let f; g W R ! R3 be smooth embeddings with disjoint closed images. Define a linking number for the pair .f; g/ and justify the definition.

Chapter 17

Cohomology

The axioms for a cohomology theory are analogous to the axioms of a homology theory. Now we consider contravariant functors. The reader should compare the two definitions, also with respect to notation. One advantage of cohomology is an additional internal product structure (called cup product) which will be explained in subsequent sections. The product structure suggests to view the family .hn.X/ j n 2 Z/ as a single object; the product then furnishes it with the structure of a ring (graded algebra). Apart from the additional information in the product structure, the ring structure is also notationally convenient (for instance, a polynomial ring has a better description than its additive group without using the multiplicative structure).

Singular cohomology is obtained from the singular chain complex by an application of the Hom-functor. We present an explicit definition of the cup product in singular cohomology (Alexander–Whitney). In more abstract terms the product can also be obtained from the Eilenberg–Zilber chain equivalences as in the case of the homology product.

We use the product structure to prove a powerful theorem (Leray–Hirsch) which says roughly that the cohomology of the total space of a fibration is a free module over the cohomology ring of the base, provided the fibre is a free module and a basis of the fibre-cohomology can be lifted to the total space. In the case of a topological product, the result is a special case of the Künneth theorem if the cohomology of the fibre is free, since the Ext-groups vanish in that case. One interesting application is to vector bundles; the resulting so-called Thom isomorphism can be considered as a twisted suspension isomorphism in that the suspension is replaced by a sphere bundle. As a specific example we determine the cohomology rings of the projective spaces.

17.1 Axiomatic Cohomology

17.1.1 The axioms. A cohomology theory for pairs of spaces with values in the category of R-modules consists of a family .hn j n 2 Z/ of contravariant functors hn W TOP(2) ! R- MOD and a family .ın j n 2 Z/ of natural transformations ın W hn 1 ı ! hn. These data are required to satisfy the following axioms.

(1)Homotopy invariance. Homotopic maps f0 and f1 between pairs of spaces induce the same homomorphism, hn.f0/ D hn.f1/.

17.1.3 Reduced cohomology. The reduced cohomology groups of a non-empty

denotes the unique map to a point. The functors Q . / are homotopy invariant. For a pointed space .X; / we have the canonical split exact sequence

with the quotient map q. The coboundary operator ın W hn 1.A/ ! h.X; A/ factors

17.1.4 Mayer–Vietoris sequence. A triad .XI A; B/ is excisive for the cohomology theory if the inclusion induces an isomorphism h .A [ B; A/ Š h .B; A \ B/. This property can be characterized in different ways as in the case of a homology theory, see (10.7.1) and (10.7.5). In particular the property is symmetric in A; B.

We have exact Mayer–Vietoris sequences for excisive triads. As in the case of a homology theory one can derive some MV-sequences by diagram chasing. For the general case of two excisive triads we use a method which we developed in the case of homology theory; the MV-sequence was obtained as the exact sequence of a triad of auxiliary spaces by some rewriting (see (10.7.6)). This procedure also works for cohomology.

Let .AI A0; A1/ .XI X0; X1/ be excisive triads. Set X01 D X0 \ X1 and A01 D A0 \ A1. Then there exists an exact Mayer–Vietoris sequence of the following form

.2/ are the restrictions. The connecting morphism in the case A D A0 D A1 is the composition