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

388 Chapter 15. Manifolds

Suppose now that .x/ D 0. We compute T.x;s/G at .v; w/ 2 Tx M Ts S D Tx X Rm. Let

m W M S ! M S; .x; s/ 7!.x; .x/s/:

Then

T.x;s/m.v; w/ D .v; .x/w C Tx .v/s/:

The chain rule, applied to G D F ı m, yields

T.x;s/G.v; w/ D Tm.x;s/F ı T.x;s/m.v; w/ D T.x;0/F .v; 0/ D Tx f .v/;

since .x/ D 0; Tx D 0 and F .x; 0/ D f .x/. Since .x/ D 0, by choice of

W and , f is transverse to Z in x, hence – since T.x;s/G and Tx f have the same image – also G is transverse to Z in .x; s/. A similar argument is applied to @G.

Then one finishes the proof as in the case C D ;.

15.10 Gluing along Boundaries

We use collars in order to define a smooth structure if we glue manifolds with boundaries along pieces of the boundary. Another use of collars is the definition of a smooth structure on the product of two manifolds with boundary (smoothing of corners).

15.10.1 Gluing along boundaries. Let M1 and M2 be @-manifolds. Let Ni @Mi be a union of components of @Mi and let ' W N1 ! N2 be a diffeomorphism. We denote by M D M1 [' M2 the space which is obtained from M1 C M2 by the identification of x 2 N1 with '.x/ 2 N2. The image of Mi in M is again denoted by Mi . Then Mi M is closed and Mi X Ni M open. We define a smooth structure on M . For this purpose we choose collars ki W R Ni ! Mi with open image Ui Mi . The map

is an embedding with image U D U1 [' U2. We define a smooth structure (depending on k) by the requirement that Mi X Ni ! M and k are smooth embeddings. This is possible since the structures agree on .Mi X Ni / \ U . Þ

15.10.2 Products. Let M1 and M2 be smooth @-manifolds. We impose a canonical smooth structure on M1 M2 X .@M1 @M2/ by using products of charts for Mi as

Here W R2 ! R1 R1 , .r; '/ 7!.r; 12 ' C 34 /, written in polar coordinates

.r; '/, and (1) interchanges the second and third factor. There exists a unique smooth structure on M1 M2 such that M1 M2 X .@M1 @M2/ M1 M2 and are diffeomorphisms onto open parts of M1 M2. Þ

15.10.3 Boundary pieces. Let B and C be smooth n-manifolds with boundary. Let M be a smooth .n 1/-manifold with boundary and suppose that

'B W M ! @B; 'C W M ! @C

are smooth embeddings. We identify in B C C the points 'B .m/ with 'C .m/ for each m 2 M . The result D carries a smooth structure with the following properties:

(1)B X 'B .M / D is a smooth submanifold.

(2)C X 'C .M / D is a smooth submanifold.

(3)W M ! D, m 7!'B .m/ 'C .m/ is a smooth embedding as a submanifold of type I.

(4) The boundary of D is diffeomorphic to the gluing of @B X 'B .M /ı with

@C X 'C .M /ı via 'B .m/ 'C .m/; m 2 @M .

The assertions (1) and (2) are understood with respect to the canonical embeddings B D C . We have to define charts about the points of .M /, since the conditions (1) and (2) specify what happens about the remaining points. For points of .M X @M / we use collars of B and C and proceed as in 15.10.1. For .@M / we use the following device.

Choose collars B W R @B ! B and W R @M ! M and an embeddingB W R @M ! @B such that the next diagram commutes,

['B

Here B can essentially be considered as a tubular map, the normal bundle of '.@M / in @B is trivial. And is “half” of this normal bundle.

Then we form ˆB D B ı .id B / W R R @M ! B. For C we choose in a similar manner C and C , but we require 'C ı D C where .m; t/ D

390 Chapter 15. Manifolds

.m; t/. Then we define ˆC from C and C . The smooth structure in a neighbourhood of .@M / is now defined by the requirement that ˛ W R R @M ! D is a smooth embedding where

15.10.4 Connected sum. Let M1 and M2 be n-manifolds. We choose smooth embeddings si W Dn ! Mi into the interiors of the manifolds. In M1 X s1.En/ C

M2 X s2.En/ we identify s1.x/ with s2.x/ for x 2 Sn 1. The result is a smooth manifold (15.10.1). We call it the connected sum M1#M2 of M1 and M2. Suppose

M1; M2 are oriented connected manifolds, assume that s1 preserves the orientation and s2 reverses it. Then M1#M2 carries an orientation such that the Mi X si .En/ are oriented submanifolds. One can show by isotopy theory that the oriented diffeomorphism type is in this case independent of the choice of the si . Þ

15.10.5 Attaching handles. Let M be an n-manifold with boundary. Furthermore, let s W Sk 1 Dn k ! @M be an embedding and identify in M C Dk Dn k the points s.x/ and x. The result carries a smooth structure (15.10.3) and is said to be obtained by attaching a k-handle to M .

Attaching a 0-handle is the disjoint sum with Dn. Attaching an n-handle means that a “hole” with boundary Sn 1 is closed by inserting a disk. A fundamental result asserts that each (smooth) manifold can be obtained by successive attaching of handles. A proof uses the so-called Morse theory (see e.g., [134], [137]). A handle decomposition of a manifold replaces a cellular decomposition, the advantage is that the handles are themselves n-dimensional manifolds. Þ

15.10.6 Elementary surgery. If M 0 arises from M by attaching a k-handle, then @M 0 is obtained from @M by a process called elementary surgery. Let

h W Sk 1 Dn k ! X be an embedding into an .n 1/-manifold with image U . Then X X U ı has a piece of the boundary which is via h diffeomorphic to Sk 1 Sn k 1. We glue the boundary of Dk Sn k 1 with h; in symbols

X0 D .X X U ı/ [h Dk Sn k 1:

The transition from X to X0 is called elementary surgery of index k at X via h. The method of surgery is very useful for the construction of manifolds with prescribed topological properties. See [191], [162], [108] to get an impression of surgery theory. Þ

Problems

1. The subsets of SmCnC1 RmC1 RnC1

D1 D f.x; y/ j kxk2 12 ; kyk2 12 g; D2 D f.x; y/ j kxk2 12 ; kyk2 12 g

are diffeomorphic to D1 Š Sm DnC1, D2 Š DmC1 Sn. They are smooth submanifolds with boundary of SmCnC1. Hence SmCnC1 can be obtained from Sm DnC1 and

DmC1 Sn by identifying the common boundary Sm Sn with the identity. A diffeomor- p

phism D1 ! Sm DnC1 is .z; w/ 7!.kzk 1z; 2w/.

2. Let M be a manifold with non-empty boundary. Identify two copies along the boundary with the identity. The result is the double D.M / of M . Show that D.M / for a compact M

is the boundary of some compact manifold. (Hint: Rotate M about @M about 180 degrees.)

3.Show M #Sn Š M for each n-manifold M .

4.Study the classification of closed connected surfaces. The orientable surfaces are S2 and connected sums of tori T D S1 S1. The non-orientable ones are connected sums of

projective planes P D RP 2. The relation T #P D P #P #P holds. The connected sum with T is classically also called attaching of a handle.

Chapter 16

Homology of Manifolds

The singular homology groups of a cell complex vanish above its dimension. It is an obvious question whether the same holds for a manifold. It is certainly technically complicated to produce a cell decomposition of a manifold and also an artificial structure. Locally the manifold looks like a Euclidean space, so there arises no problem locally. The Mayer–Vietoris sequences can be used to paste local information, and we use this technique to prove the vanishing theorem.

The homology groups of an n-manifold M in dimension n also have special properties. They can be used to define and construct homological orientations of a manifold. A local orientation about x 2 M is a generator of the local homology group Hn.M; M X xI Z/ Š Z. In the case of a surface, the two generators correspond to “clockwise” and “counter-clockwise”. If you pick a local orientation, then you can transport it along paths, and this defines a functor from the fundamental groupoid and hence a twofold covering. If the covering is trivial, then the manifold is called orientable, and otherwise (as in the case of a Möbius-band) non-orientable.

Our first aim in this chapter will be to construct the orientation covering and use it to define orientations as compatible families of local orientations.

In the case of a closed compact connected manifold we can define a global homological orientation to be a generator of Hn.M I Z/; we show that this group is either zero or Z. In the setting of a triangulation of a manifold, the generator is the sum of the n-dimensional simplices, oriented in a coherent manner. In a non-orientable manifold it is impossible to orient the simplices coherently; but in that case their sum still gives a generator in Hn.M I Z=2/, since Z=2-coefficients mean that we can ignore orientations.

Once we have global orientations, we can define the degree of a map between oriented manifolds. This is analogous to the case of spheres already studied.

16.1 Local Homology Groups

Let h . / be a homology theory and M an n-dimensional manifold. Groups of the type hk .M; M X x/ are called local homology groups. Let ' W U ! Rn be a chart

of M centered at x. We excise M X U and obtain an isomorphism

h .M; M x/ h .U; U x/ ' h . n; n 0/: k X Š k X ! k R R X

For singular homology with coefficients in G we see that Hn.M; M X xI G/ Š G, and the other local homology groups are zero. Let R be a commutative ring. Then

Hn.M; M X xI R/ Š R is a free R-module of rank 1. A generator, corresponding to a unit of R, is called a local R-orientation of M about x. We assemble the

totality of local homology groups into a covering.

Let K L M . The homomorphism rKL W hk .M; M XL/ ! hk .M; M XK/, induced by the inclusion, is called restriction. We write rxL in the case that K D fxg.

(16.1.1) Lemma. Each neighbourhood W of x contains an open neighbourhood U of x such that the restriction ryU is for each y 2 U an isomorphism.

with morphisms induced by inclusion. The maps (1) and (2) are excisions, and (3) is an isomorphism, because V X U V X y is for each y 2 U an h-equivalence

and hk .M; M X x/ is the fibre of ! over x (with discrete topology). Let U be an open neighbourhood of x such that ryU is an isomorphism for each y 2 U . We define bundle charts

We give hk .M; M X / the topology which makes 'x;U a homeomorphism onto an open subset. We have to show that the transition maps

'y;V1 'x;U W .U \ V / hk .M; M X x/ ! .U \ V / hk .M; M X y/

are continuous. Given z 2 U \ V , choose z 2 W U \ V such that rwW is an isomorphism for each w 2 W . Consider now the diagram

is independent of w 2 W , and this shows the continuity of the transition map.

We take advantage of the fact that the fibres are groups. For A M we denote by .A/ the set of continuous (D locally constant) sections over A of ! W hk .M; M X / ! M . If s and t are sections, we can define .s C t/.a/ D s.a/ C t.a/. One uses the bundle charts to see that s C t is again continuous. Hence.A/ is an abelian group. We denote by c .A/ .A/ the subgroup of sections with compact support, i.e., of sections which have values zero away from a compact set.

(16.1.2) Proposition. Let z 2 hk .M; M X U /. Then y 7!ryU z is a continuous section of !.

Problems

1.Sn 1 Rn X En and Rn X e ! Sn 1, y 7!.y e/=ky ek are h-equivalences (e 2 En). The map Sn 1 ! Sn 1, y 7!.y e/=ky ek is homotopic to the identity. These facts imply that Rn X E Rn X e is an h-equivalence.

2.Let M be an n-dimensional manifold with boundary @M . Show that x 2 @M if and only if Hn.M; M Xx/ D 0. From this homological characterization of boundary points one obtains: Let f W M ! M be a homeomorphism. Then f .@M / D @M and f .M X @M / D M X @M .

3.Let ' W .C; 0/ ! .D; 0/ be a homeomorphism between open neighbourhoods of 0 in Rn. Then ' W Hn.C; C X 0I R/ ! Hn.D; D X 0I R/ is multiplication by ˙1.

16.2 Homological Orientations

Let M be an n-manifold and A M . An R-orientation of M along A is a section s 2 .AI R/ of ! W Hn.M; M X I R/ ! M such that s.a/ 2 Hn.M; M X aI R/ Š R is for each a 2 A a generator of this group. Thus s combines the local orientations in a continuous manner. In the case that A D M , we talk about an R-orientation of M , and for R D Z we just talk about orientations. If an orientation exists, we call M (homologically) orientable. If M is orientable along A and B A, then M is orientable along B.

(16.2.1) Note. Let Ori.M / Hn.M; M X I Z/ be the subset of all generators of all fibres. Then the restriction Ori.M / ! M of ! is a 2-fold covering of M , called

(1)M is orientable.

(2)M is orientable along compact subsets.

(3)The orientation covering is trivial.

(4)The covering ! W Hn.M; M X I Z/ ! M is trivial.

Proof. (1) ) (2). Special case.

(2) ) (3). The orientation covering is trivial if and only if the covering over each component is trivial. Therefore let M be connected. Then a 2-fold covering

Q ! is trivial if and only if Q is not connected, hence the components of Q

M M M M are also coverings.

Suppose Ori.M / ! M is non-trivial. Since Ori.M / is then connected, there exists a path in Ori.M / between the two points of a given fibre. The image S of such a path is compact and connected, and the covering is non-trivial over S, since we can connect two points of a fibre in it. By the assumption (2), the orientation covering is trivial over the compact set S, hence it has a section over S. Contradiction.

(3)) (4). Let s be a section of the orientation covering. Then M Z ! Hn.M; M X I Z/, .x; k/ 7!ks.x/ is a trivialization of !: It is a map over M , bijective on fibres, continuous, and a morphism between coverings.

(4)) (1). If ! is trivial, then it has a section with values in the set of generators.

(16.2.3) Note. Ori.M / ! M is a twofold principal covering with automorphism group C D f1; t j t2 D 1g. Let t act on G as multiplication by 1. Then the associated covering Ori(M) C G is isomorphic to Hn.M; M X I G/.

Proof. The map

Ori.M / G ! Hn.M; M X I Z/ ˝ G Š Hn.M; M X I G/; .u; g/ 7!u ˝ g

induces the isomorphism. (The isomorphism is the fibrewise isomorphism

Hn.M; M X xI Z/ ˝ G Š Hn.M; M X xI G/ from the universal coefficient for-

The sections .AI G/ of ! over A correspond bijectively to the

W Ori.M /jA ! G with the property ı t D . This is a

Þ

1.Let M be a smooth n-manifold with an orienting atlas. Then there exists a unique

homological Z-orientation such that the local orientations in Hn.M; M X xI Z/ are mapped via positive charts to a standard generator of Hn.Rn; Rn X 0I Z/. Conversely, if M is Z- oriented, then M has an orienting atlas which produces the given Z-orientation.

2.Every manifold has a unique Z=2-orientation.

396 Chapter 16. Homology of Manifolds

16.3 Homology in the Dimension of the Manifold

Let M be an n-manifold and A M a closed subset. We use in this section singular homology with coefficients in the abelian group G and sometimes suppress G in the notation.

(16.3.1) Proposition. For each ˛ 2 Hn.M; M X AI G/ the section

J A.˛/ W A ! Hn.M; M X I G/; x 7!rxA.˛/ of ! (over A) is continuous and has compact support.

Proof. Let the chain c 2 Sn.M I G/ represent the homology class ˛. There exists a compact set K such that c is a chain in K. Let x 2 A X K. Then the image of c under

Sn.KI G/ ! Sn.M I G/ ! Sn.M; KI G/ ! Sn.M; M X xI G/

(16.3.2) Theorem. Let A M be closed.

(1)Then Hi .M; M X A/ D 0 for i > n.

(2)The homomorphism J A W Hn.M; M X A/ ! c .A/ is an isomorphism.

Proof. Let D.A; 1/ and D.A; 2/ stand for the statement that (1) and (2) holds for the subset A, respectively. We use the fact that J A is a natural transformation between contravariant functors on the category of closed subsets of M and their inclusions. The proof is a kind of induction over the complexity of A. It will be divided into several steps.

(1) D.A; j /; D.B; j /; D.A \ B; j / imply D.A [ B; j /. For the proof we use the relative Mayer–Vietoris sequence for .M X A \ BI M X A; M X B/ and an analogous sequence for sections. This leads us to consider the diagram

The Five Lemma and the hypotheses now show D.A [ B; 2/. The Mayer–Vietoris sequence alone yields D.A [ B; 1/.

(2) D.A; j / holds for compact convex subsets A in a chart domain U , i.e.,

The maps (3) and (4) are excisions. The maps (1) and (2) are isomorphisms, because Sn 1 Rn X B is an h-equivalence. The isomorphism rxA shows, firstly, that D.A; 1/ holds; and, secondly, D.A; 2/, since a section of a covering over a

connected set is determined by a single value.

(3) Suppose A U , ' W U ! Rn a chart, A D K1 [ [ Kr , '.Ki / compact convex. We show D.A; j / by induction on r. Let B D K1 [ [ Kr 1 and C D Kr . Then B and B \ C are unions of r 1 sets of type (2), hence D.B; j /

where the colimit is taken over the directed set of neighbourhoods V of K of type

(3). What does this isomorphism statement mean in explicit terms? Firstly, an element xK in the image has the form rKV xV for a suitable V ; and secondly, if xV and xW have the same image xK , then they become equal under a restriction to a suitable smaller neighbourhood. From this description it is then easy to verify that J K is indeed an isomorphism. Suppose . / and . / are isomorphisms. Then we obtain D.K; 1/, and the isomorphisms J V yield D.K; 2/.

The isomorphism . / holds already for the singular chain groups. It uses the fact that a chain has compact support; if the support is contained in M X K, then already in M X V for a suitable neighbourhood V of K.

./ is an isomorphism: See Problem 1.

(5) D.K; j / holds for arbitrary compact subsets K, for K is a union of a finite number of sets of type (4). Then we can use induction as in case (3).