ANÁLISIS DE RESULTADOS

L e m m a 6 .1 Let V , W be vector bundles over B with Riemannian metrics,

let V C B be open,

B

=

£: V \v

* W |(;

phism which preserves the metric on each fibre. Then £ induces an isomor­

phism

£ : r ( ( B , f l o ) , ( ( V x fl F T ) + . o o ) ) - * r ( ( B . B o ) , ( ( W x B F V T ) + . o o ) ) .

P r o o f: To simplify notations, we identify B with the oo-sections of (V' x B

F V ) + and ( W x B F W )+ . We construct a map

£+ : ( V x B FV')|* U B — ► ( W x g F W ) \ l U B

l>y defining it in the same way as before on ( V x B FV’ )| and as the identity on II. This map is clearly well-defined. We show that it is continuous. This is clear except possibly for points in d l l , Let p € HU. Choose a neighbourhood

U of n in B such that V’ and IT are trivial. Let A i be a system of

' I// l "u

neighbourhoods of p in B. For N 6 A and £ > 0. let •r > V * ., = {.r € ( T x B F V )

The se ts VN,e for £ > 0, N € A form a system of neighbourhoods of p 6 ( T x g F V ) + . Let = V,v,e O ( ( V x B F V )\ * U and be defined in the analogous way, replacing V by IT. One verifies that, for N € A and

s > 0, £+ maps VJv.j to Wjy f , so £ + is continuous at /». An inverse map is constructed in the same way. Now we define the map £ as composing a

section with £+ . □

L e m m a 6 .2 Lit V' In a vector bundle with a Riemannian mt trie over the

compact bast span B. li t I ' € B bt open and let l\ = B \ C . Then the

inclusion

*: r ( l l , ( T x B FV')|(+ ) V { ( B . I \ ) , ( ( V x H F V ) + ,o o ))

C HAPTER (i. APPROXIMATION BY CONFIGURATION SPACES 85

P r o o f: W e construct a homotopy inverse j of i as follows. Define a map

li: [0, oo] x [0. 1] —♦ [0, oo] by the formula

H : ( V x * F V ) + x [0. 1] - ( V xg F V ) +

by the formula H ( ( v , f ) , I) = (/i(||v||, t) • v , / ) . Write / / , for / / ( ■ ,f ) . N o­ tice that //,, is the identity and that H\(v) = oo for ||n|| > 1. Define the

H' is open. I\ C W , and ||II,(<r(j-))|| — oo for all ,r £ W , so the support of / / 1 o a is contained in B \ \ V , which is a compact subset o f V . This shows that j ( n) has compact support.

The composition i o j is just the map given by a >—► //■ o it. This map is homotopic to iti—» //(, o it, which is the identity. The reverse composition

j o / is given by composing a section with the appropriate restriction of Il\.

which is just in I he sam e wav seen to be homotopic to the identity, so j is a

homotopy inverse of t. O

Let M be a closed, orientable »-dimensional manifold and let /•’ —* E —» M be a fibre bundle where the fibre /•’ is (» — 1 J-connected and tr,,(/■’ ) — Z. Suppose that there is a ‘ zero section’ tr": M —► E, i. e. a section such that

a “( w ) = oom, the basepoint of the fibre over in for all in 6 M . In this case it is possible to define t he liiijrir deg(ir) £ Z for each section rrof the bundle,

and the space V ( M , E ) has countably many path components, labelled by One checks that li is continuous. Now define a map

map

j : r ( ( B , K ) , ( ( V x hF V ) + . oo)) r ( u , ( V x B FV ’ )|(+,)

CHAPTER (i. APPROXIMATION BY CONFIGURATION SPACES 86

the degree. This is done in the following way, which directly generalizes the concept of the degree of a map between spaces. From the Leray-Serre spectral sequence o f the fibration we get a short-exact sequence

0 --- ► H0( M ; H n( F) ) --- ► II, AE) --- ► H ,A \ 1 :H 0( F ) ) --- ► 0

ii? lie

I T A P ) Hn( M )

The map it“ :

H

M

II,AE)

I I, AM)' Now let it be any section M —* E. Define the degree of it by the

formula

(6.1)

d o <r.(/i) = deg(<r) • 0.

This definition, of course, depends on t he choice o f the zero section it', which

corresponds to tin* choice of a basepoint in t bo pointed set jr()(F (A /. /•.')). and clearly deg(ir") = 0. It is also clear that two homotopic sections have the same degree, l b«* converse can be proved using obstruction theory. We will writ«1 I\ { M . E) for the space til sections of degre«’ k.

In particular, the above applies to the situation where l is a rauk-n vector bundle wit h Kiemaiiuian metric over M and F. — (1 x \f F V y . bet ♦ be the bas«*point of M. Then, according to lemma 6.2. the inclu­ sion i: \' ( M \ *. !•') —> \ ’ ( M, /•.’ ) is a homotopy e«piivaleuce, so both spaces have I lie same path components. For rr £ F' ( A / \ *, we deline deg(ir) to be <leg(i(it)) and write l’( ( M \ *, !■',) for tile space of sections of degree k with compact support. Similarly, the concept of the degre«' of a section can be extemled to compact manifolds with boundary and the degree of a compactly supported se« l ion to open paracompai t manifolds.

C HAPTER 6. APPROXIMATION B Y CONFIGURATION SPACES 87