ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 25

GLC(X) ^- GL(X) -^- G

is a principal fibre bundle and hence has the homotopy lifting property

(see [Hu]). Define a map p : $Q(X)-^G by p(T)= ir(T+ K), where

K € K(X,Y) and T + K € GL(X,Y). Notice that p is properly defined.

Indeed, if K,K'eK(X) with T + K, T + K'eGLCX), then (T+K)"1(T+K')

€ GLp(X), so that TC(T + K) = ?r(T+ K'). Also, p is continuous, since, in

a neighborhood of each T € $ (X), p coincides with the composition of

n : GL(X) G with a constant mapping. Let a = pa

G.

There is a map a1 GL(X) so that the following diagram

commutes:

GL(X)

- G

Indeed, by the homotopy lifting property, the existence of a lifting will

depend only on the homotopy class of a. But, since A is contractible,

a is homotopic to a constant map, which certainly lifts.

Choose a to be a lifting of a = pa. Then it follows from the

definition of p that a.(A) - a(A) is a compact operator for each X € A

and so A i— (a (A)) is a parametrix for a. •

Proposition 2.2 Let A be a paracompact, contractible Hausdorff space and

let a:A —- » $0(X,Y) be continuous. Suppose that AQ Q A is a retract of

A and that 0 : A

GL(Y,X) is a parametrix for a : A

0

Then 0n extends to a parametrix 0:A — GL(Y,X) for a : A

•0(X.Y).

VX.Y).