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).
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