14
PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ
at x X, then f'(x) - L(x) is compact. Moreover, we also show that each
quasilinear Fredholm map may be represented with its principal part being a
family of isomorphisms. Finally, we show that quasilinear Fredholm maps,
when restricted to closed, bounded subsets of X are proper.
In Section 3, we turn to a discussion of degree and orientation.
Using the fact that GLp(X), the compact vector fields in GL(X), has two
connected components determined by the sign of the Leray-Schauder degree,
we show that if GL(X,Y) is nonempty, then GL(X,Y) has an orientation.
With respect to a choice of orientation, we show that (1.2) is properly
defined. The following section is devoted to deriving the existence,
additivity and Borsuk-Ulam property of the degree. We also show that along
quasilinear Fredholm homotopies, the absolute value of the degree is in-
variant. More precisely, we show that if the quasilinear Fredholm homotopy
F : [a,b] x X —» Y is represented by F(t,x) = M (x)(x-(C(t,x)), where
M : [a,b] x X GL(X,Y) is a family of isomorphisms and 0 £ X is open
and bounded with 0 tf F([a,b] x 80), then the following primitive form of
(1.4) holds:
deg(F(a,«),0,O) = e(Ma(0))e(Mfe(0))deg(F(b,•),0,0). (1.9)
I
Moreover, if f : X—Y is C , f(*n) = 0 and f'(x ) is an isomorphism,
and if f : X —» Y is represented by f(x) = M(x)(x-C(x)), where
M : X GL(Y,X) is a family of isomorphisms with f(xQ) = M(xQ), then
the following primitive form of (1.5) holds:
ind(f,xn) = e(M(0)). (1.10)
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