ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 7

most significant feature of the degree theory for quasilinear Fredholm maps

developed here is that the sign-switching in degree during a quasilinear

Fredholm homotopy can be completely described in a relatively elementary

way by means of an invariant associated with the homotopy class of an

admissible path of linear Fredholm operators. Due to its relevance in

bifurcation theory for nonlinear Fredholm maps, this invariant, which we

have called the parity, is of independent interest and was considered also

in [Fi-Pe; 2, 3, 4].

Given an admissible path a: [a,b] — $ (X,Y), let 0: [a,b] —-

GL(Y,X) be a parametrix for a. The parity of the path a on [a,b] is

defined by

r(cc, [a,b]) = degL

g

(|3(a)a(a)) degL

g

(0(b)a(b)), (1.3)

where the right-hand side is the product of the Leray-Schauder degrees of

two linear compact vector fields. Formula (1.3) does not depend on the

choice of parametrix.

From the geometric viewpoint, the parity of an admissible path can be

interpreted as a mod 2 intersection number of the path with the one

codimensional "analytic" subset S of all noninvertible Fredholm operators

from X to Y. More precisely, generically an admissible path

a: [a,b] — $n(X,Y) has only a finite number of singular points at each of

which x(A) has a one-dimensional kernel, and r(a, [a,b]) is the mod 2

count of the number of such singular points ([Fi-Pe,3]). From a homotopy

viewpoint, we will show that the parity of an admissible path is one if and

only if the path can be deformed (relative to the boundary) to a path in

GL(X,Y).