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).
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