An orientation distinguishes the components of each Calkin equivalence
class. We always insist that an orientation of GL(X) assign 1 to the
Once an orientation e is chosen, the degree of f on 0 is defined
deg (f,0,O) = e(M(0)) degL
(Id-C,0,O), (1.2)
where M and C are as in (1.1). The right-hand side of (1.2) is
independent of the representation (1.1).
For a topological space A, a mapping F : A x X Y is said to be a
quasilinear Fredholm family (parametrized by A) provided that F may be
represented as F(A,x) = L.(x)x + C(A,x), where L is the restriction of a
continuous mapping L : A x X $n(X,Y) and C : A x X Y is compact.
We call L : A x X $n(X,Y) a principal part of F and note that
principal parts are unique modulo families of compact operators. If A Q R
is an interval, a quasilinear Fredholm family parametrized by A is called
a quasilinear Fredholm homotopy. If A = [a,b], a continuous mapping
a : A —-» $n(X,Y) such that x(a) and a(b) are isomorphisms will be
called an admissible path of linear Fredholm operators.
The degree defined by (1.2) has the usual additivity, existence and
Borsuk-Ulam properties. With respect to homotopy dependence of the degree
and the regular value formula, if X = Y and GL(X) is connected, then
any integer-valued degree theory defined on a class of maps which includes
all linear isomorphisms and which coincides with the Leray-Schauder degree
on the class of compact vector fields can neither be homotopy invariant nor
can the classical regular value formula hold. As already mentioned, the
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