6

PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ

An orientation distinguishes the components of each Calkin equivalence

class. We always insist that an orientation of GL(X) assign 1 to the

identity.

Once an orientation e is chosen, the degree of f on 0 is defined

by

deg (f,0,O) = e(M(0)) degL

g

(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