16

PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ

kernel bundle. In Theorem 6.43, we provide a generalization to paths of

linear Fredholm maps of the fact that for a path of compact vector fields

a(A) = I-AK with *(*Q) singular, if TJ is sufficiently small,

m

—1

r(a, [A -7), A0+TJ]) = (-1) , where m is the algebraic multiplicity of A

as an eigenvalue of K. We close Section 6 with a brief description of the

parity of a closed curve a: S — $ (X,Y). The parity of a closed curve,

1 1 1

r(a,S ), has the interesting property that r(oc,S ) = r(a+k,S ) for any

continuous path k: S — L(X,Y) of compact operators. For families of

differential equations parametrized by S , this means that the parity is a

homotopy invariant which only depends on the highest order derivatives. We

show that the parity gives an explicit isomorphism of the Poincare group of

*Q(X,Y) with Z2 = {+1, -1}.

In Section 7, we use the parity to prove the Homotopy Dependence

Formula, (1.4), and also prove the Regular Value Formula, (1.5).

Suppose that N : X —-» X is compact, N(0) = 0 and N is Frechet

differentiable at 0. Then (An,0) is a global bifurcation point of the

nontrivial (i.e., u * 0) solutions of the equation u - A N(u) =0 if A

is an eigenvalue of odd algebraic multiplicity of N'(0). The local result

is due to Krasnosel'skii [Kr]; the global result is due to Rabinowitz [Ra].

Theorem 8.6 is a generalization of this result: Let F : I R x X —• Y be a

parametrized family of quasilinear Fredholm mappings, such that F(A,0) = 0,

and D F(A,0) exists and depends continuously on A€lR. If ab, D F(A,a)

and D F(A,b) are isomorphisms and rp F(A,0),[a,b] = -1, then there is

global bifurcation of the nontrivial solutions of F(A,x) = 0 from

[a,b] x {0}. In particular, if A € I R is an isolated singular point of