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
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