PARITY AND GENERALIZED MULTIPLICITY

Assuming that X and Y are Banach spaces and alpha:[a, b] --> L(X, Y) is a path of linear Fredholm operators with invertible endpoints, in [F-P1] we defined a homotopy invariant of alpha, sigma-(alpha, I) is-an-element-of Z2, the parity of alpha on I. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a mod 2 count of the number of transversal intersections of alpha-(I) with the set of singular operators, (b) that if lambda-0 is an isolated singular point of alpha, then the local parity [GRAPHICS] remains invariant under Lyapunov-Schmidt reduction, and (c) that sigma-(alpha, lambda-0) = (-1)M(G)(lambda-0), where M(G)(lambda-0) is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.