Parity and generalized multiplicity

Assuming that X and Y are Banach spaces and that T is a path of linear Fredholm operators with invertible endpoints, in [F-Pl] we defined a homotopy invariant "the parity of T . The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove that, generically, the parity is a mod 2 count of the number of transversal intersections of T with the set of singular operators, that at an isolated singular point of x of T the local parity remains invariant under Lyapunov-Schmidt reduction, and that it coincides with the mod 2 reduction of any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.