Parity and generalized multiplicity

Abstract:

Assuming that $X$ and $Y$ are Banach spaces and $\alpha :[a,b] \to \mathcal {L}(X,Y)$ is a path of linear Fredholm operators with invertible endpoints, in $[{\text {F}} - \text {P}1]$ we defined a homotopy invariant of $\alpha ,\sigma (\alpha ,I) \in {{\mathbf {Z}}_2}$, 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 $\bmod 2$ 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 \[ \sigma (\alpha ,{\lambda _0}) \equiv \lim \limits _{\varepsilon \to 0} \sigma (\alpha ,[{\lambda _0} - \varepsilon ,{\lambda _0} + \varepsilon ])\] 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.