New Topological Invariants in Multiparameter Bifurcation Theory

This research project has three aims. Firstly, we will construct a new bifurcation invariant for multiparameter difference equations. This invariant is an element of a relative cohomology group and consequently less often trivial for topological reasons than previous invariants. Secondly, we will substantially broaden the applicability of our previous work by lifting an assumption that often is too strong in applications. To be more precise, we will require instead of asymptotic hyperbolicity the existence of exponential dichotomies, which is a far more common assumption for difference equations. Moreover, we will study topological properties of Nemitskii operators in this setting and investigate new methods for computing their degree. Finally, our third aim is to initiate a long-term project in which we want to generalise Pejsachowicz’ K-theoretic bifurcation invariants for general nonlinear Fredholm maps to relative K-theory. Let us emphasise that this is a pioneering step which will open-up methods from K-theory to bifurcation settings appearing in science.