An Index Theorem in Relative K-Theory for First-Order Systems

TL;DR

This paper extends index theorems in relative K-theory to families of first-order systems on the real line, removing asymptotic hyperbolicity assumptions and considering multi-parameter spaces, leading to a simplified index formula.

Contribution

It generalizes previous index theorems by relaxing asymptotic conditions and incorporating multi-parameter families, resulting in a more versatile index formula in relative K-theory.

Findings

Developed a new index theorem for non-asymptotically hyperbolic systems.

Extended the index formula to families over general compact parameter spaces.

Simplified the index calculation in relative K-theory for first-order systems.

Abstract

Motivated by bifurcation of branches of homoclinic orbits of dynamical systems, we consider families of first-order equations on the real line and introduce a generalisation of previous index theorems by Pejsachowicz, and by Hu and Portaluri. The main novelties of our approach firstly concern the analytical setting, where we lift the common assumption that the equations are asymptotically hyperbolic. Secondly, we consider general compact parameter spaces instead of a single parameter, which results in a remarkably simple index formula in relative $K$-theory.