On the Uniqueness of the $G$-Equivariant Spectral Flow

TL;DR

This paper extends the concept of spectral flow to a $G$-equivariant setting, proving its unique characterization under elementary properties and linking it to a $G$-equivariant Maslov index.

Contribution

It introduces a $G$-equivariant spectral flow and proves its uniqueness, also providing an alternative definition via a $G$-equivariant Maslov index.

Findings

$G$-equivariant spectral flow is uniquely characterized by elementary properties.

Established a connection between $G$-spectral flow and $G$-Maslov index.

Provided an alternative definition of $G$-spectral flow.

Abstract

The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators. Lesch as well as Pejsachowicz, Fitzpatrick and Ciriza independently showed that it is uniquely characterised by its elementary properties. The authors recently introduced a $G$-equivariant spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the action of a compact Lie group $G$. The purpose of this paper is to show that the $G$-equivariant spectral flow is uniquely characterised by the same elementary properties when appropriately restated. As an application, we introduce an alternative definition of the $G$-equivariant spectral flow via a $G$-equivariant Maslov index.