Axiomatic Description of Lefschetz Type Equivariant Homotopy Invariants

TL;DR

This paper develops an axiomatic framework for equivariant Lefschetz numbers, identifying a unique invariant that detects fixed orbits in equivariant homotopy theory, with applications to specific fixed orbit indices.

Contribution

It introduces a normalization axiom for equivariant Lefschetz numbers and characterizes the fixed orbit detecting invariant uniquely.

Findings

Established generators of universal Lefschetz groups for equivariant 1-spheres

Formulated a normalization axiom to determine equivariant Lefschetz numbers

Identified the fixed orbit detecting Lefschetz number with the fixed orbit index

Abstract

We describe generators of universal Lefschetz groups consisting of self-maps of equivariant 1-spheres. This allows to formulate a normalization axiom which, together with the usual axioms, determines an equivariant Lefschetz number uniquely. We apply this technique to identify the fixed orbit detecting equivariant Lefschetz number of Chorny with the fixed orbit index of Dzedzej.