Equivariant Lefschetz and Fuller indices via topological intersection theory

TL;DR

This paper develops a homological approach to define equivariant intersection products, Lefschetz numbers, and Fuller indices for compact Lie group actions, providing new tools for fixed point theory in equivariant topology.

Contribution

It introduces a homological construction of equivariant Lefschetz and Fuller indices using G-equivariant Poincaré duality, offering a new perspective and simplified proofs in equivariant fixed point theory.

Findings

Defined an equivariant intersection product via G-equivariant Poincaré duality.

Provided a homological construction of the equivariant Lefschetz number.

Constructed an equivariant Fuller index with values in the rationalized Burnside ring.

Abstract

For a compact Lie group G, we use G-equivariant Poincar\'e duality for ordinary RO(G)-graded homology to define an equivariant intersection product, the dual of the equivariant cup product. Using this, we give a homological construction of the equivariant Lefschetz number and a simple proof of the equivariant Lefschetz fixed point theorem. With similar techniques, an equivariant Fuller index with values in the rationalized Burnside ring is constructed.