Equivariant Lefschetz and Fuller indices via topological intersection theory

Abstract

For a compact Lie group G, we use G-equivariant Poincaré 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. We relate this invariant to existing notions of equivariant Lefschetz numbers and give an explicit computational formula. Using similar techniques, we also construct an equivariant Fuller index with values in the rationalized Burnside ring.