Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

TL;DR

This paper extends the classical Lefschetz number to equivariant K-homology for spaces with group actions, providing formulas for simplicial complexes and smooth manifolds, and relating to the equivariant Lefschetz Fixed Point Theorem.

Contribution

It introduces equivariant Lefschetz maps in Kasparov theory, enabling computation of invariants for self-maps of complexes and manifolds, unifying different structures.

Findings

Computed Lefschetz invariants for simplicial complexes and smooth manifolds.

Established independence of invariants from the chosen geometric structure.

Derived formulas linking triangulations and smooth structures for the same invariant.

Abstract

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers to equivariant K-homology classes. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and of self-maps of smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in these cases. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Luck and Rosenberg.