A Lefschetz fixed-point formula for certain orbifold C*-algebras

TL;DR

This paper establishes a Lefschetz fixed-point formula for certain orbifold C*-algebras using K-theoretic Poincaré duality, linking fixed points and representation data to the trace of endomorphisms.

Contribution

It introduces a new Lefschetz fixed-point formula for endomorphisms of crossed product C*-algebras derived from manifolds and group actions, utilizing noncommutative Poincaré duality.

Findings

Formulated a Lefschetz fixed-point formula for orbifold C*-algebras.

Connected the Lefschetz number to fixed orbits and isotropy subgroup data.

Applied noncommutative Poincaré duality and Lefschetz lemma techniques.

Abstract

Using Poincar\'e duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of cross product C*-algebras $C_0(X)\cross G$ coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and $X\cross G$ cocompact and proper. The formula in question expresses the graded trace of the map on rationalized K-theory of $C_0(X)\cross G$ induced by the endomorphism, \emph{i.e.} the Lefschetz number, in terms of fixed orbits and representation-theoretic data connected with certain isotropy subgroups of the isotropy group at that point. The technique is to use noncommutative Poinca\'e duality and the formal Lefschetz lemma of the second author.