# A Lefschetz fixed-point formula for certain orbifold $C_{∗}$-algebras

### Siegfried Echterhoff

Unversity of Münster, Germany### Heath Emerson

University of Victoria, Canada### Hyun Jeong Kim

University of Victoria, Canada

## Abstract

Using Poincaré duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of crossed product $C_{∗}$-algebras $C_{0}(X)⋊G$ coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and $X⋊G$ cocompact and proper. The formula in question describes the graded trace of the map induced by the automorphism on K-theory of $C_{0}(X)⋊G$, i.e. the Lefschetz number, in terms of fixed orbits of the spatial map. Each fixed orbit contributes to the Lefschetz number by a formula involving twisted conjugacy classes of the corresponding isotropy group, and a secondary construction that associates, by way of index theory, a group character to any finite group action on a Euclidean space commuting with a given invertible matrix.

## Cite this article

Siegfried Echterhoff, Heath Emerson, Hyun Jeong Kim, A Lefschetz fixed-point formula for certain orbifold $C_{∗}$-algebras. J. Noncommut. Geom. 4 (2010), no. 1, pp. 125–155

DOI 10.4171/JNCG/51