A Lefschetz fixed-point formula for certain orbifold ${\mathrm{C}}^{\ast }$-algebras

Abstract

Using Poincaré duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of crossed product ${\mathrm{C}}^{\ast }$-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.