Index theory for manifolds with Baas–Sullivan singularities

Abstract

We study index theory for manifolds with Baas–Sullivan singularities using geometric $K$-homology with coefficients in a unital $C^{\ast }$-algebra. In particular, we define a natural analog of the Baum–Connes assembly map for a torsion-free discrete group in the context of these singular spaces. The cases of singularities modelled on $k$-points (i.e., $\mathbb Z/k\mathbb Z-manifolds) and the circle are discussed in detail. In the case of the former, the associated index theorem is related to the Freed–Melrose index theorem; in the case of the latter, the index theorem is related to work of Rosenberg.