We study index theory for manifolds with Baas–Sullivan singularities using geometric -homology with coefficients in a unital -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 -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.
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Robin J. Deeley, Index theory for manifolds with Baas–Sullivan singularities. J. Noncommut. Geom. 12 (2018), no. 1, pp. 1–28DOI 10.4171/JNCG/269