# Journal of Noncommutative Geometry

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**Volume 12, Issue 1, 2018, pp. 1–28**

**DOI: 10.4171/JNCG/269**

Published online: 2018-03-23

Index theory for manifolds with Baas–Sullivan singularities

Robin J. Deeley^{[1]}(1) University of Colorado, Boulder, USA

We study index theory for manifolds with Baas–Sullivan singularities using geometric $K$-homology with coefficients in a unital $C^*$-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.

*Keywords: *$K$-homology, manifolds with Baas–Sullivan singularities, the Freed–Melrose index theorem

Deeley Robin: Index theory for manifolds with Baas–Sullivan singularities. *J. Noncommut. Geom.* 12 (2018), 1-28. doi: 10.4171/JNCG/269