Abstract
The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification ∂M ≅ βM × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S 1, we show how to attach to such a space a noncommutative C *-algebra that captures the extra structure. We then use this C *-algebra to give a new proof of the Freed–Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S 1 singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.
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Rosenberg, J. Groupoid C *-Algebras and Index Theory on Manifolds with Singularities. Geometriae Dedicata 100, 65–84 (2003). https://doi.org/10.1023/A:1025802811202
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DOI: https://doi.org/10.1023/A:1025802811202