Geometric cycles, index theory and twisted K-homology

  • Bai-Ling Wang

    Mathematical Sciences Institute, Canberra, Australia

Abstract

We study twisted -manifolds over a paracompact Hausdorff space with a twisting . We introduce the topological index and the analytical index on the bordism group of -twisted -manifolds over , taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this article is to establish the equality between the topological index and the analytical index for closed smooth manifolds. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of analogous to Baum–Douglas's geometric cycles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold with a twisting , which generalizes the Connes–Skandalis index theorem for foliations and the Atiyah–Singer families index theorem to twisted cases.

Cite this article

Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology. J. Noncommut. Geom. 2 (2008), no. 4, pp. 497–552

DOI 10.4171/JNCG/27