Geometric cycles, index theory and twisted K-homology
By Bai-Ling Wang. 2007
51 citations recorded by the source corpus.
We study twisted Spin c -manifolds over a paracompact Hausdorff space X with a twisting � : X ! K(Z, 3). We introduce the topological index and the analytical index on the bordism group of �-twisted Spin c -manifolds over (X,�), taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topological index and the analyt- ical index for smooth manifolds. We also define a notion of geo metric twisted K-homology, whose cycles are geometric cycles of (X,�) analogous to Baum-Douglas's geometric cy- cles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold (X,F ) with a twisting � : X ! K(Z, 3), which generalizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer fami- lies index theorem to twisted cases.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/28bcf81eabadc990078b9cb010e59a355accc0d8
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- Twists of K-theory and TMF (2010, background): “The data of a configuration like (12.3) together with the homotopy c was studied by Wang [Wan08], who calls it a twisted String structure.”
- Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map (2011, methodology): “rphism R∗+dX ∼=R∗(X−Tf ), we obtain an Umkehr map R∗+d(X) −→R∗(B). Recently it has become important in a number of contexts to consider twisted generalizations of these constructions (see for example [28, 51, 18]). In our context, we explain how to provide twisted Umkehr maps for any sufficiently mult”
- Twisted Differential String and Fivebrane Structures (2009, background): “e, now by analogy with (2.8), we say that a c 2(E)- twisted String structure is a choice of homotopy H3 filling the diagram (2.11). This notion of twisted String structures was originally suggested in [61]. For it to be useful, we need to say what homotopies of twisted String structures are, homotopi”