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Published 2014 | Submitted
Journal Article Open

Projective Dirac operators, twisted K-theory, and local index formula

Zhang, Dapeng

Abstract

We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern character for K-theories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincaré dual of the A-genus of the manifold.

Additional Information

© 2014 European Mathematical Society. Received September 22, 2010; revised February 17, 2012. This article is the author's dissertation in publication form. Supported in part by International Max-Planck Research School (IMPRS). I am very grateful to Bai-LingWang. He foresaw the possibility that the projective spin Dirac operator defined by [20] in formal sense can be realized by a certain spectral triple, and introduced his interesting research project to me in 2008. The spectral triple in his mind turned out to be the projective spectral triple constructed in this paper. Without his insight, I would not have been writing this thesis. I would like to thank Adam Rennie for his very helpful remarks about Morita equivalence between various presentations of Kasparov's fundamental class on reading a draft of this paper. I also wish to thank my advisor, Matilde Marcolli, for her many years of encouragement, support, and many helpful suggestions on both this research and other aspects.

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