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 -genus of the manifold.
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Dapeng Zhang, Projective Dirac operators, twisted K-theory, and local index formula. J. Noncommut. Geom. 8 (2014), no. 1, pp. 179–215DOI 10.4171/JNCG/153