Differential K-theory. A survey

Details Differential K-theory. A survey Differential K-theory. A survey

2010-11-30

arxiv.org/abs/1011.6663v2

Global differential geometry, Springer, Heidelberg (2012), 303-358

Mathematics

K-Theory and Homology

50 pages, report based in particular on work done sponsored the DFG SSP "Globale Differentialgeometrie". v2: final version (on

Scientific paper

10.1007/978-3-642-22842-1

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.

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