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Differential K-Theory: A Survey

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Global Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 17))

Abstract

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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Acknowledgements

Partially funded by the Courant Research Center “Higher order structures in Mathematics” within the German initiative of excellence.

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Authors and Affiliations

  1. Johannes-Kepler-Universität Regensburg, 93040, Regensburg, Germany

    Ulrich Bunke

  2. Georg-August-Universität Göttingen, 37073, Göttingen, Germany

    Thomas Schick

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  1. Ulrich Bunke
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Correspondence to Ulrich Bunke .

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  1. Inst. Mathematik, Universität Potsdam, Potsdam, Brandenburg, Germany

    Christian Bär

  2. , Mathematisches Institut, Universität Münster, Einsteinstraße 62, Münster, 48149, Germany

    Joachim Lohkamp

  3. , Fakultät für Mathematik und Informatik, Universität Leipzig, Johannisgasse 26, Leipzig, 04103, Sachsen, Germany

    Matthias Schwarz

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Bunke, U., Schick, T. (2012). Differential K-Theory: A Survey. In: Bär, C., Lohkamp, J., Schwarz, M. (eds) Global Differential Geometry. Springer Proceedings in Mathematics, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22842-1_11

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