Communications in Analysis and Geometry

Volume 27 (2019)

Number 7

Cheeger–Simons differential characters with compact support and Pontryagin duality

Pages: 1473 – 1522

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n7.a2

Authors

Christian Becker (Institut für Mathematik, Universität Potsdam, Germany; and Institut für Mathematik und Informatik, Universität Greifswald, Germany)

Marco Benini (Institut für Mathematik, Universität Potsdam, Germany; and Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland, United Kingdom)

Alexander Schenkel (Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland, United Kingdom; Maxwell Institute for Mathematical Sciences and the Higgs Centre for Theoretical Physics, Edinburgh; and University of Nottingham, United Kingdom)

Richard J. Szabo (Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland, United Kingdom; Maxwell Institute for Mathematical Sciences, Edinburgh; Higgs Centre for Theoretical Physics, Edinburgh)

Abstract

By adapting the Cheeger–Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. We prove an excision theorem for differential cohomology using a suitable relative version. Furthermore, we use our model to give an independent proof of Pontryagin duality for differential cohomology recovering a result of [Harvey, Lawson, Zweck. Amer. J. Math. 125 (2003), 791]: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology.

Full Text (PDF format)

Received 19 May 2016

Accepted 30 July 2017

Published 30 December 2019