Cheeger-Simons differential characters with compact support and Pontryagin duality

TL;DR

This paper develops a differential cohomology theory with compact support using Cheeger-Simons methods, establishing functoriality, exact sequences, and proving Pontryagin duality for differential cohomology on manifolds.

Contribution

It introduces a new model of differential cohomology with compact support and proves Pontryagin duality, extending classical results to this refined setting.

Findings

Established functoriality with respect to open embeddings

Proved an excision theorem for differential cohomology

Revealed isomorphism between differential cohomology and its Pontryagin dual

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.