Bunke, Kreck, Schick: A geometric description of differential cohomology

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Ulrich Bunke; Matthias Kreck; Thomas Schick
A geometric description of differential cohomology
(Une description géométrique de la cohomologie différentielle)
Annales mathématiques Blaise Pascal, 17 no. 1 (2010), p. 1-16, doi: 10.5802/ambp.276
Article PDF | Reviews MR 2674652 | Zbl 1200.55007
Class. Math.: 55N20, 57R19
Keywords: differential cohomology, smooth cohomology, geometric cycles, cobordism

Résumé - Abstract

In this paper we give a geometric cobordism description of differential integral cohomology. The main motivation to consider this model (for other models see [5, 6, 7, 8]) is that it allows for simple descriptions of both the cup product and the integration. In particular it is very easy to verify the compatibilty of these structures. We proceed in a similar way in the case of differential cobordism as constructed in [4]. There the starting point was Quillen’s cobordism description of singular cobordism groups for a differential manifold $X$. Here we use instead the similar description of integral cohomology from [11]. This cohomology theory is denoted by $SH^*(X)$. In this description smooth manifolds in Quillen’s description are replaced by so-called stratifolds, which are certain stratified spaces. The cohomology theory $SH^*(X)$ is naturally isomorphic to ordinary integral cohomology $H^*(X)$, thus we obtain a cobordism type definition of the differential extension of ordinary integral cohomology.

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