A geometric description of differential cohomology

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.