A geometric description of differential cohomology

Bunke, Ulrich; Kreck, Matthias; Schick, Thomas

Annales mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 1-16.

Nous donnons une définition géométrique de la cohomologie intégrale différentielle. Nous utilisons des cycles de cobordisme avec singularités, et des formes différentielles distributionnelles. Avec cette description, la construction de la multiplication et de l’intégration avec toutes les proprietés désirées est particulièrement simple.

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 $S{H}^{*}\left(X\right)$. In this description smooth manifolds in Quillen’s description are replaced by so-called stratifolds, which are certain stratified spaces. The cohomology theory $S{H}^{*}\left(X\right)$ is naturally isomorphic to ordinary integral cohomology $H^{*}\left(X\right)$, thus we obtain a cobordism type definition of the differential extension of ordinary integral cohomology.

Classification : 55N20, 57R19
Keywords: differential cohomology, smooth cohomology, geometric cycles, cobordism
Mot clés : cohomologie différentielle, cycles géométriques, cobordisme

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