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A geometric description of differential cohomology. (English) Zbl 1200.55007
This paper gives a geometric description of differential integral homology. A differential extension of ordinary integral cohomology \(H^*\) is a functor \(X \mapsto \widehat{H}^*(X)\) from the category of smooth manifolds to the category of \({\mathbb Z}\)-graded groups together with natural transformations
\[ \begin{aligned} R : \widehat{H}^*(X) &\rightarrow \Omega_{cl}^*(X),\\ I : \widehat{H}^*(X) &\rightarrow H^*(X),\\ a:\Omega^{*-1}(X)/im(d) &\rightarrow \widehat{H}^*(X) \end{aligned} \]
where \(\Omega\) denotes differential forms.
The authors show how the theory \(\widehat{H}^*(X)\) can be constructed by using stratifolds (essentially stratified manifolds) as “cycles” and cobordism to give a geometric description of the theory. That this can be done depends on the fact that Stokes’ theorem applies in this situation.

MSC:
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
57R19 Algebraic topology on manifolds and differential topology
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