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.


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