## 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

### Keywords:

stratifold; Stokes’ theorem; integration along the fiber
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### References:

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