On the de Rham theorem in the globally subanalytic setting

TL;DR

This paper establishes that in the globally subanalytic setting, constructible de Rham cohomology groups are isomorphic to classical cohomology groups, extending the de Rham theorem to this context and even in the $C^1$-setting.

Contribution

It introduces constructible de Rham complexes for globally subanalytic manifolds and proves the de Rham theorem holds for these in full generality.

Findings

Constructible de Rham cohomology is isomorphic to classical cohomology.

The de Rham theorem holds for constructible forms in the globally subanalytic setting.

Results apply in the $C^1$-setting, broadening applicability.

Abstract

For globally subanalytic manifolds we define de Rham complexes of globally subanalytic differential forms and of constructible differential forms. Whereas the de Rham theorem does not hold for the former in the non-compact case, it does hold for the latter in full generality. We deduce that the constructible de Rham cohomology groups are canonically isomorphic to the classical ones. We stress that our results apply already in the $C^1$-setting.