De Rham Theorem for L^\infty forms and homology on singular spaces

TL;DR

This paper extends De Rham's theorem to singular semialgebraic sets by introducing smooth L^ifferential forms and establishing an isomorphism between singular cohomology and cohomology of these forms.

Contribution

It defines smooth L^ifferential forms on singular spaces and proves a De Rham type theorem linking singular cohomology with these forms.

Findings

Established a natural isomorphism between singular cohomology and L^ifferential form cohomology.

Defined a new class of differential forms suitable for singular semialgebraic sets.

Connected geometric homology with algebraic cohomology via the De Rham theorem.

Abstract

We introduce smooth L^\infty differential forms on a singular (semialgebraic) set X in R^n. Roughly speaking, a smooth L^\infty differential form is a certain class of equivalence of 'stratified forms', that is, a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential components on the adjacent strata and bounded size (in the metric induced from R^n). We identify the singular homology of X as the homology of the chain complex generated by semialgebraic singular simplices, i.e. continuous semialgebraic maps from the standard simplices into X. Singular cohomology of X is defined as the homology of the Hom dual to the chain complex of the singular chains. Finally, we prove a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth L^\infty forms.