2010/02/22 by Leonid Shartser, L. Shartser, Shartser, L. +3
Mathematics · #14P10 #14P25 #55N20 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #math.DG #math.MG #msc:14P10 #msc:14P25 #msc:55N20
paper · pdf · doi:10.48550/arxiv.1002.4143
36 pages
arxiv created 2010/02/22 · openalex publication_date 2010/02/22 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce smooth L^∞ differential forms on a singular (semialgebraic) set X in Rn. Roughly speaking, a smooth L^∞ 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 Rn). 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^∞ forms.