Leonid Shartser, Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4; e-mail: shartl@math.toronto.edu

Guillaume Valette, Instytut Matematyczny PAN, 31-027 Krakow, Poland; email: gvalette@impan.pl

Abstract/Résumé:

We introduce a notion of a smooth \(L^{\infty}\) form on singular (semialgebraic) spaces \(X\) in \(\mathbb{R}^n\). An \(L^\infty\) form is the data of a stratification \(\Sigma\) of \(X\) and a collection of smooth forms \(\omega\) on the nonsingular strata with matching tangential components on the adjacent strata and bounded size (in the metric induced from \(\mathbb{R}^n\)). We prove Stokes’ Theorem and Poincaré’s Lemma for \(L^\infty\) forms. As a result we obtain a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth \(L^{\infty}\) forms.

On introduit la notion d’une forme \(L^\infty\) pour des espaces singuliers semialgébriques. Une forme lisse \(L^\infty\) est la donnée d’une stratification et d’une famille de forme lisses sur les strates coincidant le long des strates adjacentes. On prouve la formule de Stokes et le lemme de Poincaré pour les formes \(L^\infty\). On en déduit un théorème de type De Rham établissant un isomorphisme naturel entre la cohomologie des formes \(L^\infty\) et la cohomologie singulière.

Keywords: De Rham theory, definable sets, homology theory, metric invariants

AMS Subject Classification: Semialgebraic sets and related spaces 14P10

PDF(click to download): A De Rham Theorem for $L^infty$ Forms and Homology on Singular Spaces