On the homology of algebras of Whitney functions over subanalytic sets


In this article we study several homology theories of the algebra $\mathcal{E}^\infty (X)$ of Whitney functions over a subanalytic set $X\subset\mathbb{R}^n$ with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for $\mathcal{E}^\infty (X)$, when $X$ is a regular subset of $\mathbb{R}^n$ having regularly situated diagonals. This includes the case of subanalytic $X$. We also compute the Hochschild cohomology of $\mathcal{E}^\infty (X)$ for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories. It is shown that the periodic cyclic homology coincides with the de Rham cohomology, thus generalizing a result of Feigin-Tsygan. Motivated by the algebraic de Rham theory of Grothendieck we finally prove that for subanalytic sets the de Rham cohomology of $\mathcal{E}^\infty (X)$ coincides with the singular cohomology. For the proof of this result we introduce the notion of a bimeromorphic subanalytic triangulation and show that every bounded subanalytic set admits such a triangulation.


Jean-Paul Brasselet

Institut de Mathématiques de Luminy
13288 Marseille

Markus J. Pflaum

Department of Mathematics
University of Colorado
Boulder, CO 80309
United States