On the homology of algebras of Whitney functions over subanalytic sets

Abstract

In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ 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 E ∞ (X), when X is a regular subset of R n having regularly situated diagonals. This includes the case of subanalytic X. We also compute the Hochschild cohomology of E ∞ (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 E ∞ (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.

Cyclic cocycles on deformation quantizations and higher index theorems

“…For k = R and O the sheaf of smooth functions on M the claim has been proven in [5]. For k = RJhK the claim follows by a spectral sequence argument.…”

De Rham theorem for Whitney functions

“…Then, using sheaf theoretical arguments we can easily prove that, given a bounded complex of constructible sheaves F , the Whitney-de Rham complex associated to F is quasi-isomorphic to F . As a corollary we obtain a theorem of [4]. (Another proof was given in [5] using deep results on D-modules.…”

A Poincaré lemma for Whitney–de Rham complex

“…Let M be a real analytic manifold, by Poincaré Lemma, it is well-known that the de Rham complex over M is isomorphic to C M . The aim of this paper is to show that a theorem of [BP08] follows easily from a deep result of Kashiwara on regular holonomic D-module [K84] and the Whitney functor of [KS96]. More precisely, we show that Main theorem(=Theorem 3.3.…”