zbMATH — the first resource for mathematics

Homological properties of stratified spaces. (English) Zbl 0792.57009
This paper is about a de Rham theorem in intersection cohomology: given a stratified space \(A\), the author exhibits a complex of differential forms on \(A - \Sigma\) (nonsingular part of \(A\)) whose cohomology is the intersection cohomology of \(A\). The approach followed here has two special features:
1. A more general notion of perversity (introduced by R. MacPherson in “Intersection cohomology and perverse sheaves”, Colloquium Lectures, Annual Meeting of the Am. Math. Soc., San Francisco, June 1991. In this context, the axiomatic presentation of the intersection cohomology does not hold. So, the author gives a direct proof of the de Rham theorem.
2. A more general notion of intersection differential form. The original one uses a neighborhood of \(\Sigma\) in order to describe the allowability condition of a differential form \(\omega\) on \(A - \Sigma\). The author replaces this germ condition in such a way that the allowability of \(\omega\) is seen directly on \(\Sigma\) (by means of a blow up \(\pi: \widetilde{A} \to A\)).
The work ends by showing how to express the Poincaré duality in terms of the wedge product.

57N80 Stratifications in topological manifolds
57R19 Algebraic topology on manifolds and differential topology
55N33 Intersection homology and cohomology in algebraic topology