Refined intersection homology on non-Witt spaces

TL;DR

This paper extends intersection homology theory to non-Witt spaces, unifying sheaf-theoretic and analytic approaches and broadening the applicability of intersection cohomology.

Contribution

It generalizes the intersection homology framework to non-Witt spaces by describing all compatible sheaf complexes, linking sheaf theory with analytic de Rham cohomology.

Findings

Unified sheaf-theoretic and analytic intersection cohomology on non-Witt spaces.

Extended the class of spaces where intersection homology applies.

Established equivalence with analytic de Rham theory on Thom-Mather spaces.

Abstract

We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of these cohomologies by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. On Thom-Mather stratified spaces this refined intersection cohomology theory coincides with the analytic de Rham theory.