Stratified Interpretation for De Rham Cohomology and Non-Witt Spaces

TL;DR

This paper develops a comprehensive theory connecting sheaf theory, stratified spaces, and de Rham cohomology, providing new tools for analyzing singular spaces and advancing the understanding of their topological and geometric properties.

Contribution

It introduces stratified sheaf-correspondence filtered spaces and stratified de Rham complexes, establishing their cohomology and duality theories, and applies these to non-Witt spaces.

Findings

Proves finiteness of stratified de Rham cohomology.

Establishes isomorphism to intersection cohomology.

Develops stratified Poincaré duality and Künneth theorem.

Abstract

In this paper, we mainly build up the theory of sheaf-correspondence filtered spaces and stratified de Rham complexes for studying singular spaces. We prove the finiteness of a stratified de Rham cohomology and obtain its isomorphism to intersection cohomology through establishing a proper duality theory. Additionally, we present the stratified Poincar\'e duality, the K\"unneth decomposition theorem and develop stratified structure theory to non-Witt spaces as an application of a theory of stratified mezzoperversities. Our results connect differential forms, sheaf theory and intersection homology and pave the way for new approaches to study singular geometries, as well as topological invariants on them. Extensions to conical singularities and fibration of complex curves provide examples of the power of this method. This development will be foundational to new tools in stratified calculus and a strengthening of Hodge theory, advancing research in the Cheeger-Goresky-MacPherson conjecture.