On the cohomology of $L^2$-harmonic forms of an incomplete Riemannian manifold

TL;DR

This paper extends the concept of harmonic form cohomology from compact manifolds with boundary to Hilbert complexes, with applications to incomplete Riemannian manifolds and stratified spaces.

Contribution

It introduces a new framework for cohomology of harmonic forms in Hilbert complexes, generalizing previous notions to incomplete manifolds and stratified spaces.

Findings

Extended cohomology notions to Hilbert complexes.

Applied the framework to incomplete Riemannian manifolds.

Provided geometric insights into stratified Thom-Mather spaces.

Abstract

Motivated by the work of Cappell, Deturck, Gluch and Miller, we extend the notion of cohomology of harmonic forms (of a compact manifold with boundary) to the abstract setting of Hilbert complexes. Then, we present some geometric applications of our construction to incomplete Riemannian manifolds with particular interest to the case of smoothly stratified Thom-Mather spaces.