Hodge Theory for Intersection Space Cohomology

TL;DR

This paper explores the cohomology of intersection spaces using Hodge theory, providing a new description for certain stratified spaces and discussing implications for the signature, distinct from traditional intersection homology.

Contribution

It offers a novel Hodge-theoretic description of intersection space cohomology for two-strata pseudomanifolds with product link bundles, connecting it to weighted harmonic forms.

Findings

Cohomology described via weighted $L^2$ harmonic forms with fibred scattering metric

Provides insights into the signature of intersection spaces

Clarifies the relationship between intersection space homology and traditional intersection homology

Abstract

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincar\'e duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well.