Intersection Space Cohomology of Three-Strata Pseudomanifolds

TL;DR

This paper generalizes the intersection space cohomology theory to three-strata pseudomanifolds with flat link bundles, establishing Poincaré duality and broadening the applicability of the theory using differential forms on manifolds with corners.

Contribution

It extends intersection space cohomology to 3-strata pseudomanifolds with flat link bundles, proving Poincaré duality in this new setting.

Findings

Established Poincaré duality for the new class of spaces.

Generalized differential form techniques to manifolds with corners.

Provided examples demonstrating the theory's application.

Abstract

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincar\'e duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincar\'e duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory.