Foliated Stratified Spaces and a De Rham Complex Describing Intersection Space Cohomology

TL;DR

This paper introduces a de Rham complex for intersection space cohomology in stratified pseudomanifolds, establishing a differential form-based approach that complements existing intersection cohomology theories and relates to mirror symmetry.

Contribution

It provides a de Rham-theoretic description of intersection space cohomology for certain stratified spaces, including a wedge product structure for all perversities.

Findings

De Rham complex models intersection space cohomology for depth 1 stratifications.

Wedge product induces a cup product on intersection space cohomology.

Applicable to foliated stratified spaces and Borel-Serre compactifications.

Abstract

The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincar\'e duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with stratification depth 1 and flat link bundles, the present paper provides a de Rham-theoretic description of the theory HI by a complex of global smooth differential forms on the top stratum. We prove that the wedge product of forms introduces a perversity-internal cup product on HI, for every perversity. Flat link bundles arise for example in foliated stratified spaces and in reductive Borel-Serre compactifications of locally symmetric spaces. A precise topological definition of the notion of a stratified foliation is given.