Hodge theory on Cheeger spaces

TL;DR

This paper generalizes Hodge theory to stratified pseudomanifolds with ideal boundary conditions, establishing Fredholm properties, cohomology isomorphisms, and Poincare Duality on Cheeger spaces.

Contribution

It introduces mezzoperversities and ideal boundary operators for arbitrary stratified spaces, extending Hodge theory beyond isolated conic singularities.

Findings

De Rham operator with boundary conditions is Fredholm with compact resolvent.

Hodge and L2 de Rham cohomology groups are isomorphic and metric-independent.

Poincare Duality holds on Cheeger spaces with self-dual boundary conditions.

Abstract

We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L2 de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call `Cheeger spaces', we show that these Hodge/de Rham cohomology groups satisfy Poincare Duality.