Hodge cohomology of some foliated boundary and foliated cusp metrics

TL;DR

This paper generalizes the understanding of $L^2$ harmonic forms on fibred boundary and cusp metrics to cases involving Riemannian foliations with compact leaves, connecting foliation geometry with stratified space cohomology.

Contribution

It extends previous results to Riemannian foliations with compact leaves, linking harmonic forms to intersection cohomology without requiring resolutions for Witt spaces.

Findings

Generalization of $L^2$ harmonic form characterization to foliated boundary metrics.

Identification of conditions where no resolution is needed for Witt spaces.

Connection between foliation structures and stratified space cohomology.

Abstract

For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of $L^2$ harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of the foliation) is a Witt space and if the metric considered is a foliated cusp metric, then no such resolution is required.