Do the Hodge spectra distinguish orbifolds from manifolds? Part 1

TL;DR

This paper investigates whether the Hodge spectra can differentiate orbifolds with singularities from smooth manifolds, demonstrating that combined heat invariants of 0- and 1-forms suffice in low dimensions.

Contribution

It shows that heat invariants of the Hodge Laplacian on 0- and 1-forms can distinguish orbifolds from manifolds with certain singularities in low dimensions.

Findings

Heat invariants distinguish orbifolds from manifolds with singularities of codimension ≤ 3.

The method works for all dimensions ≤ 3.

Spectral data on 0- and 1-forms are sufficient for differentiation.

Abstract

We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$