Do the Hodge spectra distinguish orbifolds from manifolds? Part 1

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 from manifolds as long as the singular sets have codimension ≤ 3. This is enough to distinguish orbifolds from manifolds for dimension ≤ 3. We also give both positive and negative inverse spectral results for the individual p-spectra. For example, we give conditions on the codimension of the singular set which guarantee that the volume of the singular set is determined, and in many cases we show by providing counterexamples that the conditions are sharp.