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 with singularities from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$

Part 2, joint with Juan Pablo Rossetti, will appear. V1 contained in addition to the results in Part 1 sections that examined information contained in the individual $p$-spectra. After realizing that Rossetti had overlapping results, we split the original paper into the current Part 1 together with a strengthened Part 2, joint with Rossetti, that combines our work on the individual $p$-spectra