Do the Hodge spectra distinguish orbifolds from manifolds? I (Q6590157)

Inverse Spectral Geometry studies in what extent the spectral information determines the geometry and topology of a space. The most usual setting concerning spectral information is the spectrum of the Laplace-Beltrami operator (acting on smooth functions) associated to a (closed) Riemannian manifold. However, almost everything extend smoothly for Riemannian orbifolds and one of the most fundamental open question is whether the spectrum of the Laplace-Beltrami operator detect singularities on a (closed) Riemannian orbifold: \N\NCan a closed Riemannian orbifold with non-trivial singularities be isospectral to a closed Riemannian manifold? \N\NThe article under review considers this question by using also the spectrum of the Hodge-Laplace operators acting on smooth \(p\)-forms (the case \(p=0\) coincides with the Laplace-Beltrami operator), called \textit{\(p\)-spectrum}.\N\NMain Theorem: The \(0\)-spectrum and \(1\)-spectrum together distinguish closed Riemannian orbifolds with singular sets of codimension \(\leq 3\) from closed Riemannian manifolds.\N\NThat is, any Riemannian orbifold \(\mathcal O\) with singular sets of codimension \(\leq 3\) cannot share the \(0\)-spectrum and \(1\)-spectrum simultaneously with a closed Riemannian manifold.\N\NCorollary: The \(0\)-spectrum and \(1\)-spectrum together distinguish closed Riemannian orbifolds of dimension \(\leq 3\) from closed Riemannian manifolds. \N\NThe strategy used to distinguish the spectra of orbifolds and manifolds is comparing the spectral invariants coming from an asymptotic expansion of the heat trace called \textit{heat invariants}. Following the calculation of the heat invariants of the \(0\)-spectrum done in [\textit{E. B. Dryden} et al., Michigan Math. J. 56, 205--238 (2008; Zbl 1175.58010)], the authors compute the \(p\)-th heat invariants for every \(p\) (Theorem~3.15). This result is interesting in itself. In fact, there is a second part available in [\textit{K. Gittins} et al., ``Do the Hodge spectra distinguish orbifolds from manifolds? Part 2'', Preprint, \url{arXiv:2311.00337}] obtaining further consequences.