Universal topological statistics on triangulated singular spaces

Abstract

We prove a universality theorem for random persistent homology over a class of triangulable spaces. More precisely, let $M\subset {\mathbb{R}}^D$ be a compact $C^2$-triangulable space satisfying a geometric quality condition and let $f:M\to \mathbb{R}$ be a probability density. Then the expected persistence ratio measure computed from the Čech or Vietoris-Rips complex of a Poisson point process with intensity $nf$ has a universal limit independent of $(M,f)$. Since smooth manifolds, algebraic varieties, semialgebraic sets and Whitney stratified spaces are all triangulable spaces, our theorem applies to a large class of non-Euclidean spaces. Beyond persistent homology, our proof covers a general class of scale-invariant functionals. It relies on a geometric transfer method that adapts constructions in Euclidean space to triangulable spaces through successive approximations by Freudenthal-Kuhn triangulations, and control of interference across singular strata.

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