Tilings, packings and expected Betti numbers in simplicial complexes

TL;DR

This paper investigates the asymptotic behavior of expected Betti numbers in random subcomplexes of barycentric subdivisions of simplicial complexes, introducing tiling concepts to analyze packing problems and universal limits.

Contribution

It establishes convergence of normalized expected Betti numbers, develops a monotony theorem for codimension one, and introduces tiling notions to study packing of simplices in high barycentric subdivisions.

Findings

Expected Betti numbers converge to universal limits as subdivision degree increases.

A monotony theorem improves estimates of Betti number limits based on packings.

Tiling concepts enable analysis of maximal packing of disjoint simplices in subdivisions.

Abstract

Let $K$ be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its $d$-th barycentric subdivision $\text{Sd}^d (K)$ converge to universal limits as $d$ grows to $+ \infty$. In codimension one, we use canonical filtrations of $\text{Sd}^d (K)$ to upper estimate these limits and get a monotony theorem which makes it possible to improve these estimates given any packing of disjoint simplices in $\text{Sd}^d (K)$. We then introduce a notion of tiling of simplicial complexes having the property that skeletons and barycentric subdivisions of tileable simplicial complexes are tileable. This enables us to tackle the problem: How many disjoint simplices can be packed in $\text{Sd}^d (K)$, $d \gg 0$?