Tilings, packings and expected Betti numbers in simplicial complexes

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 ${\text{Sd}}^d(K)$, $d\gg 0$?

28 pages, 4 figures

References in corpus (2)

• Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
• Universal components of random nodal sets

Cited by in corpus (2)

• Morse shellings on products
• Shellable tilings on relative simplicial complexes and their h-vectors