Shellable tilings on relative simplicial complexes and their h-vectors

TL;DR

This paper introduces shellable h-tilings on finite simplicial complexes, demonstrating their existence after subdivisions, and explores their properties, including h-vectors, critical tiles, and behavior under stellar subdivisions.

Contribution

It proves the existence of shellable h-tilings on all finite simplicial complexes after finitely many stellar subdivisions and relates h-vectors to critical tiles, extending understanding of complex tilings.

Findings

h-tilings exist on all finite simplicial complexes after subdivisions

h-vectors are determined by critical tiles and satisfy palindromic properties in manifolds

tilings are shellable and behave predictably under stellar subdivisions

Abstract

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.