Some combinatorial properties of flag simplicial pseudomanifolds and spheres

TL;DR

This paper investigates combinatorial properties of flag simplicial pseudomanifolds and spheres, establishing connectivity and subgraph structures, and characterizing the minimality of the $h$-vector in flag homology spheres.

Contribution

It proves new connectivity and subgraph properties for flag simplicial pseudomanifolds and characterizes the minimal $h$-vector for flag homology spheres, extending understanding of their combinatorial structure.

Findings

Graph of a flag simplicial pseudomanifold is highly connected.

Existence of a subdivision of the cross-polytope graph within these complexes.

Minimal $h$-vector occurs for the boundary of the cross-polytope.

Abstract

A simplicial complex $\Delta$ is called flag if all minimal nonfaces of $\Delta$ have at most two elements. The following are proved: First, if $\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\Delta$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $\Delta$ of dimension $d-1$ is minimized when $\Delta$ is the boundary complex of the $d$-dimensional cross-polytope.