Rank Selection and Depth Conditions for Balanced Simplicial Complexes

TL;DR

This paper establishes new theorems on rank selection and depth conditions for balanced simplicial complexes, extending prior results and providing formulas linking depth to homological properties.

Contribution

It introduces novel rank selection theorems for balanced complexes satisfying Serre's condition and relates depth to reduced homologies, broadening existing theoretical frameworks.

Findings

Rank selected subcomplexes preserve Serre's condition $(S_{ ext{ extellipsis}})$.

A formula for the depth of balanced complexes in terms of homology.

Extension of classical rank selection theorems to broader classes.

Abstract

We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that rank selected subcomplexes of balanced simplicial complexes satisfying Serre's condition $(S_{\ell})$ retain $(S_{\ell})$. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial compex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.