Rank Selection and Depth Conditions for Balanced Simplicial Complexes

  • Brent Holmes
  • Justin Lyle

Abstract

We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that if a balanced simplicial complex satisfies Serre's condition $(S_{\ell})$ then so do all of its rank selected subcomplexes.  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 complex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi. 

Published
2021-05-21
Article Number
P2.28