Lower Bound Theorems and a Generalized Lower Bound Conjecture for balanced simplicial complexes

TL;DR

This paper extends classical face enumeration theorems to balanced simplicial complexes, proposing conjectures and characterizations for their face numbers and structures, and constructing examples with minimal vertices.

Contribution

It introduces balanced analogs of key theorems and conjectures in face enumeration, including the Lower Bound Theorem and the Generalized Lower Bound Conjecture, with characterizations and new constructions.

Findings

Balanced Lower Bound Theorem proven for pseudomanifolds

Characterization of equality cases in balanced complexes

Constructed balanced manifolds with few vertices

Abstract

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; we propose the balanced analog of the Generalized Lower Bound Conjecture and establish some related results. We close with constructions of balanced manifolds with few vertices.