On a Generalization of the Flag Complex Conjecture of Charney and Davis

TL;DR

This paper introduces a stronger version of the Flag Complex Conjecture related to simplicial complexes and Coxeter groups, proves their equivalence, and advances understanding of the conjecture's validity.

Contribution

The paper proposes a stronger conjecture and demonstrates its equivalence to the original Flag Complex Conjecture, providing new insights into the structure of flag complexes and Coxeter groups.

Findings

Proposes a stronger version of the Flag Complex Conjecture.

Proves the equivalence between the new conjecture and the original.

Enhances understanding of the combinatorial properties of flag complexes.

Abstract

The Flag Complex Conjecture of Charney and Davis states that for a simplicial complex $S$ which triangulates a $(2n - 1)$-generalized homology sphere as a flag complex one has $(-1)^n \sum_{\sigma \in S} \left(\frac{-1}{2}\right)^{\dim\sigma + 1} \ge 0$, where the sum runs over all simplices $\sigma$ of $S$ (including the empty simplex). Interpreting the $1$-skeleta of $\sigma\in S$ as graphs of Coxeter groups, we present a stronger version of this conjecture, and prove the equivalence of the latter to the Flag Complex Conjecture.