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

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 $(-1)^n \sum_{σ\in S} \left(\frac{-1}{2}\right)^{\dimσ+ 1} \ge 0$, where the sum runs over all simplices $σ$ of $S$ (including the empty simplex). Interpreting the $1$-skeleta of $σ\in S$ 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.

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