Face numbers and the fundamental group

Abstract

We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $\Delta$ that represents a connected normal pseudomanifold of dimension $d\ge 3$ is at least as large as $\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)m(\Delta )$, where $m(\Delta )$ denotes the minimum number of generators of the fundamental group of $\Delta$. Furthermore, we prove that a weaker bound, $h_2(\Delta )\ge \left(\genfrac{}{}{0ex}{}{d+1}{2}\right)m(\Delta )$, applies to any $d$-dimensional pure simplicial poset $\Delta$ all of whose faces of co-dimension $\ge 2$ have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset $\Psi$ all of whose vertex links satisfy Serre's condition $(S_r)$, we establish lower bounds on $h_1(\Psi ),\dots ,h_r(\Psi )$ in terms of the $\mu$-numbers introduced by Bagchi and Datta.

13 pages