Face numbers and the fundamental group

TL;DR

This paper proves new lower bounds on face numbers of simplicial complexes related to their fundamental groups, generalizing previous results and establishing bounds for various classes of simplicial posets.

Contribution

It resolves Kalai's conjecture relating $g_2$-numbers to the fundamental group and extends bounds to broader classes of simplicial posets.

Findings

Proves $g_2$-number bound in terms of fundamental group for pseudomanifolds.

Establishes $h_2$-number bound for certain simplicial posets.

Provides lower bounds on $h_1, \,\ldots,\, h_r$ for relative simplicial posets with Serre's condition.

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 $d\geq 3$ is at least as large as ${d+2 \choose 2}m(\Delta)$, where $m(\Delta)$ denotes the minimum number of generators of the fundamental group of $\Delta$. Furthermore, we prove that a weaker bound, $h_2(\Delta)\geq {d+1 \choose 2}m(\Delta)$, applies to any $d$-dimensional pure simplicial poset $\Delta$ all of whose faces of co-dimension $\geq 2$ 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 $h_1(\Psi),\ldots,h_r(\Psi)$ in terms of the $\mu$-numbers introduced by Bagchi and Datta.