Face numbers and the fundamental group

We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $Δ$ that represents a connected normal pseudomanifold of dimension $d\geq 3$ is at least as large as ${d+2 \choose 2}m(Δ)$, where $m(Δ)$ denotes the minimum number of generators of the fundamental group of $Δ$. Furthermore, we prove that a weaker bound, $h_2(Δ)\geq {d+1 \choose 2}m(Δ)$, applies to any $d$-dimensional pure simplicial poset $Δ$ 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 $Ψ$ all of whose vertex links satisfy Serre's condition $(S_r)$, we establish lower bounds on $h_1(Ψ),\ldots,h_r(Ψ)$ in terms of the $μ$-numbers introduced by Bagchi and Datta.

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