Face numbers and the fundamental group
  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. 
Satoshi Murai and Isabella Novik
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