Face numbers and the fundamental group (Q1686314)

In this paper the authors provide the proof of the Kalai conjecture. Recall that the Kalai conjecture states that the \(g_2\)-number of any (finite) simplicial complex \(\Delta\) that represents a normal pseudomanifold of dimension \(d\geq3\) 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 \(\Delta\). In addition, the authors show a weaker bound \(h_2(\Delta)\geq{{d+1} \choose 2} m(\Delta)\) for any \(d\)-dimensional pure simplicial poset \(\Delta\) all of whose faces of co-dimension 2 have connected links. The last generalizes a result of Klee. Finally, for a pure relative simplicial poset \(\Psi\) all of whose vertex links satisfy Serre's condition \((S_r)\), the authors establish lower bounds on \(h_1(\Psi),\ldots,h_r(\Psi)\) in terms of the \(\mu\)-numbers introduced by Bagchi and Datta.