Face numbers and the fundamental group

We resolve a conjecture of Kalai asserting that the g₂-number of any (finite) simplicial complex Δ that represents a normal pseudomanifold of dimension d ≥ 3 is at least as large as (d+22)m(Δ), where m(Δ) denotes the minimum number of generators of the fundamental group of Δ. Furthermore, we prove that a weaker bound, h2(d+12)m(Δ), applies to any d-dimensional pure simplicial poset Δ all of whose faces of co-dimension ≥ 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 h₁(Ψ),..,h_r(Ψ) in terms of the μ-numbers introduced by Bagchi and Datta.