We resolve a conjecture of Kalai asserting that the g 2 -number of any simplicial complex ∆ that represents a connected normal pseudomanifold of dimension d ≥ 3 is at least as large as d+2 2 m(∆), where m(∆) denotes the minimum number of generators of the fundamental group of ∆. Furthermore, we prove that a weaker bound, h 2 (∆) ≥ d+1 2 m(∆), applies to any d-dimensional pure simplicial poset ∆ 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 h 1 (Ψ), . . . , h r (Ψ) in terms of the µ-numbers introduced by Bagchi and Datta.