Rigidity and the Lower Bound Theorem for Doubly Cohen-Macaulay Complexes

We prove that for d ≥ 3, the 1-skeleton of any (d− 1)-dimensional doubly Cohen-Macaulay (abbreviated 2-CM) complex is generically drigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes ([4],[3]) hold for every 2-CM complex of dimension ≥ 2 (see Kalai [8]). Moreover, the initial part (g0, g1, g2) of the g-vector of a 2-CM complex (of dimension ≥ 3) is an M -sequence. It was conjectured by Björner and Swartz [14] that the entire g-vector of a 2-CM complex is an M -sequence.