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

We prove that for $d\ge 3$, the 1-skeleton of any $(d-1)$-dimensional doubly Cohen Macaulay (abbreviated 2-CM) complex is generically $d$-rigid. This implies the following two corollaries (by Kalai and Lee respectively): Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes hold for every 2-CM complex (of dimension $\ge 2$). Moreover, the initial part $(g_0,g_1,g_2)$ of the $g$-vector of a 2-CM complex (of dimension $\ge 3$) is an $M$-sequence. It was conjectured by Bj"{o}rner and Swartz that the entire $g$-vector of a 2-CM complex is an $M$-sequence.