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

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

2005-05-16

arxiv.org/abs/math/0505334v2

Discrete Comput. Geom. 39 (2008), no. 1-3, 411--418.

Mathematics

Combinatorics

Revised: 9 pages, no figures, a relation to nowhere zero flows added, some minor changes. To appear in DCG

Scientific paper

We prove that for $d\geq 3$, 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 $\geq 2$). Moreover, the initial part $(g_0,g_1,g_2)$ of the $g$-vector of a 2-CM complex (of dimension $\geq 3$) 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.

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