The stresses on centrally symmetric complexes and the lower bound theorems

Abstract

In 1987, Stanley conjectured that if a centrally symmetric Cohen--Macaulay simplicial complex $\Delta$ of dimension $d-1$ satisfies $h_i(\Delta)=\binom{d}{i}$ for some $i\geq 1$, then $h_j(\Delta)=\binom{d}{j}$ for all $j\geq i$. Much more recently, Klee, Nevo, Novik, and Zheng conjectured that if a centrally symmetric simplicial polytope $P$ of dimension $d$ satisfies $g_i(\partial P)=\binom{d}{i}-\binom{d}{i-1}$ for some $d/2\geq i\geq 1$, then $g_j(\partial P)=\binom{d}{j}-\binom{d}{j-1}$ for all $d/2\geq j\geq i$. This note uses stress spaces to prove both of these conjectures.