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 $h_i(\Delta )=\left(\genfrac{}{}{0ex}{}{d}{i}\right)$ for some $i\ge 1$, then $h_j(\Delta )=\left(\genfrac{}{}{0ex}{}{d}{j}\right)$ for all $j\ge i$. Much more recently, Klee, Nevo, Novik, and Zheng conjectured that if a centrally symmetric simplicial polytope $P$ of dimension $d$ satisfies $g_i(\partial P)=\left(\genfrac{}{}{0ex}{}{d}{i}\right)-\left(\genfrac{}{}{0ex}{}{d}{i-1}\right)$ for some $d/2\ge i\ge 1$, then $g_j(\partial P)=\left(\genfrac{}{}{0ex}{}{d}{j}\right)-\left(\genfrac{}{}{0ex}{}{d}{j-1}\right)$ for all $d/2\ge j\ge i$. This note uses stress spaces to prove both of these conjectures.

arXiv commentSimplified statements and proofs of Lemmas 3.3 and 3.4, plus some minor edits. To appear in Algebraic Combinatorics

References in corpus (1)

• Combinatorial Lefschetz theorems beyond positivity