The stresses on centrally symmetric complexes and the lower bound theorems

TL;DR

This paper proves two conjectures related to the combinatorial structure of centrally symmetric complexes and polytopes using stress spaces, confirming specific lower bound properties for their face numbers.

Contribution

It introduces a method using stress spaces to prove longstanding conjectures on face number bounds in centrally symmetric complexes and polytopes.

Findings

Proves Stanley's conjecture on h-vectors of Cohen--Macaulay complexes.

Confirms Klee et al.'s conjecture on g-vectors of centrally symmetric polytopes.

Establishes a new technique using stress spaces for combinatorial topology.

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.