The stresses on centrally symmetric complexes and the lower bound theorems

Isabella Novik,Hailun Zheng

doi:10.5802/alco.168

Abstract

In 1987, Stanley conjectured that if a centrally symmetric Cohen–Macaulay simplicial complex Δ of dimension d-1 satisfies h i (Δ)=d i for some i≥1, then h j (Δ)=d j for all j≥i. Much more recently, Klee, Nevo, Novik, and Zheng conjectured that if a centrally symmetric simplicial polytope P of dimension d satisfies g i (∂P)=d i-d i-1 for some d/2≥i≥1, then g j (∂P)=d j-d j-1 for all d/2≥j≥i. This note uses stress spaces to prove both of these conjectures.

Highlights

This paper is devoted to analyzing the cases of equality in Stanley’s lower bound theorems on the face numbers of centrally symmetric Cohen–Macaulay complexes and centrally symmetric polytopes
In the fifty years since, this theory has become a major tool in the study of face numbers of simplicial complexes that resulted in a myriad of theorems and applications
Among them are a complete characterization of face numbers of Cohen– Macaulay (CM, for short) simplicial complexes [10], a complete characterization of flag face numbers of balanced CM complexes [3, 11], and a complete characterization of face numbers of simplicial polytopes [2, 12], to name just a few

Summary

Introduction

This paper is devoted to analyzing the cases of equality in Stanley’s lower bound theorems on the face numbers of centrally symmetric Cohen–Macaulay complexes and centrally symmetric polytopes. In the fifty years since, this theory has become a major tool in the study of face numbers of simplicial complexes that resulted in a myriad of theorems and applications. A simplicial complex ∆ is called centrally symmetric (or cs) if its vertex set V is endowed with a free involution α : V → V that induces a free involution on the set of all non-empty faces of ∆. Motivated by the desire to understand face numbers of cs simplicial polytopes as well as to find a complete characterization of face numbers of cs CM complexes, Stanley [13, Theorems 3.1 and 4.1] proved the following Lower Bound Theorem: Theorem 1.1. Cohen–Macaulay complexes, polytopes, centrally symmetric, face numbers, stress spaces

Setting the stage

Proof of the conjectures