The stresses on centrally symmetric complexes and the lower bound theorems

2020/08/28 by Isabella Novik, Novik, Isabella, Hailun Zheng +1
Mathematics · #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2008.12503

Abstract

In 1987, Stanley conjectured that if a centrally symmetric Cohen--Macaulay simplicial complex Δ of dimension d-1 satisfies h_i(Δ)=\binomdi for some i≥ 1, then h_j(Δ)=\binomdj 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)=\binomdi-\binomdi-1 for some d/2≥ i≥ 1, then g_j(∂ P)=\binomdj-\binomdj-1 for all d/2≥ j≥ i. This note uses stress spaces to prove both of these conjectures.

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