Algebraic shifting and strongly edge decomposable complexes

Satoshi Murai

doi:10.46298/dmtcs.3652

Abstract

Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, where $C(n,d)$ is the boundary complex of the cyclic $d$-polytope with $n$ vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.

Highlights

Algebraic shifting, which was introduced by, is a map which associates with each simplicial complex Γ another simplicial complex ∆(Γ) having a simple structure, called shifted
One of the major open problem in the theory of face vectors of simplicial complexes is the characterization of face vectors of simplicial spheres
If Γ is a (d − 1)-dimensional simplicial sphere it is known that ∆(Γ) satisfies the following properties

Summary

Introduction

Algebraic shifting, which was introduced by , is a map which associates with each simplicial complex Γ another simplicial complex ∆(Γ) having a simple structure, called shifted. To characterize face vectors of simplicial spheres by using algebraic shifting, Kalai and Sarkaria suggested the following conjecture. An important fact on this conjecture is that if Conjecture 1 is true (for either exterior shifting or symmetric shifting) it yields the characterization of face vectors of simplicial spheres (see Kalai (1991, 2002)). (Note that, for symmetric algebraic shifting, this result was essentially proved in Babson and Nevo (2008).) Second, we show that for any (d−1)-dimensional pure shifted complex Σ with n vertices satisfying Σ ⊂ ∆s(C(n, d)) and hi(Σ) = hd−i(Σ) for i = 0, 1, . If Conjecture 1 is true, this result gives the sufficiency of the characterization of algebraic shifted complexes of simplicial spheres

Strongly edge decomposable simplicial complexes

Algebraic shifting

Exterior shifting of strongly edge decomposable complex

The strong Lefschetz property

Algebraic shifting and squeezed spheres