On the generalized lower bound conjecture for polytopes and spheres

2012/03/08 by Murai, Satoshi, Nevo, Eran · 1 citation
#13F55 #52B05 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1203.1720

Abstract

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h₀,h₁,...,h_d) satisfies h₀ ≤ h₁ ≤ ... ≤ h_{\lfloor \frac d 2 \rfloor}. Moreover, if h_r-1=h_r for some r ≤ \frac d 2 then P can be triangulated without introducing simplices of dimension ≤ d-r. The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this property to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

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• Missing faces of neighborly and nearly neighborly polytopes and spheres

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