TY - JOUR

T1 - On the generalized lower bound conjecture for polytopes and spheres

AU - Murai, Satoshi

AU - Nevo, Eran

N1 - Funding Information:
Research of the first author was partially supported by KAKENHI 22740018. Research of the second author was partially supported by Marie Curie grant IRG-270923 and by an ISF grant.

PY - 2013

Y1 - 2013

N2 - 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, ..., hd) satisfies, Moreover, if hr-1 = hr for some, 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 result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

AB - 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, ..., hd) satisfies, Moreover, if hr-1 = hr for some, 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 result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

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U2 - 10.1007/s11511-013-0093-y

DO - 10.1007/s11511-013-0093-y

M3 - Article

AN - SCOPUS:84875609486

VL - 210

SP - 185

EP - 202

JO - Acta Mathematica

JF - Acta Mathematica

SN - 0001-5962

IS - 1

ER -