On the generalized lower bound conjecture for polytopes and spheres

Satoshi Murai, Eran Nevo

Research output: Contribution to journalArticle

19 Citations (Scopus)

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, ..., 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.

Original languageEnglish
Pages (from-to)185-202
Number of pages18
JournalActa Mathematica
Volume210
Issue number1
DOIs
Publication statusPublished - 2013 Apr 4
Externally publishedYes

Fingerprint

Polytopes
Lower bound
H-vector
Toric Varieties
Polytope
Generalise
Theorem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the generalized lower bound conjecture for polytopes and spheres. / Murai, Satoshi; Nevo, Eran.

In: Acta Mathematica, Vol. 210, No. 1, 04.04.2013, p. 185-202.

Research output: Contribution to journalArticle

@article{e2d603f3490d4b81a3ad060bf671b2a0,
title = "On the generalized lower bound conjecture for polytopes and spheres",
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, ..., 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.",
author = "Satoshi Murai and Eran Nevo",
year = "2013",
month = "4",
day = "4",
doi = "10.1007/s11511-013-0093-y",
language = "English",
volume = "210",
pages = "185--202",
journal = "Acta Mathematica",
issn = "0001-5962",
publisher = "Springer Netherlands",
number = "1",

}

TY - JOUR

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

AU - Murai, Satoshi

AU - Nevo, Eran

PY - 2013/4/4

Y1 - 2013/4/4

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.

UR - http://www.scopus.com/inward/record.url?scp=84875609486&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875609486&partnerID=8YFLogxK

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 -