## Abstract

Kalai proved that the simplicial polytopes with g_{2}=0 are the stacked polytopes. We characterize the g_{2}=1 case.Specifically, we prove that every simplicial d-polytope (d≥4) which is prime and with g_{2}=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d≥4) with g_{2}=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphere (d≥4) with g_{2}=1, and our proof takes advantage of working with this larger class of complexes.

Original language | English |
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Pages (from-to) | 387-395 |

Number of pages | 9 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 118 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2011 |

Externally published | Yes |

### Bibliographical note

Funding Information:E-mail addresses: [email protected] (E. Nevo), [email protected] (E. Novinsky). 1 Research partially supported by an NSF grant DMS-0757828.

## Keywords

- Graph rigidity
- Homology sphere
- Polytope

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