## Abstract

We present some enumerative and structural results for ag homology spheres. For a ag homology sphere δ, we show that its γ-vector γ^{δ} =(1, γ_{1}, γ_{2}, ...) satisfues: γ_{j} =0 for all j γ_{1}, γ_{2} ≤ (γ^{1}_{2}, γ_{γ1} ϵ 0, 1, and γ_{γ1-1} ϵ 0, 1, 2, γ_{1}, supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for δ in extremal cases. As an application, the techniques used produce infunitely many f-vectors of ag balanced simplicial complexes that are not γ-vectors of ag homology spheres (of any dimension); these are the furst examples of this kind. In addition, we prove a ag analog of Perles' 1970 theorem on k-skeleta of polytopes with "few" vertices, specifucally, the number of combinatorial types of κ-skeleta of ag homology spheres with γ_{1} ≤ b of any given dimension, is bounded independently of the dimension.

Original language | American English |
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Pages (from-to) | 2064-2078 |

Number of pages | 15 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - 2017 |

### Bibliographical note

Publisher Copyright:© 2017 Society for Industrial and Applied Mathematics.

## Keywords

- Ag complex
- Face vectors
- Homology spheres
- Simplicial complex