## Abstract

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k-1)-dimensional skeleton and facets such that H_{k}(X;ℚ)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group ℤ_{n}. The sum complex X_{A} is the pure k-dimensional complex on the vertex set ℤ_{n} whose facets are σ⊂ℤ_{n} such that {pipe}σ{pipe}=k+1 and ∑_{x∈σ}x∈A. It is shown that if n is prime, then the complex X_{A} is a k-hypertree for every choice of A. On the other hand, for n prime, X_{A} is k-collapsible iff A is an arithmetic progression in ℤ_{n}.

Original language | American English |
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Pages (from-to) | 622-636 |

Number of pages | 15 |

Journal | Discrete and Computational Geometry |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - 2010 |

### Bibliographical note

Funding Information:N. Linial and R. Meshulam are supported by ISF and BSF grants.

## Keywords

- Fourier transform
- Homology
- Hypertrees