# Sum Complexes—a New Family of Hypertrees

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DOI: 10.1007/s00454-010-9252-5

- Cite this article as:
- Linial, N., Meshulam, R. & Rosenthal, M. Discrete Comput Geom (2010) 44: 622. doi:10.1007/s00454-010-9252-5

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## Abstract

A *k*-dimensional hypertree *X* is a *k*-dimensional complex on *n* vertices with a full (*k*−1)-dimensional skeleton and \(\binom{n-1}{k}\) 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 |*σ*|=*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}.