# Sum Complexes—a New Family of Hypertrees

## Authors

- First Online:

- Received:
- Accepted:

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

## 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
}.