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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 ℤn . The sum complex X A is the pure k-dimensional complex on the vertex set ℤn whose facets are σ⊂ℤn 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 ℤn .

Sum Complexes—a New Family of Hypertrees