Sum Complexes—a New Family of Hypertrees

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 .