Sum Complexes-a New Family of Hypertrees

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∈σx∈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.