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 ( n−1 k ) facets such that Hk(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 XA 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 XA is a k-hypertree for every choice of A. On the other hand, for n prime XA is k-collapsible iff A is an arithmetic progression in Zn.