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 subsets \sigma of Z_n such that |\sigma|=k+1 and \sum_{x \in \sigma}x \in 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.