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 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 sigma subset of Z(n) such that |sigma| = k+ 1 and Sigma(x is an element of sigma) x is an element of 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).
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