Sum complexes - a new family of hypertrees

TL;DR

This paper introduces sum complexes, a new family of hypertrees constructed from cyclic groups, and explores their properties, including conditions for being hypertrees or collapsible, especially when n is prime.

Contribution

The paper defines sum complexes based on cyclic groups and characterizes when they form hypertrees or are collapsible, extending the understanding of hypertree structures.

Findings

Sum complexes are hypertrees when n is prime.

X_A is k-collapsible iff A is an arithmetic progression in Z_n.

Sum complexes generalize known hypertree constructions.

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.