Discrete & Computational Geometry

, Volume 44, Issue 3, pp 622–636

Sum Complexes—a New Family of Hypertrees


DOI: 10.1007/s00454-010-9252-5

Cite this article as:
Linial, N., Meshulam, R. & Rosenthal, M. Discrete Comput Geom (2010) 44: 622. doi:10.1007/s00454-010-9252-5


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 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 complexXA is the pure k-dimensional complex on the vertex set ℤn whose facets are σ⊂ℤn such that |σ|=k+1 and ∑xσxA. 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.


HypertreesHomologyFourier transform

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Computer ScienceHebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsTechnionHaifaIsrael