Discrete & Computational Geometry

, Volume 44, Issue 3, pp 622–636

Sum Complexes—a New Family of Hypertrees


  • N. Linial
    • Department of Computer ScienceHebrew University
    • Department of MathematicsTechnion
  • M. Rosenthal
    • Department of Computer ScienceHebrew University

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 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 σ⊂ℤ n such that |σ|=k+1 and ∑xσ xA. 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 .


Hypertrees Homology Fourier transform

Copyright information

© Springer Science+Business Media, LLC 2010