Discrete & Computational Geometry

, Volume 44, Issue 3, pp 622–636

Sum Complexes—a New Family of Hypertrees

Article

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 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.

Keywords

Hypertrees Homology Fourier transform 

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References

  1. 1.
    Kalai, G.: Enumeration of ℚ-acyclic simplicial complexes. Isr. J. Math. 45, 337–351 (1983) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Munkres, J.: Elements of Algebraic Topology. Addison-Wesley, Reading (1984) MATHGoogle Scholar
  3. 3.
    Stevenhagen, P., Lenstra, H.W.: Chebotarëv and his density theorem. Math. Intell. 18, 26–37 (1996) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

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

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