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Sum Complexes—a New Family of Hypertrees

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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 .


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Correspondence to R. Meshulam.

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N. Linial and R. Meshulam are supported by ISF and BSF grants.

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Linial, N., Meshulam, R. & Rosenthal, M. Sum Complexes—a New Family of Hypertrees. Discrete Comput Geom 44, 622–636 (2010). https://doi.org/10.1007/s00454-010-9252-5

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  • Hypertrees
  • Homology
  • Fourier transform