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- Title
Sum Complexes—a New Family of Hypertrees.
- Authors
Linial, N.; Meshulam, R.; Rosenthal, M.
- 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 σ⊂ℤ n such that | σ|= k+1 and ∑ x∈ σ x∈ 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.
- Subjects
HOMOLOGY theory; FOURIER transforms; SKELETON; MATHEMATICAL transformations; MATHEMATICS
- Publication
Discrete & Computational Geometry, 2010, Vol 44, Issue 3, p622
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-010-9252-5