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 .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Kalai, G.: Enumeration of ℚ-acyclic simplicial complexes. Isr. J. Math. 45, 337–351 (1983)
Munkres, J.: Elements of Algebraic Topology. Addison-Wesley, Reading (1984)
Stevenhagen, P., Lenstra, H.W.: Chebotarëv and his density theorem. Math. Intell. 18, 26–37 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
N. Linial and R. Meshulam are supported by ISF and BSF grants.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-010-9252-5