Sum Complexes-a New Family of Hypertrees

N. Linial, R. Meshulam*, M. Rosenthal

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k-1)-dimensional skeleton and 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 complex XA is the pure k-dimensional complex on the vertex set ℤn whose facets are σ⊂ℤn such that {pipe}σ{pipe}=k+1 and ∑x∈σx∈A. 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.

Original languageAmerican English
Pages (from-to)622-636
Number of pages15
JournalDiscrete and Computational Geometry
Issue number3
StatePublished - 2010

Bibliographical note

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


  • Fourier transform
  • Homology
  • Hypertrees


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