A Moore bound for simplicial complexes | Zendy

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A Moore bound for simplicial complexes

Author(s) -

Lubotzky Alexander,

Meshulam Roy

Publication year - 2007

Publication title -

bulletin of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 2.396

H-Index - 48

eISSN - 1469-2120

pISSN - 0024-6093

DOI - 10.1112/blms/bdm003

Subject(s) - mathematics , simplicial complex , abstract simplicial complex , h vector , simplicial approximation theorem , simplicial homology , pure mathematics , combinatorics , simplicial set , homotopy , homotopy category

Let X be a d ‐dimensional simplicial complex with N faces of dimension d − 1. Suppose that every ( d − 1)‐face of X is contained in at least k ⩾ d + 2 faces of X , of dimension d . Extending the classical Moore bound for graphs, it is shown that X must contain a ball B of radius at most ⌈log k − d N ⌉ whose d ‐dimensional homology H d ( B ) is non‐zero. The Ramanujan complexes constructed by Lubotzky, Samuels and Vishne are used to show that this upper bound on the radius of B cannot be improved by more than a multiplicative constant factor.

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