Premium
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.