Journal of Topology Advance Access published online on April 28, 2008
Journal of Topology, doi:10.1112/jtopol/jtn010
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© 2008 London Mathematical Society
Leray numbers of projections and a topological Helly-type theorem
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Departments of Computer Science and Mathematics, Yale University, New Haven, CT 06520, USA kalai@math.huji.ac.il
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel meshulam@math.technion.ac.il
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Abstract |
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Let X be a simplicial complex on the vertex set V. The rational Leray number 
of X is the minimal d, such that 
for all induced subcomplexes Y 
X and i 
d. Suppose that 
is a partition of V such that the induced subcomplexes X[Vi] are all 0-dimensional. Let 
denote the projection of X into the (m – 1)-simplex on the vertex set {1, ..., m} given by (v) = i if v 
Vi. Let r = max{|–1((x))|:x |X|}. It is shown that 
One consequence is a topological extension of a Helly-type result of Amenta. Let 
be a family of compact sets in 
such that for any 
, the intersection 
is either empty or contractible. It is shown that if 
is a family of sets such that for any finite 
, the intersection 
is a union of at most r disjoint sets in 
, then the Helly number of 
is at most r(d + 1).
Received April 2, 2007.
2000 Mathematics Subject Classification 55U10 (primary), 52A35 (secondary)
Both authors are supported by grants from the Israel Science Foundation and the US–Israel Binational Science Foundation. The first author is supported by an NSF grant.