Journal of Topology

Journal of Topology Advance Access originally published online on April 28, 2008
Journal of Topology 2008 1(3):551-556; doi:10.1112/jtopol/jtn010

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© 2008 London Mathematical Society

Leray numbers of projections and a topological Helly-type theorem

Gil Kalai

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

Roy Meshulam

Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel meshulam@math.technion.ac.il

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[V_i] 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 V_i. 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.

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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.

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Kalai, G. Articles by Meshulam, R. Social Bookmarking          What's this?