Multinerves and Helly Numbers of Acyclic Families

Abstract : The nerve of a family of sets is a simplicial complex that records the intersection pattern of its subfamilies. Nerves are widely used in computational geometry and topology, because the nerve theorem guarantees that the nerve of a family of geometric objects has the same topology as the union of the objects, if they form a good cover. In this paper, we relax the good cover assumption to the case where each subfamily intersects in a disjoint union of possibly several homology cells, and we prove a generalization of the nerve theorem in this framework, using spectral sequences from algebraic topology. We then deduce a new topological Helly-type theorem that unifies previous results of Amenta, Kalai and Meshulam, and Matousek. This Helly-type theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory.
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Communication dans un congrès
Symposium on Computational Geometry - SoCG '12, Jun 2012, Chapel Hill, United States. ACM, pp.209-218, 2012, 〈http://dl.acm.org/citation.cfm?doid=2261250.2261282〉. 〈10.1145/2261250.2261282〉
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Contributeur : Xavier Goaoc <>
Soumis le : mercredi 12 décembre 2012 - 15:01:10
Dernière modification le : mercredi 28 septembre 2016 - 16:18:46
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Éric Colin De Verdière, Grégory Ginot, Xavier Goaoc. Multinerves and Helly Numbers of Acyclic Families. Symposium on Computational Geometry - SoCG '12, Jun 2012, Chapel Hill, United States. ACM, pp.209-218, 2012, 〈http://dl.acm.org/citation.cfm?doid=2261250.2261282〉. 〈10.1145/2261250.2261282〉. 〈hal-00752073〉

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