Multinerves and Helly Numbers of Acyclic Families

Éric Colin de Verdière (1) , Grégory Ginot (2) , Xavier Goaoc (3)

1 LIENS - Laboratoire d'informatique de l'école normale supérieure
2 UPMC - Université Pierre et Marie Curie - Paris 6
3 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility

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.

Domaines: Géométrie algorithmique [cs.CG]

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. pp.209-218, ⟨10.1145/2261250.2261282⟩. ⟨hal-00752073⟩