Multinerves and helly numbers of acyclic families

Éric Colin de Verdière, Grégory Ginot, Xavier Goaoc
2012 Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12  
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
more » ... 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 Matoušek. This Hellytype theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory.
doi:10.1145/2261250.2261282 dblp:conf/compgeom/VerdiereGG12 fatcat:bhtzuklqnvhull747566j3z2hm