Helly numbers of acyclic families

Éric Colin de Verdière, Grégory Ginot, Xavier Goaoc
2014 Advances in Mathematics  
The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G ⊆ F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d Γ + 1), where d Γ is the smallest integer j such
more » ... hat every open set of Γ has trivial Q-homology in dimension j and higher. (In particular d R d = d.) This bound is best possible. We also prove a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known. In fact, our proof bounds the Leray number of the nerves of the families under consideration and thus also yields, under similar assumptions, a fractional Helly theorem, a (p, q)-theorem and the existence of small weak -nets. 1 Helly himself gave a topological extension of that theorem [34 34] (see also Debrunner [16 16]), asserting that any finite good cover in R d has Helly number at most d + 1. (For our purposes, a good cover is a finite family of open sets where the intersection of any sub-family is empty or contractible.) In this paper, we prove topological Helly-type theorems for families of non-connected sets, that is, we give upper bounds on Helly numbers for such families. Our results Let Γ be a locally arc-wise connected topological space. We let d Γ denote the smallest integer such that every open subset of Γ has trivial Q-homology in dimension d Γ and higher; in particular, when Γ is a d-dimensional manifold, we have d Γ = d if Γ is non-compact or non-orientable and d Γ = d + 1 otherwise (see Lemma 23 23); for example, d R d = d. We call a family F of open subsets of Γ acyclic if for any non-empty sub-family G ⊆ F, each connected component of the intersection of the elements of G is a Q-homology cell. (Recall that, in particular, any contractible set is a homology cell.) 1 1 We prove the following Helly-type theorem:
doi:10.1016/j.aim.2013.11.004 fatcat:3yns3opur5ghzkqspopjix32t4