Leray numbers of projections and a topological Helly-type theorem

Abstract

Let X be a simplicial complex on the vertex set V. The rational Leray number L (X) of X is the minimal d, such that H ˜ i ( Y ; ℚ ) = 0 for all induced subcomplexes Y ⊂ X and i ⩾ d. Suppose that V = U i = 1 m V i is a partition of V such that the induced subcomplexes X[Vi] are all 0-dimensional. Let π denote the projection of X into the (m − 1)-simplex on the vertex set {1, …, m} given by π(v) = i if v ∈ Vi. Let r = max{|π−1(π(x))|:x ∈ |X|}. It is shown that L ( π ( X ) ) ⩽ r L ( X ) + r − 1 One consequence is a topological extension of a Helly-type result of Amenta. Let F be a family of compact sets in ℝ d such that for any F ′ ⊂ F , the intersection ∩ F ′ is either empty or contractible. It is shown that if G is a family of sets such that for any finite G ′ ⊂ G , the intersection ∩ G ′ is a union of at most r disjoint sets in F, then the Helly number of G is at most r(d + 1).