Leray Numbers of Projections and a Topological Helly Type Theorem

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 H̃i(Y ;Q) = 0 for all induced subcomplexes Y ⊂ X and i ≥ d. Suppose V = ⋃m i=1 Vi 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)) ≤ rL(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 Rd 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). Math Subject Classification. 55U10, 52A35