| Leray numbers of projections and a topological Helly type theorem (2007) | |||||||||||||||
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 ˜ Hi(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 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). | |||||||||||||||
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