| Leray numbers of projections and a topological Helly-type theorem (2008) | |||||||||||||
Abstract | |||||||||||||
| Let X be a simplicial complex on the vertex set V. The rational Leray number of X is the minimal d, such that for all induced subcomplexes Y ⊂ X and i ⩾ d. Suppose that 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 One consequence is a topological extension of a Helly-type result of Amenta. Let be a family of compact sets in such that for any , the intersection is either empty or contractible. It is shown that if is a family of sets such that for any finite , the intersection is a union of at most r disjoint sets in , then the Helly number of is at most r(d + 1). | |||||||||||||
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