TY - JOUR
T1 - Leray numbers of projections and a topological Helly-type theorem
AU - Kalai, Gil
AU - Meshulam, Roy
N1 - Publisher Copyright:
© 2008 London Mathematical Society.
PY - 2008/7
Y1 - 2008/7
N2 - 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 (Formula presented.) for all induced subcomplexes Y ⊂ X and i ⩾ d. Suppose that (Formula presented.) 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 (Formula presented.) One consequence is a topological extension of a Helly-type result of Amenta. Let (Formula presented.) be a family of compact sets in (Formula presented.) such that for any (Formula presented.), the intersection (Formula presented.) is either empty or contractible. It is shown that if (Formula presented.) is a family of sets such that for any finite (Formula presented.), the intersection (Formula presented.) is a union of at most r disjoint sets in (Formula presented.), then the Helly number of (Formula presented.) is at most r(d + 1).
AB - 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 (Formula presented.) for all induced subcomplexes Y ⊂ X and i ⩾ d. Suppose that (Formula presented.) 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 (Formula presented.) One consequence is a topological extension of a Helly-type result of Amenta. Let (Formula presented.) be a family of compact sets in (Formula presented.) such that for any (Formula presented.), the intersection (Formula presented.) is either empty or contractible. It is shown that if (Formula presented.) is a family of sets such that for any finite (Formula presented.), the intersection (Formula presented.) is a union of at most r disjoint sets in (Formula presented.), then the Helly number of (Formula presented.) is at most r(d + 1).
UR - http://www.scopus.com/inward/record.url?scp=68449091618&partnerID=8YFLogxK
U2 - 10.1112/jtopol/jtn010
DO - 10.1112/jtopol/jtn010
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AN - SCOPUS:68449091618
SN - 1753-8416
VL - 1
SP - 551
EP - 556
JO - Journal of Topology
JF - Journal of Topology
IS - 3
ER -