Many triangulated spheres
作者： Gil Kalai
DOI: 10.1007/BF02187893
关键词: Mathematics 、 SPHERES 、 Asymptotic formula 、 Combinatorics 、 Fundamental theorem 、 Upper and lower bounds 、 Discrete mathematics 、 Theoretical computer science 、 Computational Theory and Mathematics 、 Geometry and topology 、 Discrete Mathematics and Combinatorics
摘要: Lets(d, n) be the number of triangulations withn labeled vertices ofSd?1, (d?1)-dimensional sphere. We extend a construction Billera and Lee to obtain large family triangulated spheres. Our shows that logs(d, n)?C1(d)n[(d?1)/2], while known upper bound is n)≤C2(d)n[d/2] logn. Letc(d, combinatorial types simpliciald-polytopes vertices. (Clearly,c(d, n)≤s(d, n).) Goodman Pollack have recently proved bound: logc(d, n)≤d(d+1)n logn. Combining this forc(d, with our lower bounds fors(d, n), we obtain, for everyd?5, limn??(c(d, n)/s(d, n))=0. The cased=4 left open. (Steinitz's fundamental theorem asserts thats(3,n)=c(3,n), everyn.) also prove that, everyb?4, limd??(c(d, d+b)/s(d, d+b))=0. (Mani thats(d, d+3)=c(d, d+3), everyd.) Lets(n) spheres logs(n)=20.69424n(1+o(1)). same asymptotic formula describes manifolds
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