Many triangulated spheres |
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Authors: | Gil Kalai |
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Institution: | (1) Institute of Mathematics, Hebrew University, Jerusalem, Israel |
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Abstract: | Lets(d, n) be the number of triangulations withn labeled vertices ofS
d–1, the (d–1)-dimensional sphere. We extend a construction of Billera and Lee to obtain a large family of triangulated spheres. Our construction shows that logs(d, n) C
1(d)n
(d–1)/2], while the known upper bound is logs(d, n) C
2(d)n
d/2] logn.Letc(d, n) be the number of combinatorial types of simpliciald-polytopes withn labeled vertices. (Clearly,c(d, n) s(d, n).) Goodman and Pollack have recently proved the upper bound: logc(d, n) d(d+1)n logn. Combining this upper bound forc(d, n) with our lower bounds fors(d, n), we obtain, for everyd 5, that lim
n![rarr](/content/78044667x381777g/xxlarge8594.gif) (c(d, n)/s(d, n))=0. The cased=4 is left open. (Steinitz's fundamental theorem asserts thats(3,n)=c(3,n), for everyn.) We also prove that, for everyb 4, lim
d![rarr](/content/78044667x381777g/xxlarge8594.gif) (c(d, d+b)/s(d, d+b))=0. (Mani proved thats(d, d+3)=c(d, d+3), for everyd.)Lets(n) be the number of triangulated spheres withn labeled vertices. We prove that logs(n)=20.69424n(1+o(1)). The same asymptotic formula describes the number of triangulated manifolds withn labeled vertices.Research done, in part, while the author visited the mathematics research center at AT&T Bell Laboratories. |
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