Discrete & Computational Geometry

, Volume 3, Issue 1, pp 1–14 | Cite as

Many triangulated spheres

  • Gil Kalai


Lets(d, n) be the number of triangulations withn labeled vertices ofSd−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)≥C1(d)n[(d−1)/2], while the known upper bound is logs(d, n)≤C2(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 limn→∞(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, limd→∞(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.


Partial Order Simplicial Complex Label Vertex Simplicial Polytopes Dimensional Simplicial Complex 
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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Gil Kalai
    • 1
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael

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