Discrete & Computational Geometry

, Volume 3, Issue 1, pp 1–14

# Many triangulated spheres

• Gil Kalai
Article

## Abstract

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.

## Keywords

Partial Order Simplicial Complex Label Vertex Simplicial Polytopes Dimensional Simplicial Complex
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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