Many triangulated spheres

Let s(d, n) be the number of triangulations with n labeled vertices of S^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 log s(d, n)≥C₁(d)n^[(d-1)/2], while the known upper bound is log s(d, n)≤C₂(d)n^[d/2] log n. Let c(d, n) be the number of combinatorial types of simplicial d-polytopes with n labeled vertices. (Clearly, c(d, n)≤s(d, n).) Goodman and Pollack have recently proved the upper bound: log c(d, n)≤d(d+1)n log n. Combining this upper bound for c(d, n) with our lower bounds for s(d, n), we obtain, for every d≥5, that lim_n→∞(c(d, n)/s(d, n))=0. The case d=4 is left open. (Steinitz's fundamental theorem asserts that s(3, n)=c(3, n), for every n.) We also prove that, for every b≥4, lim_d→∞(c(d, d+b)/s(d, d+b))=0. (Mani proved that s(d, d+3)=c(d, d+3), for every d.) Let s(n) be the number of triangulated spheres with n labeled vertices. We prove that log s(n)=2^{0.69424 n(1+o(1))}. The same asymptotic formula describes the number of triangulated manifolds with n labeled vertices.