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## How many different triangulations of the d-sphere are there ?

The upper bound theorem says that a \$d\$-sphere with \$n\$ vertices has at most \$O(n^\lceil d/2 \rceil)\$ facets. As a corollary, one gets that there are at most \$2^O(n^\lceil d/2 \rceil \log n)\$ combinatorially different such triangulations. On the side of lower bounds, Kalai (1988) showed a construction giving \$2^\Omega(n^\lfloor d/2 \rfloor)\$ different ones. In even dimension the upper and lower bounds differ only in the \$\log n\$ factor, but in odd dimension their difference is much bigger. Most strikingly, for \$d=3\$ the upper bound is \$2^O(n2\log n)\$ while the lower bound is \$2^\Omega(n)\$.

In this talk I will review these results and show a new construction which gives, in every odd dimension, \$2^\Omega(n^\lceil d/2 \rceil)\$ different triangulatons.

As variations and/or byproducts of the construction we also obtained the following (results are in arbitrary dimension, but we state them in dimension three for simplicity. In all the results \$n\$ is the number of vertices) :

There are \$2^\Omega(n^3/2)\$ geodesic also called star-convex) triangulations of the 3-sphere.

There are \$3\$-spheres with \$\Omega(n2)\$ facets that are not simplices.

There are \$4\$-polytopes with \$\Omega(n^3/2)\$ facets that are not simplices.

This is joint work with E. Nevo and S. Wilson, http://arxiv.org/abs/1408.3501.