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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.


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