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In Gromov's talk at the Clay Math Research from 23:23 to 25:55 Gromov says (slightly paraphrased)

I want to emphasize a problem which comes from mathematical physics which is unsolved which is indicating that we don't understanding what triangulation is. And the problem is extremely simple. You take a manifold, and you just ask how many triangulations it has with a given number of simplices. So you have your manifold $X$ and you have the number of triangulations with k simplices $N_k(X)$ and you want to know what happens to it as $k$ goes to infinity, roughly. You take triangulations up to isomorphism. It's bounded below by $(1 + \epsilon)^k$ and it's bounded from above by $k^k$, roughly. That's kind of trivial, you just keep subdividing and you see how many automorphisms you have on a $k$ element set. The question is, where is the truth? And nothing is known, just absolutely blank. For surfaces you know, it's like that [exponential] and physicists kind of made that computation.

[...]

The whole point is that the manifold must be fixed, if you vary the manifold you would have $k^k$.

The subtle point is fix a topological manifold, how does the combinatorics tell you something about the topology. We think we understand it, but when we do this problem we don't. There's absolutely not a direct link between the two. We have a zero level question in topology, we cannot answer it.

The introduction to Kontsevich's thesis Intersection theory on the moduli space of curves and the matrix Airy function gives references to the solution to the problem for closed surfaces. Are there any other papers in the literature that discuss this problem?

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See mathoverflow.net/questions/87393/… and ldtopology.wordpress.com/2012/02/05/2306 for some recent discussion –  j.c. Feb 20 '12 at 23:09
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Ben Burton works on this problem. I think he's pretty convinced that even for spheres, the growth-rate is super-exponential. He certainly has numerical evidence but I suspect he might eventually have a proof. –  Ryan Budney Feb 20 '12 at 23:52
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Following up on Ryan's pointer, Burton's paper "The Pachner graph and the simplification of 3-sphere triangulations" (arxiv.org/abs/1011.4169) includes an algorithm for "isomorph-free generation of all 3-manifold triangulations of a given size." –  Joseph O'Rourke Feb 21 '12 at 0:27
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A related result is due to Gil Kalai. He showed that the number of triangulated manifolds (of any dimension) with $n$ labelled vertices is $2^{2^{.69424\cdots n(1+o(1))}}$. See springerlink.com/content/78044667x381777g. –  Richard Stanley Feb 21 '12 at 0:34
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@Richard: does this mean triangulations which are manifolds, or DISTINCT (topologically) manifolds? –  Igor Rivin Feb 21 '12 at 3:02

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