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In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its expected topology where "expected" makes sense because there are sensible measures on the space of hypersurfaces. See Welschinger-Gayet http://arxiv.org/abs/1107.2288 and http://arxiv.org/abs/1005.3228 for recent progress on such questions (e.g. what is the expected Betti number of a random real hypersurface of degree d?).

In geometry more generally you might want to make statements like "a general manifold is aspherical" or "a general manifold has positive simplicial volume". It seems difficult to construct sensible measures for which these questions have answers: to talk about probability you need some way of producing manifolds (and then distinguishing them) in a random way.

However, Cheeger proved that for fixed L there is only a finite set $D_L$ of diffeomorphism classes of manifold admitting a Riemannian metric with sectional curvatures bounded above in norm by L, volume bounded below by 1/L and the diameter bounded above by L (see the first theorem in Peters "Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds" http://www.reference-global.com/doi/abs/10.1515/crll.1984.349.77). This means that you can ask questions like "what is the average total Betti number of a manifold in $D_L$" and "how does this increase with L?" (are there exponential upper bounds?), or one can try to make sense of the limit as $L\rightarrow\infty$ of the proportion of manifolds in $D_L$ with zero simplicial volume.

Are there any known concrete answers to these questions, or other formulations of the questions which lead to answers?

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Nice question! :D –  David Roberts Jul 19 '11 at 7:25
    
For one notion of random manifolds, see the earlier MO question, "Essentially one random metric on $\mathbb{S}^2$?": mathoverflow.net/questions/44759 –  Joseph O'Rourke Jul 19 '11 at 10:37
    
Matthew Kahle has written on the topic: "Random geometric complexes": arxiv.org/abs/0910.1649 –  Joseph O'Rourke Jul 19 '11 at 10:42
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See also Betti numbers of random manifolds by Farber and Kappeler, where they obtain random manifolds as configuration spaces of random planar linkages. intlpress.com/HHA/v10/n1/a8 –  Matthew Kahle Jul 28 '11 at 23:03
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4 Answers

up vote 17 down vote accepted

To define a random $n$-manifold you typically need to define a complexity on the set $\mathcal M_n$ of all $n$-manifolds you want to consider, which satisfies a finiteness property: for every $k$, there are only a finite number of manifolds having complexity at most $k$. There are various ways to do this in a combinatorial framework.

The minimum number of simplexes $t(M)$ in a triangulation of $M$ is a natural example. I don't think there is much known on random manifolds in this context. Since $t$ is roughly subadditive on connected sums, a random manifold might be a connected sum of many manifolds and would hence be far from being aspherical. In dimension 3 one may restrict to irreducible manifolds. It might seem reasonable to expect that in this conext a random 3-manifold is hyperbolic, but this is still unknown. The first segment $t(M)\leq 11$ shows up a huge number of graph manifolds, see the tables here and here. The hard point here is that it is very difficult to estimate such complexity from below, even for simple manifolds like lens spaces. As an example, the number of hyperbolic manifolds having complexity smaller than $t$ grows more than exponentially with $t$, and the number of lens spaces is conjecturally roughly $2^t$. However, I think we are not yet able to say that the number of lens spaces does not grow more than exponentially. Another fact that shows our ignorance in this conext is the following: we still don't know if the number of triangulation of the three-sphere grows exponentially with the number of tetrahedra, see Gromov's recent questions.

As pointed by Jean-Marc Schlenker, there is another natural complexity in dimension 3 which is easier to treat and gives various nice results: we can consider the smaller set $\mathcal M_3^g$ of all 3-manifolds decomposing into two genus-$g$ handlebodies. A manifold there is determined by an element of the mapping class group MCG of the surface $\Sigma_g$ of genus $g$, and after fixing a set of generators for MCG we can define the complexity of one such element as the minimum lenght of a word which represents it. Many results have been obtained in this context by Nathan Dunfield (with D. and W. Thurston, and H. Wong), Juan Souto and others.

In dimension 4 there are various combinatorial notions of complexity one may use, but they are difficult to treat. One can use Kirby diagrams to represent 4-manifolds and define a complexity by counting the number of crossings in the diagrams as I did here. It is very easy to produce doubles of 2-handlebodies in this context (draw a random diagram and add small 0-framed unknots encircling every component) and to perform blow-ups, so I suppose that in this context most manifolds are of this type: these manifolds are never aspherical and have simplicial norm zero. In this context, Auckly has proved that there is a big discrepancy between homeomorphic and diffeomorphic classes of manifolds: the number of simply connected manifolds of complexity up to $n$ seen up to homeomorphism grows like $n^2$, whereas the number of simply connected manifolds seen up to diffeomorphism grows more than polinomially.

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Thanks, this is fascinating stuff. –  Jonny Evans Jul 20 '11 at 15:59
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Let me add a few remarks to the very good answers above. Indeed there are two important papers by Nathan M. Dunfield and William P. Thurston with a definition of random 3-manifolds. The first paper Finite covers of random 3-manifolds discuss the advantages of the proposed model compared to other models. The paper also compares the situation with earlier notions and results about random groups. The virtual Haken conjecture is the central motivating question. (Much was recently done on this and related questions.) The second paper by them is The virtual Haken conjecture: Experiments and examples. These papers are related to earlier papers on "random groups" of various kind, and turned out to be connected to notions of "growth" of groups, to properties T and $\tau$, and to be related to sieve computations (see, e.g., this paper by Kowalski).

Another natural model of random manifolds is based on random triangulations. We have to remember that as the dimension grows it is harder and harder being a manifold. (And its also hard to be orientable.)

One possibility is to consider triangulations of d-manifolds with n vertices and alternatively we can consider triangulations with T facets (namely T d-dimensional simplices). These are quite different models. There are obvious guesses that we cannot prove: 1) it is likely that most triangulated 3-homology spheres are not spheres, (It is also plausible that most triangulated spheres are not PL, and that most PL spheres re not shellable.)

2) It is plausible that the dual-graph of a triangulated sphere (perhaps also homology-sphere) determined the entire triangulation. This is not known,

3) There is a simple computation for what the maximum Euler characteristic of a d-dimensional manifold with n vertices should be, when d is even. It is likely that most d-manifolds attain or come close to this maximum. This is not known and for dimension > 2 there are only finite example of manifolds attaining the proposed maximum.

4) It is a big mystery how many triangulations of d spheres there are with T facets. (exponential in T or more). This can also be asked for manifolds.

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Along the lines of triangulations, there's this nice site of Frank Lutz which collects a good deal of data arising from fairly impressive work on enumeration: math.tu-berlin.de/diskregeom/stellar –  j.c. Jul 20 '11 at 1:10
    
A little comment on orientability: although it is reasonable to expect that with big numbers of tetrahedra orientability should be a rare property, it turns out however that in dimension 3 non-orientable 3-manifolds seem more sporadic than orientable ones at least for small numbers of tetrahedra (say up to 10), see maths.uq.edu.au/~bab/papers/2011-genus-issac.html –  Bruno Martelli Oct 23 '11 at 15:49
    
When you have small number of tetrahedra the only manifolds you see (without boundary) are spheres. (And with boundary, balls and spheres). So you need larger number of simlices to see more complicated things but when the number of simplices increase we can still expect orientability is rare. –  Gil Kalai Oct 23 '11 at 21:23
    
I meant 'singular', not 'simplicial' triangulations: you just take N tetrahedra and glue the faces in pair according to some simplicial pairing. With such triangulations, you already see interesting manifolds even with 1 tetrahedron (the Gieseking manifold, for instance). With <=11 tetrahedra we get 13.399 closed irreducible orientable 3-manifolds, but only 366 non-orientable ones. I think that this is due to the fact that non-orientable manifolds have non-trivial second homology and hence contain an "incompressible" surface, so they tend to need more tetrahedra. –  Bruno Martelli Dec 1 '11 at 15:18
    
hmm, I bet that the non-orientable ones will be a very clear majority when you take n<=20 –  Gil Kalai Dec 1 '11 at 23:56
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In a slightly different direction there are natural (but probably not uniquely defined notions of random surfaces, and also of random 3-manifolds, as in this paper by Dunfield and Thurston, based on Heegaard splittings. For those notions of randomness no curvature hypothesis is needed. But it does not seem obvious how to generalize them in higher dimension.

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You may also want to look at the work of Nati Linial and Roy Meshulam, Eric Babson, Matt Kahle, et al, on random simplicial complexes. You can define a "random manifold" by conditioning your complex to be a manifold, though no one has succeeded in doing this in anything like a useful way.Still, this is a very natural generalization of Erdos-Renyi random graphs, and is somewhat tractable.

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