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?



Random manifolds
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](http://www.matlas.math.csu.ru/) and [here](http://www.dm.unipi.it/pages/petronio/public_html/files/3D/c9/c9_census.html). 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](http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf) 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](http://it.wikipedia.org/wiki/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](http://www.math.uiuc.edu/~nmd/preprints/index.html) (with D. and W. Thurston, and H. Wong), [Juan Souto](http://www-personal.umich.edu/~jsouto/papers.html) 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](http://en.wikipedia.org/wiki/Kirby_calculus) to represent 4-manifolds and define a complexity by counting the number of crossings in the diagrams as I did [here](http://arxiv.org/abs/math.GT/0412511). 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](http://arxiv.org/abs/math/0701269) 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.


