Topology of large random spaces
We consider simplicial complexes or manifolds which result from many independent random choices. As in the central limit theorem, their behaviour become predictable.
Complex networks. Usually theu represent pairwise interactions, but it has been realized that interactions between more actors need be taken
It will focus on classical problems in topology: Whitehead and Eilenberg-Ganea conjectures.
There are many different models, and I will survey some of them.
1. Random surfaces
Start with oriented triangles, even. Pick a pairing of sides (there are choices). The number of vertices is a random variables, hence so is the Euler characteristic. With probability , the surface is connected, its genus has expectation .
2. Random 3-manifolds
Start with two handlebodies. Launch random walk on MCG and glue handlebodies;
3. Random polygon spaces
Also known as moduli spaces of linkages.
Fix positive real vector and consider shapes of planar -gons with these edge lengths, i.e. polygons up to planar motions. For generic, , they form a manifold (since configurations with aligned edges do not occur) of dimension .
Small values of lead to classifications (Walker, Kapovich, Millson, Hausmann-Rodriguez. The number of types of manifolds obtained increases rapidly. So what happens as tends to infinity? Pick a probability measure on the unit simplex (normalize by ). Then becomes a random manifold.
We study the expected -th Betti number, for a large class of measures .
Theorem 1 (Kappeler-Dombry-Mazza-Farber)
up to an exponentially small error.
What would 3 dimensional
4. Random simplicial complexes
Linial and Meshulam, 2006.
Start with the complete graph on vertices. Add triangles independently at random, with probability . We study properties that holds asymptotically almost surely (AAS).
Theorem 2 (Linial-Meshulam, 2006) For , .
For , .
Theorem 3 (Kahle et al., 2011) For , , .
For , is infinite hyperbolic.
Theorem 4 (Costa-Farber) 2-torsion appears for and not for . Odd torsion never occurs.
Indeed, the icosaedral triangulation of real projective plane embeds in the random complex in this regime.
Proof uses uniform hyperbolicity. We show that the isoperimetric constant of random 2-complexes is bounded below in terms of only when .
4.1. Geometric and cohomological dimensions
They are known to coincide unless (possibly) and . Eilenberg-Ganea’s conjecture states that this case does not occur.
Theorem 5 (Costa-Farber) For , . For , . Because of 2-torsion, if .
Potentially, there could be counterexamples when .
Whitehead’s conjecture states that subcomplexes of aspherical 2-complexes are aspherical. In the Linial-Meshulam model, random 2-complexes are not aspherical, but have the following property: a subcomplex is aspherical iff any subcomplex with at most faces is aspherical, where . It follows that subcomplexes satisfy Whitehead’s conjecture.