## Notes of Michael Farber’s Southampton lecture 30-03-2017

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.

0.1. Motivations

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

Pippenger-Schleich 2006.

Start with ${n}$ oriented triangles, ${n}$ even. Pick a pairing of sides (there are ${(3n-1)!!}$ choices). The number ${h}$ of vertices is a random variables, hence so is the Euler characteristic. With probability ${1-5/18n+..}$, the surface is connected, its genus has expectation ${\mathop{\mathbb E}(g)=n/4+..}$.

2. Random 3-manifolds

Dunfield-Thurston 2006.

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 ${l}$ and consider shapes of planar ${n}$-gons with these edge lengths, i.e. polygons up to planar motions. For generic, ${l}$, they form a manifold ${M_l}$ (since configurations with aligned edges do not occur) of dimension ${n-3}$.

Small values of ${n}$ lead to classifications (Walker, Kapovich, Millson, Hausmann-Rodriguez. The number of types of manifolds obtained increases rapidly. So what happens as ${n}$ tends to infinity? Pick a probability measure ${\mu_n}$ on the unit simplex (normalize ${l}$ by ${\sum l_i=1}$). Then ${M_l}$ becomes a random manifold.

We study the expected ${p}$-th Betti number, for a large class of measures ${\mu_n}$.

Theorem 1 (Kappeler-Dombry-Mazza-Farber)

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(b_p)=\begin{pmatrix}n-1\\p\end{pmatrix}, \end{array}$

up to an exponentially small error.

What would 3 dimensional

4. Random simplicial complexes

Linial and Meshulam, 2006.

Start with the complete graph on ${n}$ vertices. Add triangles independently at random, with probability ${p}$. We study properties that holds asymptotically almost surely (AAS).

Theorem 2 (Linial-Meshulam, 2006) For ${p>2\log n/n}$, ${b_1=0}$.

For ${p<2\log n/n}$, ${b_1\not=0}$.

Theorem 3 (Kahle et al., 2011) For ${p=n^\alpha}$, ${\alpha<1/2}$, ${\pi_1=1}$.

For ${\alpha<1/2}$, ${\pi_1}$ is infinite hyperbolic.

Theorem 4 (Costa-Farber) 2-torsion appears for ${\alpha\in(1/2,3/5)}$ and not for ${\alpha>3/5}$. 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 ${\alpha}$ only when ${\alpha>1/2}$.

4.1. Geometric and cohomological dimensions

They are known to coincide unless (possibly) ${gd=3}$ and ${cd=2}$. Eilenberg-Ganea’s conjecture states that this case does not occur.

Theorem 5 (Costa-Farber) For ${\alpha\in(3/5,1)}$, ${gd=cd=2}$. For ${\alpha>1}$, ${gd=cd=1}$. Because of 2-torsion, ${gd=cd=\infty}$ if ${\alpha\in(1/2,3/5)}$.

Potentially, there could be counterexamples when ${\alpha=1}$.

4.2. Asphericity

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 ${Y'\subset Y}$ is aspherical iff any subcomplex ${Y''\subset Y'}$ with at most ${2/\epsilon}$ faces is aspherical, where ${\epsilon=\alpha-\frac{1}{2}}$. It follows that subcomplexes ${Y'}$ satisfy Whitehead’s conjecture.