TITLE: Are "most" spaces aspherical?
QUESTION [25 upvotes]: There's a heuristic idea that "most" closed manifolds $M$ are aspherical (i.e. $\pi_{\geq 2}(M) = 0$). Does this heuristic extend usefully to all spaces -- or at least to all finite CW complexes?
To make this question more precise, I should say something about in what sense "most" manifolds are aspherical. I don't know a lot about this heuristic, but here's where I'm coming from:

It's true in low dimensions: trivially in 0 or 1 dimensions, and by classification of surfaces in 2 dimensions. In 3 dimensions, I've heard it said that part of the upshot of Thurston's Geometrization Conjecture is that "most" 3-manifolds are hyperbolic, and in particular aspherical.
There's some discussion of this heuristic in this survey article of Luck (at the end).

How do things look if we think about CW complexes? Well, every 0 or 1-dimensional CW complex is aspherical. And the Kan-Thurston theorem tells us that every space is homology-equivalent to an aspherical space. But it's really not clear to me whether I should think of "most" spaces as being aspherical.

REPLY [7 votes]: One way to think about whether "most" spaces are aspherical is measure-theoretically. Here a few examples and non-examples of random topological spaces being aspherical.
Examples

Presentation complexes of density random groups are aspherical for every density $d < 1/2$, and for density $d> 1/2$ these groups collapse, so this is essentially the entire interesting range of parameter.

Random 3-manifolds. N. Dunfield and W. Thurston introduced a model for random 3-manifold using a random walk on the mapping class group to generate a random Heegaard splitting. Joseph Maher showed that these random 3-manifolds are hyperbolic with high probability, so in particular their universal cover hyperbolic space $\mathbb{H}^3$ is contractible.

Let $Y(n,p)$ denote the Linial-Meshulam random 2-dimensional simplicial complex. This complex has vertex set $[n]$, complete $1$-skeleton, and each $2$-face appears independently with probability $p=p(n)$. Costa and Farber showed that if $p \ll n^{-1/2 - \epsilon}$, $Y(n,p)$ is nearly aspherical, in the following sense: if you delete one 2-face from every sufficiently small sphere, pinched sphere (along a vertex, or an edge), or projective plane, the resulting complex is aspherical. It is easy to check that the expected number of these local obstructions is much smaller than the expected total number of 2-faces. So you can delete one face from each one to result in an aspherical complex and have almost all the faces remaining.

In a similar spirit, Andrew Newman and I recently showed that random 2-dimensional hypertrees (random Q-acyclic complexes) according to a certain "determinantal measure" are are aspherical, in Topology and geometry of random 2-dimensional
hypertrees.


Non-examples

If one considers the random 2-complex $Y(n,p)$ with $p \ge (\gamma n)^{-1/2}$ and $\gamma = 4^4 / 3^3$, Luria and Peled showed that $Y(n,p)$ is simply connected, so at that this point is homotopy equivalent to a bouquet of $2$-spheres, and is not aspherical. It is not "nearly aspherical" in the sense of Costa and Farber either, so there is a phase transition near $p = n^{-1/2}$ from nearly aspherical to not.

What if we just count homotopy types of simplicial complexes on $n$ vertices? Andrew Newman showed that there are doubly exponentially many homotopy types, at least $2^{2^{0.02n}}.$ On the other hand, there are most $2^{n \choose 3}$ different fundamental groups, a much smaller number, so somehow "most" homotopy types of simplicial complexes can not be aspherical.