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Andrew Newman and I recently showed that random 2-dimensional hypertrees (Q-acyclic complexes) are aspherical, in Topology and geometry of random 2-dimensional hypertrees.

This is based, in part, on earlier work of Costa and Farber, who showed that for a wide range of parameter, the Linial–Meshulam random 2-complex (with complete 1-skeleton and independent 2-dimensional faces) is almost aspherical, in the sense that if you delete one face from every tetrahedron boundary, you get an aspherical complex.

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.

Several different models of random topological spaces have been studied by now, including several models of random simplicial complex.

Andrew Newman and I recently showed that random 2-dimensional hypertrees (Q-acyclic complexes) are aspherical, in Topology and geometry of random 2-dimensional hypertrees.

This is the first case I am aware of where a random space is shown to be aspherical with high probability. It is based, in part, on earlier work of Costa and Farber, who showed that for a wide range of parameter, the Linial–Meshulam random 2-complex (with complete 1-skeleton and independent 2-dimensional faces) is almost aspherical, in the sense that if you delete one face from every tetrahedron boundary, you get an aspherical complex.

Andrew Newman and I recently showed that random 2-dimensional hypertrees (Q-acyclic complexes) are aspherical, in Topology and geometry of random 2-dimensional hypertrees.

This is based, in part, on earlier work of Costa and Farber, who showed that for a wide range of parameter, the Linial–Meshulam random 2-complex (with complete 1-skeleton and independent 2-dimensional faces) is almost aspherical, in the sense that if you delete one face from every tetrahedron boundary, you get an aspherical complex.

Several different models of random topological spaces have been studied by now, including several models of random simplicial complex.

Andrew Newman and I recently showed that random 2-dimensional hypertrees (Q-acyclic complexes) are aspherical. https://arxiv.org/abs/2004.13572

This is the first case I am aware of where a random space is shown to apsherical with high probability. It is based, in part, on earlier work of Costa and Farber, who showed that for a wide range of parameter, the Linial-Meshulam random 2-complex (with complete 1-skeleton and independent 2-dimensional faces) is almost aspherical, in the sense that if you delete one face from every tetrahedron boundary, you get an aspherical complex.

Several different models of random topological spaces have been studied by now, including several models of random simplicial complex.

Andrew Newman and I recently showed that random 2-dimensional hypertrees (Q-acyclic complexes) are aspherical, in Topology and geometry of random 2-dimensional hypertrees.

This is the first case I am aware of where a random space is shown to be aspherical with high probability. It is based, in part, on earlier work of Costa and Farber, who showed that for a wide range of parameter, the Linial–Meshulam random 2-complex (with complete 1-skeleton and independent 2-dimensional faces) is almost aspherical, in the sense that if you delete one face from every tetrahedron boundary, you get an aspherical complex.