TITLE: Is being simply connected very rare?
QUESTION [27 upvotes]: Essentially, my question is how strong a restriction it is to be simply connected.
Here is a way of making this precise: Let's say we want to count simplicial complexes (of dimension 2, though that does not matter much, any fixed dimension is fine) on N simplices that are subject to the following restrictions:
A: every vertex is contained only in a bounded number of simplices (say, 10000).
B: the complex is simply connected.
So properly: How many distinct complexes like this are there? In fact, I only want a rough answer: is it exponential in N, or is it superexponential. Note that if I remove either restriction, the answer is superexponential.

REPLY [6 votes]: The following does not answer your question, but adding just in case it is helpful.
If you weaken "simply connected" to $H_1(\Delta, \mathbb{Q}) = 0$, and weaken "every vertex is in a bounded number of edges" to "the average number of triangles containing an edge is bounded", then there are super-exponentially many such complexes. This is an old result of Kalai, on enumerating $\mathbb{Q}$-acyclic complexes. https://link.springer.com/article/10.1007/BF02804017
There are $\exp( cn^2)$ such complexes, where $c>0$ is some constant not depending on $n$.
For the latest on enumerating $\mathbb{Q}$-complexes, see Linial and Peled:
https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.20841
Andrew Newman and I recently showed that a typical $\mathbb{Q}$-acyclic complex (according to a natural `determinantal measure') is not simply connected. With high probability, the fundamental group is a hyperbolic group of cohomological dimension 2.
https://arxiv.org/abs/2004.13572