MathOverflow

In the paper linked here, I used monte carlo simulations to measure that the average distance between two random points in a hypersphere converges to Sqrt[N]/e for larger N. Can this be confirmed? Is this finding already known?

I am seeking advice regarding a submission to a Top 5 mathematics journal (analytic number theory).
Timeline:
September 2024: Article submitted, immediate acknowledgment from editor E1

Prove that the functor $J\rightarrow S(|J|)$ is conservative but does not satisfy the RLP with respect to $\Delta^1\rightarrow J$ where $J$ is the walking isomorphism. Well-known fact: $|J|$ is the infinity sphere $S^{\infty}$.
It is obvious that the functor $J\rightarrow S(|J|)$ is conservative because $J$ and $S(|J|)$ are both Kan complexes. To understand the absence of RLP we should consider the diagram...

The Lusternik-Schnirelmann category of closed 3-manifolds is fully determined by their fundamental group (Gómez-Larrañaga & González-Acuña). In dimension 2, topological complexity is also fully classified for closed surfaces.
Therefore, I wonder if there exists a complete classification for the topological complexity of closed 3-manifolds. I presume not, since topological complexity is usually significantly harder to compute....