A universal non-embedding theorem for 3-manifolds

TL;DR

The paper proves that for any two compact oriented 3-manifolds, one can find a hyperbolic 3-manifold closely related to the first that does not embed into the second, using advanced quantum topology techniques.

Contribution

It establishes a universal non-embedding theorem for 3-manifolds employing Frohman--Kania-Bartoszyńska ideals and quantum representations.

Findings

Existence of hyperbolic 3-manifolds not embedding into certain 3-manifolds.

Construction of 3-manifolds with complex Frohman--Kania-Bartoszyńska ideals.

Application of strong approximation in quantum topology.

Abstract

We prove that given two compact oriented $3$-manifolds $N$ and $M,$ with $M$ satisfying only a mild hypothesis, there is a hyperbolic $3$-manifold $N'$ arbitrarily ``closely related'' to $N,$ and such that $N'$ does not embed in $M.$ For instance, as a weak version of our main theorem, if $M$ is a rational homology sphere then for any $k\geq 1$ the $3$-manifold $N'$ can be chosen to be $Y_k$-equivalent to $N.$ Our techniques rely on the construction of $3$-manifolds with complicated Frohman--Kania-Bartoszy\'nska ideals, using the strong approximation for $\mathrm{SO}_3$-Witten-Reshetikhin-Turaev quantum representations of mapping class groups of surfaces.