A universal non-embedding theorem for 3-manifolds

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 $N^{\mathrm{\prime }}$ arbitrarily ``closely related’’ to $N,$ and such that $N^{\mathrm{\prime }}$ does not embed in $M\mathrm{.}$ For instance, as a weak version of our main theorem, if $M$ is a rational homology sphere then for any $k\ge 1$ the $3$-manifold $N^{\mathrm{\prime }}$ can be chosen to be $Y_k$-equivalent to $N\mathrm{.}$ Our techniques rely on the construction of $3$-manifolds with complicated Frohman–Kania-Bartoszyńska ideals, using the strong approximation for ${\mathrm{S}\mathrm{O}}_3$-Witten-Reshetikhin-Turaev quantum representations of mapping class groups of surfaces.