Embeddability between 3-manifolds is not stable

Giulio Belletti; Renaud Detcherry

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arXiv · 2604.22387

Embeddability between 3-manifolds is not stable

Giulio Belletti·Renaud Detcherry

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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-Bartoszyńska ideals, using the strong approximation for $\mathrm{SO}_3$-Witten-Reshetikhin-Turaev quantum representations of mapping class groups of surfaces.

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BibTeX RIS

Giulio Belletti, Renaud Detcherry. 2026-09-01. Embeddability between 3-manifolds is not stable. https://arxiv.org/abs/2604.22387

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