A universal non-embedding theorem for 3-manifolds

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.

• The paper introduces quantum FKB ideals and profinite techniques to systematically obstruct embeddings in 3-manifolds.
• It employs mapping class group representations and Yₖ-equivalence to construct hyperbolic manifolds that resist embedding.
• Probabilistic analysis yields explicit non-embedding bounds, challenging the intuition that embeddability is a generic property.

Universal Non-Embedding for 3-Manifolds: Quantum and Profinite Techniques

Problem Formulation and Classical Obstructions

The paper "A universal non-embedding theorem for 3-manifolds" (2604.22387) provides a systematic investigation of the embeddability problem for compact oriented 3-manifolds. Given two manifolds $N$ and $M$, it addresses the fundamental question: under which circumstances does $N$ embed inside $M$? The authors survey classical obstructions derived from homology, geometric decomposition, and hyperbolic invariants. These invariants yield necessary constraints, such as:

• Homological bounds for the first Betti numbers (Lemma 1.3 of [Ton11]),
• Gromov norm comparisons under Dehn fillings (e.g., $||M|| \leq ||N||$ for irreducible, atoroidal manifolds with incompressible boundary),
• Casson invariant congruences in knot complements,
• Constraints from JSJ decompositions for connected and irreducible manifolds.

Despite algorithmic advances (cf. embeddability in $S^3$ [MST18]), these classical and algorithmic methods are insufficient for resolving embeddability outside specific classes (e.g., homology spheres, manifolds with low-genus boundary). This motivates the introduction of quantum invariants and profinite-completion techniques to more robustly obstruct embeddings.

Quantum Obstructions: Frohman–Kania-Bartoszyńska Ideals

The principal machinery in this work is the utilization of Frohman–Kania-Bartoszyńska (FKB) ideals, which leverage the Witten–Reshetikhin–Turaev (WRT) topological quantum field theory (TQFT) invariants. For each odd prime $p$, the FKB ideal $I_p(N)$ is generated in $Z[p^{-1}, \zeta_p]$ (with $\zeta_p$ a primitive $M$0-th root of unity) from WRT invariants of closed 3-manifolds containing $M$1 as a submanifold. Embedding $M$2 into $M$3 requires $M$4, so constructing $M$5 with $M$6 quantum obstructs the embedding.

To produce such $M$7, the authors invoke strong approximation results for quantum representations of the mapping class group $M$8 of a surface $M$9, specifically at prime levels for $N$0 WRT TQFT. These representations, at infinitely many primes $N$1, surject onto $N$2 for $N$3 [MR12], making randomized Torelli twists along Heegaard surfaces powerful in constructing non-embedding hyperbolic manifolds.

Profinite Equivalence and Rigidity

Alongside quantum obstructions, the paper explores equivalence via profinite completions of fundamental groups. By employing Dijkgraaf-Witten TQFTs, the authors establish that "closeness" can be formalized through profinite convergence: constructing sequences $N$4 of hyperbolic 3-manifolds whose fundamental groups $N$5 converge (in the sense of finite quotients) to $N$6. Profinite rigidity results for hyperbolic manifolds (cf. [Liu23], [Xu25]) and Seifert fibered spaces ([Wilkes]) imply that for sufficiently "close" manifolds, embeddability is highly restricted.

$N$7-Equivalence, $N$8-Equivalence, and Probabilistic Non-Embeddability

A salient technical innovation is the use of $N$9-equivalence: two manifolds $M$0 are $M$1-equivalent if $M$2 arises from $M$3 by cutting along a surface and regluing via mapping class group elements in $M$4, the $M$5-th term of the lower central series of the Torelli group. This provides a filtration quantifying the "closeness" of manifolds. The authors further refine this with $M$6-equivalence, incorporating additional constraints from Dehn twist powers and the kernels of Dijkgraaf-Witten representations. Notably, they conjecture all compact oriented 3-manifolds are "very good," i.e., $M$7-good for all but finitely many primes.

The main theorems assert:

• For any $M$8 and any compact oriented $M$9 satisfying mild conditions ("very good"), there exists a sequence $||M|| \leq ||N||$0 of hyperbolic manifolds, $||M|| \leq ||N||$1-equivalent and profinitely converging to $||M|| \leq ||N||$2, such that none embed in $||M|| \leq ||N||$3.
• For probabilistic constructions in the Dunfield–Thurston random walk model, a positive proportion of $||M|| \leq ||N||$4-equivalent random manifolds fail to embed in $||M|| \leq ||N||$5.

Explicit numerical bounds are given: the lim inf of non-embedding probability $||M|| \leq ||N||$6 satisfies

$||M|| \leq ||N||$7

where $||M|| \leq ||N||$8 is a chosen prime and $||M|| \leq ||N||$9, $S^3$0 are dimensions of TQFT spaces.

Technical Implications and Contradictory Claims

The paper's central claim is contradictory to the intuition that embeddability is a generic property for close manifolds: for nearly any compact oriented $S^3$1 and any $S^3$2, one can construct arbitrarily nearby hyperbolic $S^3$3 that quantum-obstruct and profinitely-obstruct embedding in $S^3$4. Furthermore, it is shown that for any $S^3$5 (resp., countable infinity), one can find $S^3$6 (resp., infinitely many) manifolds hyperbolic and $S^3$7-equivalent to $S^3$8 but pairwise non-embeddable (Corollary).

The probabilistic lower bounds imply these non-embedding phenomena are not rare but occur with explicitly computable positive density among random walks in mapping class group subgroups.

Practical and Theoretical Outlook, Future Perspectives

Quantum obstructions, combined with profinite rigidity and mapping class group representation theory, yield a potent arsenal for non-embedding in 3-manifold topology. The results suggest that classical topological and homological invariants are insufficient for full classification of embeddability and motivate the widespread use of quantum and profinite invariants. Moreover, the formalism of $S^3$9-equivalence, along with random walk models and TQFT representation theory, could be adapted to broader settings in higher-dimensional topology and geometric group theory.

Future developments may include computational algorithms for explicit FKB ideal computations, systematic classification of very good manifolds, and further refinement of TQFT-based invariants. The probabilistic techniques are likely adaptable to other random manifold models, with implications for topological sampling and statistical topology.

Conclusion

This paper establishes that, for virtually any pair of compact oriented 3-manifolds $p$0 and $p$1 (with mild constraints, satisfied by rational homology spheres), quantum and profinite methods enable the construction of arbitrarily near hyperbolic $p$2 which cannot embed in $p$3, irrespective of classical topological proximity. The strong numerical and probabilistic results reinforce the non-genericity of embeddability among 3-manifolds. The techniques, grounded in TQFT and mapping class group representations, extend the landscape of embedding obstructions and herald continued interplay between quantum topology and classical manifold theory (2604.22387).