LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Boundary fiber slopes are unique outside the small Seifert exceptions

Statement

Assume full AC. Let $M$ be a compact connected orientable Seifert-fibered $3$ -manifold other than $T^{2} \times I$ and the orientable twisted $I$ -bundle over the Klein bottle. Let $A$ be a finite family of connected, two-sided essential annuli in $M$ such that every boundary torus of $M$ contains a boundary circle of some annulus in $A$ . Then any two Seifert fibrations of $M$ induce isotopic regular-fiber slopes on each boundary torus.

Statement references

Allen Hatcher, Notes on Basic 3-Manifold Topology, Lemma 1.15

Walter Neumann and Gadde Swarup, Canonical decompositions of 3-manifolds

Facts & Assumptions

Given: Full AC, a compact connected orientable Seifert-fibered $3$ -manifold $M$ outside the two orientable exceptional manifolds in the Statement, a finite family $A$ of essential annuli meeting every boundary torus, and two Seifert fibrations of $M$ .

[A1]

Full AC is the axiom that every set-indexed family of nonempty sets has a choice function (The Axiom of Choice, Definition).

[F1]

An essential smooth boundary-preimage annulus in a Seifert piece can be isotoped through smooth embeddings to a vertical annulus for a given fibration; for neat annuli this is a smooth ambient isotopy, with a fibration change needed only in the exceptional underlying manifolds (Essential annuli in Seifert pieces can be made vertical, Statement).

[F2]

The fibered solid-torus models and regular and exceptional fibers are defined in Incompressibility, Seifert fibrations, and JSJ systems. On a boundary torus these models give a circle bundle over a circle. Its fibers have a common unoriented isotopy class, called the regular-fiber slope: local bundle charts carry fibers to fibers along a base path. A vertical annulus has fiber circles as its boundary. Restricting a proper embedding isotopy to a boundary circle preserves that circle's isotopy class on the boundary torus. These are the conventions used below, not additional definitions asserted by the cited item.

Proof

1.1A1F1givenconstruct

Fix an annulus $A \in A$ whose boundary meets a chosen boundary torus $T$ . Apply [F1] to the first fibration. Since $M$ is outside the exceptional cases, a smooth embedding isotopy makes $A$ vertical without changing that fibration. Its boundary circle on $T$ therefore represents the regular-fiber slope of the first fibration.

2.1F1F2step 1.1construct∎

Apply [F1] to the second fibration. Again the annulus becomes vertical by a smooth embedding isotopy without replacing that second fibration. Restricting this embedding isotopy to the boundary gives a smooth isotopy of boundary circles on $T$ , so it preserves the free isotopy class of its boundary circle on $T$ , so that circle also represents the regular-fiber slope for the second fibration. The two primitive slopes are therefore isotopic on $T$ . Repeating this argument for each boundary torus, using the annulus family $A$ , proves the claim for all boundary components.

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Sources

Allen Hatcher, Notes on Basic 3-Manifold Topology (standard reference, not scraped)
Walter Neumann and Gadde Swarup, Canonical decompositions of 3-manifolds (standard reference, not scraped)