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Property P for nontrivial knots

Statement

Assume the Axiom of Choice. Let $K \subset S^{3}$ be a nontrivial smooth knot, and let $p, q$ be coprime integers with $q \neq 0$ . For the Dehn filling $Y_{p / q} = E_{K} (p / q)$ whose attached solid-torus meridian is identified with $p [μ] + q [λ]$ , the fundamental group $π_{1} (Y_{p / q})$ is nontrivial.

Statement references

Chosen knot exterior

Meridian and preferred longitude

Filling slope convention

First homology of knot fillings

Noncyclic SU(2) representations for numerator-one slopes

Kronheimer–Mrowka, Dehn surgery, the fundamental group and SU(2), Theorem 1

Kronheimer–Mrowka, Knots, sutures and excision, §7.8

Motegi, Dehn surgery on knots: survey, pp. 19–21 and 26

Kronheimer–Mrowka, published Theorem 1, p. 741

Facts & Assumptions

Given: Full AC, a nontrivial smooth knot $K \subset S^{3}$ , and a nonmeridional slope $p [μ] + q [λ]$ with $\gcd (p, q) = 1$ and $q \neq 0$ .

[F1]

The filling has $H_{1} (E_{K} (p / q); Z) ≅ Z / ∣ p ∣ Z$ for $p \neq 0$ , and $H_{1} (E_{K} (p / q); Z) ≅ Z$ for $p = 0$ . First homology of knot fillings

[F2]

For a path-connected space, $H_{1}$ is naturally the abelianization of its fundamental group. The first Hurewicz map is abelianization

[F3]

If $∣ p ∣ = 1$ and $q \neq 0$ , the filling group admits a noncyclic homomorphism to $S U (2)$ . The supplier derives this exact interface from the explicitly imported published Kronheimer–Mrowka Theorem 1, p. 741; its gauge-theoretic theorem is not claimed as locally reconstructed. Noncyclic SU(2) representations of numerator-one surgeries

[F4]

The filling is obtained by attaching a solid torus to $E_{K}$ along its entire boundary torus, with meridian identified with $p [μ] + q [λ]$ . Oriented slopes and knot Dehn filling

[F5]

The chosen exterior is the complement of the interior of the closed tubular disk bundle over the connected knot $K$ , with boundary $T_{K} = \partial N (K)$ . Knot exterior

[A1]

Full AC is the assertion that every set-indexed family of nonempty sets has a choice function. The Axiom of Choice

[L1]

Applying [A1] to the set of members of any countable family gives the countable choice function required by [F1]. The Axiom of Choice

Proof

1.1givenF4F5construct

By [F5], the chosen tubular neighbourhood is a disk bundle over the connected knot $K$ , so its boundary $T_{K}$ is a circle bundle over $K$ with connected fibres and is connected. Indeed, a separation of $T_{K}$ would, since every fibre is connected, split it into two saturated open sets; their projections are open by the local bundle charts, disjoint, and cover the connected base $K$ . Since $T_{K}$ is a connected smooth manifold, it is path-connected. The exterior $E_{K}$ is a compact smooth $3$ -manifold with boundary $T_{K}$ . Because $T_{K}$ is connected, it lies in one component of $E_{K}$ . Any other component is boundaryless, hence lies in $int E_{K}$ and is open in $S^{3}$ ; it is also closed in $S^{3}$ , since it is closed in $E_{K}$ and $E_{K}$ is closed in $S^{3}$ by [F5]. This contradicts connectedness of $S^{3}$ . Thus $E_{K}$ is connected and, being a smooth manifold, path-connected. By [F4], $Y_{p / q}$ is the union of this path-connected exterior and an attached path-connected solid torus along $T_{K}$ , so $Y_{p / q}$ is path-connected.

1.2givenF3algebra

If $∣ p ∣ = 1$ , the nonmeridional condition gives $q \neq 0$ , and [F3] supplies a homomorphism with noncyclic image. The trivial group has only the trivial, hence cyclic, image, so $π_{1} (Y_{p / q})$ is nontrivial.

2.1givenA1L1F1F2step 1.1algebra

By [A1] and [L1], the countable-choice hypothesis of [F1] follows from the Statement, and step 1.1 verifies its connected-exterior hypothesis. If $p = 0$ , [F1] gives $H_{1} (Y_{p / q}; Z) ≅ Z$ . If $∣ p ∣ \geq 2$ , it gives the nonzero finite group $Z / ∣ p ∣ Z$ . Choose any basepoint $y_{0}$ of the path-connected filling. By [F2], $H_{1} (Y_{p / q}; Z) ≅ π_{1} (Y_{p / q}, y_{0})^{a b}$ , so in either case the fundamental group is nontrivial.

3.1givenstep 2.1step 1.2algebra∎

Coprimality leaves exactly the cases $p = 0$ , $∣ p ∣ = 1$ , or $∣ p ∣ \geq 2$ , all covered by steps 2.1 and 1.2. Thus every nonmeridional filling has nontrivial fundamental group.

External dependency

The numerator-one case uses the external Kronheimer–Mrowka theorem through [F3], with its exact hypotheses and reading limits recorded in that supplier. The homology cases and the deduction of nontriviality from a noncyclic image are the local argument above.

Depends on

Used by

An S3 filling detects the meridian slope Lemma

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and SU(2), Mathematical Research Letters 11 (2004), 741–754, Theorem 1, p. 741 (standard reference, not scraped)
P. B. Kronheimer and T. S. Mrowka, Dehn surgery, the fundamental group and SU(2), Theorem 1 (standard reference, not scraped)
P. B. Kronheimer and T. S. Mrowka, Knots, sutures and excision, §7.8 (standard reference, not scraped)
Kimihiko Motegi, Dehn surgery on knots: survey, pp. 19–21 and 26 (standard reference, not scraped)