The Akbulut–Kirby conjecture on 0-surgery and concordance

Akbulut and Kirby conjectured that a knot in S³ is determined up to concordance by its 0-surgery: if the 0-framed surgeries on two knots give the same 3-manifold, then the knots are concordant (Kirby's problem 1.19). Yasui disproved it, constructing knots with the same 0-surgery that are not concordant for any choice of orientations.

The conjecture is recorded here as a proposition, TauCeti.AkbulutKirbyConjecture, about the knots of TauCeti.SmoothCircleEmbedding in the unit sphere S³ of ℝ⁴, and Yasui's theorem as its negation, TauCeti.YasuiTheorem; neither is proved.

• The 0-surgery on a knot K with solid torus neighbourhood Φ is a Dehn filling (TauCeti.DehnFilling) of the exterior of K along the longitude of the preferred framing (TauCeti.IsSolidTorusNeighborhood.preferredFramedBoundaryTorus), the longitude that is null-homologous in the knot complement.
• "The same 3-manifold" means an orientation-preserving homeomorphism. The filled spaces are topological 3-manifolds, oriented by homological orientations (TauCeti.HomologicalOrientation) compatible with one orientation of S³ on the open exteriors (TauCeti.DehnFilling.IsOrientationCompatible). Orientations matter: the 0-surgeries on a knot and on its mirror image are homeomorphic by an orientation-reversing map, while the right-handed trefoil and its mirror image are not concordant, as their signatures differ. The sphere S³ is simply connected (TauCeti.simplyConnectedSpace_sphere_euclideanSpace), so it has orientations (TauCeti.HomologicalOrientation.nonempty_of_simplyConnectedSpace); on a connected filled space with nonempty open exterior (always the case for a knot in S³) the compatible orientation is unique (TauCeti.DehnFilling.IsOrientationCompatible.unique).
• Concordance is smooth concordance of the parametrized knots (TauCeti.SmoothEmbedding.Concordant), and the conclusion allows reversing the orientation of one knot (TauCeti.SmoothCircleEmbedding.reverse). The 0-surgery does not see the orientation of the knot, and there are knots that are not concordant to their reverses (Livingston), so a conclusion about oriented concordance would fail for a knot and its reverse.

Main definitions

• TauCeti.AkbulutKirbyConjecture: knots in S³ with orientation-preservingly homeomorphic 0-surgeries are smoothly concordant, up to orientation.
• TauCeti.YasuiTheorem: the Akbulut–Kirby conjecture is false.

Main results

• TauCeti.AkbulutKirbyConjecture.concordant_or_concordant_reverse: applying the conjecture.
• TauCeti.yasuiTheorem_iff: Yasui's theorem spelled out, as the existence of two knots with orientation-preservingly homeomorphic 0-surgeries that are not concordant for any orientations.

References

• R. Kirby (ed.), Problems in Low-Dimensional Topology, Problem 1.19, in Geometric Topology, AMS/IP Stud. Adv. Math. 2.2 (1997).
• K. Yasui, Corks, exotic 4-manifolds and knot concordance, arXiv:1505.02551 (2015).
• C. Livingston, Knots which are not concordant to their reverses, Quart. J. Math. Oxford Ser. (2) 34 (1983), 323–328.
• D. Rolfsen, Knots and Links, Publish or Perish (1976), Chapter 9 (Dehn surgery and the 0-framing).

def TauCeti.AkbulutKirbyConjecture : Prop

The Akbulut–Kirby conjecture, Kirby's problem 1.19: if the 0-surgeries on two knots K K' : S¹ → S³ are homeomorphic by an orientation-preserving homeomorphism, then K is smoothly concordant to K' or to its reverse.

The 0-surgery on K, for a solid torus neighbourhood Φ, is a Dehn filling F of its exterior along the longitude of the preferred framing (TauCeti.IsSolidTorusNeighborhood.preferredFramedBoundaryTorus), defined because the reduced integral homology of S³ vanishes in degrees 1 and 2 (TauCeti.isZero_reducedSingularHomologyFunctor_one_sphere_euclideanSpace_fin_four and TauCeti.isZero_reducedSingularHomologyFunctor_two_sphere_euclideanSpace_fin_four). The filled spaces Y, Y' are topological 3-manifolds oriented by the orientations oY, oY' compatible with an orientation o of S³, and the homeomorphism e : Y ≃ₜ Y' preserves orientation when it pulls oY' back to oY. Yasui proved that the conjecture is false (TauCeti.YasuiTheorem).

def TauCeti.YasuiTheorem : Prop

Yasui's theorem: the Akbulut–Kirby conjecture is false. There are knots in S³ whose 0-surgeries are homeomorphic by an orientation-preserving homeomorphism but which are not smoothly concordant, even after reversing the orientation of one of them (TauCeti.yasuiTheorem_iff).

theorem TauCeti.AkbulutKirbyConjecture.concordant_or_concordant_reverse (hAK : AkbulutKirbyConjecture) (o : HomologicalOrientation (↑(Metric.sphere 0 1)) (EuclideanSpace ℝ (Fin 3)) 3) {K K' : SmoothCircleEmbedding (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) ↑(Metric.sphere 0 1)} {Φ Φ' : SolidTorus → ↑(Metric.sphere 0 1)} (h : IsSolidTorusNeighborhood (⇑K) Φ) (h' : IsSolidTorusNeighborhood (⇑K') Φ') (h₁ : CategoryTheory.Limits.IsZero ((reducedSingularHomologyFunctor (↧ℤ) 1).obj ↧↑(Metric.sphere 0 1))) (h₂ : CategoryTheory.Limits.IsZero ((reducedSingularHomologyFunctor (↧ℤ) 2).obj ↧↑(Metric.sphere 0 1))) {Y Y' : Type} [TopologicalSpace Y] [T2Space Y] [ChartedSpace (EuclideanSpace ℝ (Fin 3)) Y] [TopologicalSpace Y'] [T2Space Y'] [ChartedSpace (EuclideanSpace ℝ (Fin 3)) Y'] (F : DehnFilling h (IsSolidTorusNeighborhood.preferredFramedBoundaryTorus h₁ h₂ h).longitude Y) (F' : DehnFilling h' (IsSolidTorusNeighborhood.preferredFramedBoundaryTorus h₁ h₂ h').longitude Y') {oY : HomologicalOrientation Y (EuclideanSpace ℝ (Fin 3)) 3} {oY' : HomologicalOrientation Y' (EuclideanSpace ℝ (Fin 3)) 3} (hF : F.IsOrientationCompatible o oY) (hF' : F'.IsOrientationCompatible o oY') (e : Y ≃ₜ Y') (he : oY'.comap ⋯ = oY) : SmoothEmbedding.Concordant K K' ∨ SmoothEmbedding.Concordant K K'.reverse

Applying the Akbulut–Kirby conjecture: knots in S³ whose 0-surgeries, oriented compatibly with one orientation of S³, are homeomorphic by an orientation-preserving homeomorphism are smoothly concordant up to orientation. The 0-surgeries are read in the preferred framing defined from any proofs that the reduced integral homology of S³ vanishes in degrees 1 and 2.

theorem TauCeti.yasuiTheorem_iff : YasuiTheorem ↔ ∃ (o : HomologicalOrientation (↑(Metric.sphere 0 1)) (EuclideanSpace ℝ (Fin 3)) 3) (K : SmoothCircleEmbedding (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) ↑(Metric.sphere 0 1)) (K' : SmoothCircleEmbedding (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) ↑(Metric.sphere 0 1)) (Φ : SolidTorus → ↑(Metric.sphere 0 1)) (Φ' : SolidTorus → ↑(Metric.sphere 0 1)) (h : IsSolidTorusNeighborhood (⇑K) Φ) (h' : IsSolidTorusNeighborhood (⇑K') Φ') (Y : Type) (Y' : Type) (x : TopologicalSpace Y) (x_1 : T2Space Y) (x_2 : ChartedSpace (EuclideanSpace ℝ (Fin 3)) Y) (x_3 : TopologicalSpace Y') (x_4 : T2Space Y') (x_5 : ChartedSpace (EuclideanSpace ℝ (Fin 3)) Y') (F : DehnFilling h (IsSolidTorusNeighborhood.preferredFramedBoundaryTorus ⋯ ⋯ h).longitude Y) (F' : DehnFilling h' (IsSolidTorusNeighborhood.preferredFramedBoundaryTorus ⋯ ⋯ h').longitude Y') (oY : HomologicalOrientation Y (EuclideanSpace ℝ (Fin 3)) 3) (oY' : HomologicalOrientation Y' (EuclideanSpace ℝ (Fin 3)) 3) (e : Y ≃ₜ Y'), F.IsOrientationCompatible o oY ∧ F'.IsOrientationCompatible o oY' ∧ oY'.comap ⋯ = oY ∧ ¬SmoothEmbedding.Concordant K K' ∧ ¬SmoothEmbedding.Concordant K K'.reverse

Yasui's theorem spelled out: there are knots K K' : S¹ → S³ with 0-surgeries Y, Y', oriented compatibly with an orientation of S³, and an orientation-preserving homeomorphism Y ≃ₜ Y', such that K is smoothly concordant neither to K' nor to its reverse.