Gabai's theorem on the genus of the zero-surgery

Let K be a knot in S³, and S³₀(K) its 0-surgery: the Dehn filling of the knot exterior along the preferred longitude, the boundary of the 4-manifold obtained from B⁴ by attaching a 2-handle along K with framing 0. Its second homology is infinite cyclic, generated by the class of a Seifert surface of K capped off by a meridian disc of the filling solid torus, so a generator is represented by a closed surface of genus g(K), the Seifert genus of K. Giffen asked for the minimal genus of a surface representing a generator, conjecturing that it is g(K) (Kirby's problem 1.41 (B)), and Gabai proved this as a consequence of his construction of taut foliations: the minimal genus of a surface representing a generator of H₂(S³₀(K); ℤ) is the genus of K (Gabai, Foliations and the topology of 3-manifolds III, Corollary 8.3). For the unknot, whose 0-surgery is S¹ × S², both sides are 0.

TauCeti.GabaiZeroSurgeryGenusTheorem records this statement, not proved here. The knot is a smooth circle embedding K : S¹ → S³ (TauCeti.SmoothCircleEmbedding), its 0-surgery any TauCeti.DehnFilling of the exterior of a solid torus neighbourhood of K along the longitude of the preferred framing (TauCeti.IsSolidTorusNeighborhood.preferredFramedBoundaryTorus), the minimal genus that of TauCeti.minimalGenus, and the genus of the knot its Seifert genus TauCeti.seifertGenus. The filled space is only a topological space, so the surfaces are topologically embedded; in a 3-manifold this changes nothing (see TauCeti.AlgebraicTopology.Singular.MinimalGenus). Every Dehn filling along the slope is admitted, not only those whose gluing is isotopic to a gluing induced by a matrix, so the statement does not depend on the well-definedness of Dehn filling. Generators are those c with Submodule.span ℤ {c} = ⊤.

Applied to the unknot, the theorem says that every generator of H₂(S¹ × S²; ℤ) has minimal genus 0 (TauCeti.GabaiZeroSurgeryGenusTheorem.minimalGenus_circle_prod_sphere), as the sphere {1} × S² shows directly: the 0-surgery on the unknot is S¹ × S² (TauCeti.unknotZeroDehnFilling), and the unknot has Seifert genus 0 (TauCeti.seifertGenus_unknot).

Main definitions

• TauCeti.GabaiZeroSurgeryGenusTheorem: a generator of the second homology of the 0-surgery on a knot in S³ has minimal genus the Seifert genus of the knot.

Main results

• TauCeti.GabaiZeroSurgeryGenusTheorem.minimalGenus_circle_prod_sphere: under the theorem, every generator of H₂(S¹ × S²; ℤ) has minimal genus 0.

References

• R. Kirby (ed.), Problems in Low-Dimensional Topology, in Geometric Topology, AMS/IP Stud. Adv. Math. 2.2 (1997), Problem 1.41 (B) (Giffen).
• D. Gabai, Foliations and the topology of 3-manifolds III, J. Differential Geom. 26 (1987), 479–536, Corollary 8.3.
• D. Rolfsen, Knots and Links, Publish or Perish (1976), Chapter 9 (the 0-surgery and its homology).

def TauCeti.GabaiZeroSurgeryGenusTheorem : Prop

Gabai's theorem on the genus of the 0-surgery (Giffen's conjecture, Kirby's problem 1.41 (B)): for a smooth knot K in S³, every generator of the second integral homology of a 0-surgery on K, a Dehn filling of the exterior of a solid torus neighbourhood of K along the longitude of the preferred framing, has minimal genus the Seifert genus of K. Stated, not proved.

theorem TauCeti.GabaiZeroSurgeryGenusTheorem.minimalGenus_circle_prod_sphere (hG : GabaiZeroSurgeryGenusTheorem) {c : ↑(secondHomology (Circle × ↑(Metric.sphere 0 1)))} (hc : ℤ ∙ c = ⊤) : minimalGenus c = 0

Under Gabai's theorem, a generator of H₂(S¹ × S²; ℤ) has minimal genus 0: S¹ × S² is the 0-surgery on the unknot, which has Seifert genus 0.