Torus maps fixing the meridian extend over the solid torus

A homeomorphism of the boundary torus S¹ × S¹ of the solid torus V = D² × S¹ induced by an invertible integral matrix (Matrix.GeneralLinearGroup.torusHomeomorph) whose matrix fixes the meridian (1, 0) up to sign, (a, b) ↦ (a ^ ε * b ^ k, b ^ δ) with ε, δ = ±1, extends to V as (z, b) ↦ (z * b ^ k, b ^ δ), with z replaced by its complex conjugate when ε = -1. This is the step of the well-definedness of Dehn filling (Rolfsen, Knots and Links, 9F) that concerns the filling solid torus.

Main results

• TauCeti.SolidTorus.exists_homeomorph_boundaryInclusion_torusHomeomorph: matrices fixing the meridian up to sign act on the boundary torus by maps that extend over the solid torus.

References

• D. Rolfsen, Knots and Links, Publish or Perish (1976), Chapter 9, Section 9F (a homeomorphism of the boundary torus preserving the meridian extends over the solid torus).

theorem TauCeti.SolidTorus.exists_homeomorph_boundaryInclusion_torusHomeomorph (A : GL (Fin 2) ℤ) (hA : ↑A 1 0 = 0) : ∃ (ψ : SolidTorus ≃ₜ SolidTorus), ∀ (q : Circle × Circle), ψ (boundaryInclusion q) = boundaryInclusion (A.torusHomeomorph q)

Torus maps fixing the meridian extend over the solid torus. If an invertible integral matrix A has A₁₀ = 0, so that it fixes the meridian class (1, 0) up to sign, the induced homeomorphism (a, b) ↦ (a ^ A₀₀ * b ^ A₀₁, b ^ A₁₁) of the boundary torus S¹ × S¹ is the restriction of a homeomorphism of the solid torus D² × S¹.