The Heegaard surface of a splitting

A Heegaard splitting TauCeti.HeegaardSplitting M records its two handlebodies as subsets of M meeting in the frontier of each, so for a general space M that frontier need not be the boundary surface of the handlebodies. When M is a 3-manifold without boundary it is: by invariance of domain (Homeomorph.coe_apply_mem_frontier_iff), any homeomorphism from one handlebody of the splitting onto the standard handlebody TauCeti.handlebody g ⊆ ℂ × ℝ maps the points where the two handlebodies meet exactly onto the frontier of TauCeti.handlebody g in ℂ × ℝ, its boundary surface of genus g.

This is the identification through which results about the standard handlebody and its boundary surface, such as the computation of fundamental groups, apply to the two sides of a splitting and to the surface along which they meet.

Main results

• TauCeti.HeegaardSplitting.coe_mem_right_iff: a point of the first handlebody lies in the second exactly when it corresponds to a point of the frontier of the standard handlebody.
• TauCeti.HeegaardSplitting.coe_mem_left_iff: the same, with the two handlebodies exchanged.

References

• M. Scharlemann, Heegaard splittings of compact 3-manifolds, in Handbook of Geometric Topology, North-Holland (2002), Section 2.

theorem TauCeti.HeegaardSplitting.coe_mem_right_iff {E : Type u_1} {M : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace M] [ChartedSpace E M] (s : HeegaardSplitting M) (hE : Module.finrank ℝ E = 3) (e : ↑s.left ≃ₜ ↑(handlebody s.genus)) (x : ↑s.left) : ↑x ∈ s.right ↔ ↑(e x) ∈ frontier (handlebody s.genus)

The two handlebodies of a splitting of a 3-manifold meet in their boundary surface. In a Heegaard splitting of a manifold without boundary charted on a three-dimensional real normed space, a point of the first handlebody lies in the second exactly when, under a homeomorphism of the first handlebody onto the standard one, it corresponds to a point of the frontier of the standard handlebody in ℂ × ℝ.

theorem TauCeti.HeegaardSplitting.coe_mem_left_iff {E : Type u_1} {M : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace M] [ChartedSpace E M] (s : HeegaardSplitting M) (hE : Module.finrank ℝ E = 3) (e : ↑s.right ≃ₜ ↑(handlebody s.genus)) (x : ↑s.right) : ↑x ∈ s.left ↔ ↑(e x) ∈ frontier (handlebody s.genus)

The two handlebodies of a splitting of a 3-manifold meet in their boundary surface. In a Heegaard splitting of a manifold without boundary charted on a three-dimensional real normed space, a point of the second handlebody lies in the first exactly when, under a homeomorphism of the second handlebody onto the standard one, it corresponds to a point of the frontier of the standard handlebody in ℂ × ℝ.