The minimal genus of a degree-two homology class

A closed, connected, oriented surface S embedded in a space Y by f : S → Y represents the integral class f_*[S] ∈ H₂(Y; ℤ), the image of its fundamental class (TauCeti.HomologicalOrientation.mapFundamentalClass). The minimal genus of a class c ∈ H₂(Y; ℤ) (TauCeti.minimalGenus c) is the least genus of an embedded surface representing it, or ⊤ if there is none. It is the invariant of Kirby's problem 1.41 (B): Giffen conjectured, and Gabai proved, that a generator of H₂(S³₀(K); ℤ) ≅ ℤ has minimal genus the genus of the knot K; on a 3-manifold it is also the quantity measured, up to the correction by spheres and tori, by the Thurston norm.

A representing surface (TauCeti.RepresentingSurface c) bundles a compact, connected, Hausdorff topological surface without boundary, modelled on ℝ², a homological orientation of it (TauCeti.HomologicalOrientation), and a topological embedding into Y carrying its fundamental class to c. Its genus (TauCeti.RepresentingSurface.genus) is read off its rational singular homology: a closed, connected, orientable surface of genus g has H₁(S; ℚ) ≅ ℚ²ᵍ, so its genus is half the dimension of H₁(S; ℚ), the same convention as for Seifert and slice surfaces.

The embeddings are only asked to be topological, so that the definition applies to topological spaces such as Dehn fillings. In a 3-manifold nothing is lost: by Bing's approximation theorem every embedded surface is approximated by a tame one, which is isotopic to a smooth one and, being homotopic to the first, represents the same class with the same genus.

Reversing the orientation negates the represented class, so a class and its negative have the same minimal genus (TauCeti.minimalGenus_neg). In particular the two generators of an infinite cyclic H₂(Y; ℤ) have the same minimal genus.

Main definitions

• TauCeti.RepresentingSurface c: a closed, connected, oriented surface embedded in Y representing c ∈ H₂(Y; ℤ).
• TauCeti.RepresentingSurface.genus: its genus dim H₁(S; ℚ) / 2 (TauCeti.surfaceGenus).
• TauCeti.RepresentingSurface.neg: the same surface with the reversed orientation, representing -c.
• TauCeti.minimalGenus c: the least genus of a surface representing c, in ℕ∞.

Main results

• TauCeti.minimalGenus_le_genus, TauCeti.le_minimalGenus_iff: the minimal genus is the infimum of the genera of the representing surfaces.
• TauCeti.minimalGenus_lt_top_iff, TauCeti.exists_genus_eq_minimalGenus: it is finite exactly when a representing surface exists, and is then attained.
• TauCeti.minimalGenus_neg: a class and its negative have the same minimal genus.

References

• R. Kirby (ed.), Problems in Low-Dimensional Topology, in Geometric Topology, AMS/IP Stud. Adv. Math. 2.2 (1997), Problem 1.41.
• W. P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986).
• R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. 65 (1957).

@[reducible, inline]

noncomputable abbrev TauCeti.secondHomology (Y : Type) [TopologicalSpace Y] : ModuleCat ℤ

The second integral singular homology H₂(Y; ℤ) of a space Y, as a ℤ-module: the home of the classes represented by embedded closed oriented surfaces.

structure TauCeti.RepresentingSurface {Y : Type} [TopologicalSpace Y] (c : ↑(secondHomology Y)) : Type 1

A representing surface of a class c ∈ H₂(Y; ℤ): a compact, connected, Hausdorff topological surface without boundary, modelled on ℝ², with a homological orientation and a topological embedding into Y carrying its fundamental class to c.

• carrier : Type

  The underlying surface.

• topologicalSpace : TopologicalSpace self.carrier

  The topology of the surface.

• t2Space : T2Space self.carrier

  The surface is Hausdorff.

• chartedSpace : ChartedSpace (EuclideanSpace ℝ (Fin 2)) self.carrier

  The charts of the surface, modelled on the plane.

• compactSpace : CompactSpace self.carrier

  The surface is compact.

• connectedSpace : ConnectedSpace self.carrier

  The surface is connected.

• orientation : HomologicalOrientation self.carrier (EuclideanSpace ℝ (Fin 2)) 2

  The orientation of the surface.

• toFun : C(self.carrier, Y)

  The embedding of the surface into Y.

• isEmbedding : Topology.IsEmbedding ⇑self.toFun

  The map is a topological embedding.

• mapFundamentalClass_eq : self.orientation.mapFundamentalClass ⋯ self.toFun = c

  The embedded surface represents c.

noncomputable def TauCeti.RepresentingSurface.genus {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} (S : RepresentingSurface c) : ℕ

The genus of a representing surface: half the dimension of its first rational homology (TauCeti.surfaceGenus), which is ℚ²ᵍ for a closed, connected, orientable surface of genus g.

theorem TauCeti.RepresentingSurface.genus_def {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} (S : RepresentingSurface c) : S.genus = surfaceGenus S.carrier

The genus of a representing surface is the genus TauCeti.surfaceGenus of its carrier.

noncomputable def TauCeti.RepresentingSurface.neg {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} (S : RepresentingSurface c) : RepresentingSurface (-c)

A surface representing c, with its orientation reversed, represents -c.

@[simp]

theorem TauCeti.RepresentingSurface.genus_neg {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} (S : RepresentingSurface c) : S.neg.genus = S.genus

Reversing the orientation of a representing surface keeps its genus.

noncomputable def TauCeti.minimalGenus {Y : Type} [TopologicalSpace Y] (c : ↑(secondHomology Y)) : ℕ∞

The minimal genus of a class c ∈ H₂(Y; ℤ): the least genus of a closed, connected, oriented surface embedded in Y representing c, or ⊤ if there is none.

theorem TauCeti.minimalGenus_def {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} : minimalGenus c = ⨅ (S : RepresentingSurface c), ↑S.genus

The minimal genus is the infimum of the genera of all representing surfaces.

theorem TauCeti.minimalGenus_le_genus {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} (S : RepresentingSurface c) : minimalGenus c ≤ ↑S.genus

The minimal genus of a class is at most the genus of any surface representing it.

@[simp]

theorem TauCeti.le_minimalGenus_iff {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} {m : ℕ∞} : m ≤ minimalGenus c ↔ ∀ (S : RepresentingSurface c), m ≤ ↑S.genus

A lower bound for the minimal genus is a lower bound for the genus of every representing surface.

theorem TauCeti.exists_genus_eq_minimalGenus {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} [Nonempty (RepresentingSurface c)] : ∃ (S : RepresentingSurface c), ↑S.genus = minimalGenus c

The minimal genus is attained when a representing surface exists.

@[simp]

theorem TauCeti.minimalGenus_lt_top_iff {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} : minimalGenus c < ⊤ ↔ Nonempty (RepresentingSurface c)

The minimal genus is finite exactly when a representing surface exists.

@[simp]

theorem TauCeti.minimalGenus_neg {Y : Type} [TopologicalSpace Y] {c : ↑(secondHomology Y)} : minimalGenus (-c) = minimalGenus c

A class and its negative have the same minimal genus: reversing the orientation of a representing surface negates the class it represents.