Rank versus Heegaard genus

The fundamental group of a closed orientable 3-manifold M with a Heegaard splitting of genus g is generated by g elements: it is a quotient of the free fundamental group of either handlebody. So the rank of π₁(M), its least number of generators (Group.rank), is at most the Heegaard genus of M (TauCeti.heegaardGenus). Waldhausen asked whether these two invariants are always equal. They are not: Boileau and Zieschang found Seifert-fibred 3-manifolds of rank two and Heegaard genus three, and Li later constructed closed orientable hyperbolic 3-manifolds whose Heegaard genus exceeds the rank of their fundamental group by an arbitrarily large amount.

This file states Waldhausen's rank-versus-genus conjecture and Li's theorem. A closed orientable 3-manifold is represented, as in TauCeti.GeometrizationConjecture, by a compact connected orientable smooth manifold without boundary, with charts in EuclideanSpace ℝ (Fin 3) and an ambient metric. The conjecture asks, at every base point, for exactly what Waldhausen's equation needs in order to make sense: that π₁(M) is finitely generated, that M has a Heegaard splitting, and that the Heegaard genus equals the rank. The first two are theorems for every closed orientable 3-manifold, so they are part of the conclusion rather than hypotheses. Li's theorem provides, for every n, a closed orientable hyperbolic 3-manifold (TauCeti.IsHyperbolic) whose Heegaard genus exceeds the rank by more than n; already its case n = 0 refutes the conjecture.

Both statements are recorded, not proved. On the 3-sphere the two invariants agree: it is simply connected and has Heegaard genus zero.

Main definitions

• TauCeti.RankGenusConjecture: Waldhausen's conjecture that the Heegaard genus of a closed orientable 3-manifold equals the rank of its fundamental group.
• TauCeti.LiRankGenusTheorem: Li's theorem that the Heegaard genus of a closed orientable hyperbolic 3-manifold can exceed the rank of its fundamental group by any amount.

Main results

• TauCeti.RankGenusConjecture.heegaardGenus_eq_rank: applying the conjecture to a closed orientable 3-manifold.
• TauCeti.not_rankGenusConjecture_of_heegaardGenus_ne: a closed orientable 3-manifold whose Heegaard genus differs from the rank of its fundamental group refutes the conjecture.
• TauCeti.LiRankGenusTheorem.not_rankGenusConjecture: Li's theorem refutes the conjecture.
• TauCeti.exists_heegaardGenus_sphere_eq_rank: the 3-sphere satisfies the conclusion of the conjecture.

References

• F. Waldhausen, Some problems on 3-manifolds, in Algebraic and Geometric Topology, Proc. Sympos. Pure Math. 32, Part 2, Amer. Math. Soc. (1978), 313–322.
• M. Boileau, H. Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds, Invent. Math. 76 (1984), 455–468.
• T. Li, Rank and genus of 3-manifolds, J. Amer. Math. Soc. 26 (2013), 777–829.

Waldhausen's conjecture

def TauCeti.RankGenusConjecture : Prop

Waldhausen's rank-versus-genus conjecture: every closed orientable 3-manifold M, that is a compact connected orientable smooth 3-manifold without boundary, has a finitely generated fundamental group and a Heegaard splitting, and its Heegaard genus equals the rank of its fundamental group at every base point.

The conjecture is false, by Boileau–Zieschang and, for hyperbolic manifolds, by Li (TauCeti.LiRankGenusTheorem.not_rankGenusConjecture).

theorem TauCeti.rankGenusConjecture_iff : RankGenusConjecture ↔ ∀ (M : Type u) [inst : MetricSpace M] [inst_1 : ChartedSpace (EuclideanSpace ℝ (Fin 3)) M] [IsManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) (↑⊤) M] [CompactSpace M] [ConnectedSpace M] [Orientable (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) M] (x : M), ∃ (x_1 : Group.FG (FundamentalGroup M x)) (x_2 : Nonempty (HeegaardSplitting M)), heegaardGenus M = Group.rank (FundamentalGroup M x)

The defining characterization of the rank-versus-genus conjecture.

theorem TauCeti.RankGenusConjecture.nonempty_heegaardSplitting {M : Type u} [MetricSpace M] [ChartedSpace (EuclideanSpace ℝ (Fin 3)) M] [IsManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) (↑⊤) M] [CompactSpace M] [ConnectedSpace M] [Orientable (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) M] (h : RankGenusConjecture) : Nonempty (HeegaardSplitting M)

Under the rank-versus-genus conjecture, every closed orientable 3-manifold has a Heegaard splitting.

theorem TauCeti.RankGenusConjecture.heegaardGenus_eq_rank {M : Type u} [MetricSpace M] [ChartedSpace (EuclideanSpace ℝ (Fin 3)) M] [IsManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) (↑⊤) M] [CompactSpace M] [ConnectedSpace M] [Orientable (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) M] (h : RankGenusConjecture) (x : M) [Group.FG (FundamentalGroup M x)] [Nonempty (HeegaardSplitting M)] : heegaardGenus M = Group.rank (FundamentalGroup M x)

Under the rank-versus-genus conjecture, the Heegaard genus of a closed orientable 3-manifold equals the rank of its fundamental group.

theorem TauCeti.not_rankGenusConjecture_of_heegaardGenus_ne {M : Type u} [MetricSpace M] [ChartedSpace (EuclideanSpace ℝ (Fin 3)) M] [IsManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) (↑⊤) M] [CompactSpace M] [ConnectedSpace M] [Orientable (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) M] (x : M) [Group.FG (FundamentalGroup M x)] [Nonempty (HeegaardSplitting M)] (hne : heegaardGenus M ≠ Group.rank (FundamentalGroup M x)) : ¬RankGenusConjecture

A closed orientable 3-manifold whose Heegaard genus differs from the rank of its fundamental group refutes the rank-versus-genus conjecture.

Li's theorem

def TauCeti.LiRankGenusTheorem : Prop

Li's rank-versus-genus theorem: for every n, some closed orientable hyperbolic 3-manifold M, that is a compact connected orientable smooth 3-manifold without boundary admitting a complete Riemannian metric of constant curvature -1, has a finitely generated fundamental group and a Heegaard splitting, and its Heegaard genus exceeds the rank of its fundamental group by more than n.

The case n = 0 already refutes Waldhausen's conjecture (TauCeti.LiRankGenusTheorem.not_rankGenusConjecture).

theorem TauCeti.liRankGenusTheorem_iff : LiRankGenusTheorem ↔ ∀ (n : ℕ), ∃ (M : Type u) (x : MetricSpace M) (x_1 : ChartedSpace (EuclideanSpace ℝ (Fin 3)) M) (x_2 : IsManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) (↑⊤) M) (_ : CompactSpace M) (x_4 : ConnectedSpace M) (_ : Orientable (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin 3))) M) (x_6 : M) (x_7 : Group.FG (FundamentalGroup M x_6)) (x_8 : Nonempty (HeegaardSplitting M)), IsHyperbolic ∧ Group.rank (FundamentalGroup M x_6) + n < heegaardGenus M

The defining characterization of Li's rank-versus-genus theorem.

theorem TauCeti.LiRankGenusTheorem.not_rankGenusConjecture (h : LiRankGenusTheorem) : ¬RankGenusConjecture

Li's theorem refutes Waldhausen's conjecture: some closed orientable 3-manifold has Heegaard genus different from the rank of its fundamental group.

The 3-sphere

theorem TauCeti.exists_heegaardGenus_sphere_eq_rank {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (Module.finrank ℝ E = 4)] (x : ↑(Metric.sphere 0 1)) : ∃ (x_1 : Group.FG (FundamentalGroup (↑(Metric.sphere 0 1)) x)) (x_2 : Nonempty (HeegaardSplitting ↑(Metric.sphere 0 1))), heegaardGenus ↑(Metric.sphere 0 1) = Group.rank (FundamentalGroup (↑(Metric.sphere 0 1)) x)

The 3-sphere satisfies Waldhausen's equation: it is simply connected and has Heegaard genus zero, so its fundamental group is finitely generated and its Heegaard genus equals the rank of its fundamental group.