Hilbert's Fifth Problem and the Hilbert–Smith Conjecture

The Hilbert–Smith conjecture states that a locally compact topological group acting continuously and faithfully on a connected finite-dimensional topological manifold must be a Lie group. It remains open in general; Pardon proved it for 3-manifolds in 2013. An equivalent formulation: no p-adic integer group ℤ_[p] can act faithfully on any connected finite-dimensional topological manifold.

References:

• Wikipedia
• Tao's blog
• Pardon 2013, arXiv:1112.2324
• arXiv:math/0103145

def Hilbert5.AdmitsLieGroupStructure (G : Type u_3) [Group G] [TopologicalSpace G] : Prop

A topological group G admits a Lie group structure if there exists a finite-dimensional smooth manifold structure on G making it a real Lie group.

Equations

• Hilbert5.AdmitsLieGroupStructure G = ∃ (k : ℕ) (cs : ChartedSpace (EuclideanSpace ℝ (Fin k)) G), LieGroup (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin k))) ⊤ G

theorem Hilbert5.admitsLieGroupStructure_of_lieGroup {G : Type u_1} [Group G] [TopologicalSpace G] {n : ℕ} [ChartedSpace (EuclideanSpace ℝ (Fin n)) G] [LieGroup (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) ⊤ G] : AdmitsLieGroupStructure G

Every Lie group trivially admits a Lie group structure.

theorem Hilbert5.locallyCompact_of_admitsLieGroupStructure {G : Type u_1} [Group G] [TopologicalSpace G] (h : AdmitsLieGroupStructure G) : LocallyCompactSpace G

A group admitting a Lie group structure is locally compact.

theorem Hilbert5.hilbert_smith_conjecture {G : Type u_1} [Group G] [TopologicalSpace G] {X : Type u_2} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] [IsTopologicalGroup G] [LocallyCompactSpace G] [MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X] : AdmitsLieGroupStructure G

Hilbert–Smith conjecture: every locally compact topological group acting continuously and faithfully on a connected finite-dimensional topological manifold is a Lie group.

theorem Hilbert5.hilbert_smith_conjecture.variants.riemannian {G : Type u_1} [Group G] [TopologicalSpace G] {n : ℕ} {X : Type u_2} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] [ChartedSpace (EuclideanSpace ℝ (Fin n)) X] [IsTopologicalGroup G] [LocallyCompactSpace G] [MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X] [MetricSpace X] [IsManifold (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) (↑⊤) X] (hiso : ∀ (g : G), Isometry fun (x : X) => g • x) : AdmitsLieGroupStructure G

The conjecture holds when G acts by isometries on a Riemannian manifold, since G embeds as a closed subgroup of the isometry group, which is a Lie group by Myers–Steenrod.

theorem Hilbert5.hilbert_smith_conjecture.variants.dimension_three {G : Type u_1} [Group G] [TopologicalSpace G] {X : Type u_3} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] [ChartedSpace (EuclideanSpace ℝ (Fin 3)) X] [IsTopologicalGroup G] [LocallyCompactSpace G] [MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X] : AdmitsLieGroupStructure G

Pardon (2013): the Hilbert–Smith conjecture holds for 3-dimensional manifolds. See arXiv:1112.2324.

theorem Hilbert5.hilbert_smith_padic_formulation {X : Type u_2} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] (p : ℕ) [Fact (Nat.Prime p)] [AddAction ℤ_[p] X] [ContinuousVAdd ℤ_[p] X] [FaithfulVAdd ℤ_[p] X] : False

Equivalent p-adic formulation: the p-adic integers ℤ_[p] cannot act continuously and faithfully on any connected finite-dimensional topological manifold. By the Gleason–Yamabe theorem, this is equivalent to hilbert_smith_conjecture.

theorem Hilbert5.hilbert_fifth_problem {G : Type u_1} [Group G] [TopologicalSpace G] {n : ℕ} [IsTopologicalGroup G] [ChartedSpace (EuclideanSpace ℝ (Fin n)) G] : LieGroup (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) ⊤ G

Hilbert's fifth problem (Gleason–Montgomery–Zippin, 1952): every locally Euclidean topological group is a Lie group.