5.lean`

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7 declarations, 5 contain sorry, 0 declared as axioms

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.

Main statements

hilbert_smith_conjecture: the Hilbert–Smith conjecture.
hilbert_smith_padic_formulation: the equivalent formulation for ℤ_[p].
hilbert_smith_conjecture.variants.dimension_three: Pardon's theorem for 3-manifolds.
hilbert_smith_conjecture.variants.riemannian: the case of isometric actions on Riemannian manifolds.
hilbert_fifth_problem: Hilbert's fifth problem, solved by Gleason, Montgomery and Zippin.

Implementation notes

AdmitsLieGroupStructure G, defined in FormalConjecturesForMathlib.Geometry.Manifold.LieGroupPresentation, says that G is continuously isomorphic to a finite-dimensional real-analytic Lie group. Lie groups are Hausdorff but not assumed second countable: every discrete group is a 0-dimensional Lie group (admitsLieGroupStructure_of_discreteTopology). This matters for the Hilbert–Smith conjecture, since uncountable discrete groups act continuously and faithfully on connected manifolds, for instance ℝ with the discrete topology acting on ℝ by translations.

The acting group is not assumed Hausdorff: a topological group acting continuously and faithfully on a Hausdorff space is Hausdorff, because the closure of the identity acts trivially.

The hypotheses on the manifold X are written in each theorem. A section variable [ChartedSpace (EuclideanSpace ℝ (Fin n)) X] is included only in a theorem that mentions n, so it would be dropped silently from a statement that mentions only X.

References:

Wikipedia
Tao's blog
[Pardon 2013] J. Pardon, The Hilbert–Smith conjecture for three-manifolds, J. Amer. Math. Soc. 26 (2013), 879–899. https://doi.org/10.1090/S0894-0347-2013-00766-3, arXiv:1112.2324
[Myers–Steenrod 1939] S. B. Myers, N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. 40 (1939), 400–416. https://doi.org/10.2307/1968928
[van den Dries–Goldbring 2015] L. van den Dries, I. Goldbring, Hilbert's 5th problem, Enseign. Math. 61 (2015), 3–43. https://doi.org/10.4171/LEM/61-1/2-2

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Wikipedia/HilbertFifthProblem.lean

theorem Hilbert5.admitsLieGroupStructure_circle The circle group admits a Lie group structure line 77 source
Source documentation
The circle group admits a Lie group structure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AdmitsLieGroupStructure, Circle
Proof uses: admitsLieGroupStructure_of_lieGroup, OfNat.ofNat, Nat
13 instances, coercions and auxiliary constants left out.
```
theorem admitsLieGroupStructure_circle : AdmitsLieGroupStructure Circle :=
…
```
Mentions: AdmitsLieGroupStructure
theorem Hilbert5.admitsLieGroupStructure_multiplicative_int The discrete group ℤ admits a Lie group structure line 82 source
Source documentation
The discrete group ℤ admits a Lie group structure.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AdmitsLieGroupStructure, Multiplicative, Int
Proof uses: admitsLieGroupStructure_of_discreteTopology
10 instances, coercions and auxiliary constants left out.
```
theorem admitsLieGroupStructure_multiplicative_int :
    AdmitsLieGroupStructure (Multiplicative ℤ) :=
…
```
Mentions: AdmitsLieGroupStructure
theorem Hilbert5.hilbert_smith_conjecture Hilbert–Smith conjecture: every locally compact topological group acting continuously and faithfully on a connected finite-dimensional topological manifold is a Lie group sorry line 89 source
Source documentation
Hilbert–Smith conjecture: every locally compact topological group acting continuously and faithfully on a connected finite-dimensional topological manifold is a Lie group.

depends on sorry what this means
Statement uses: Group, TopologicalSpace, Nat, T2Space, ConnectedSpace, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, IsTopologicalGroup, LocallyCompactSpace, MulAction, ContinuousSMul, FaithfulSMul, AdmitsLieGroupStructure
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hilbert_smith_conjecture {X : Type*}
    [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] :
    AdmitsLieGroupStructure G := by
…
```
Mentions: AdmitsLieGroupStructure
theorem Hilbert5.hilbert_smith_conjecture.variants.riemannian Documented declaration: riemannian sorry line 104 source
Source documentation
The Hilbert–Smith conjecture holds for actions by isometries of a connected smooth Riemannian manifold X: the isometry group of X is a Lie group by the Myers–Steenrod theorem, so G has no small subgroups and is a Lie group by the Gleason–Montgomery–Zippin theorem.
Here X carries its Riemannian distance (IsRiemannianManifold), so the metric topology of X is its manifold topology.

depends on sorry what this means
Statement uses: Group, TopologicalSpace, Nat, EMetricSpace, ConnectedSpace, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, IsManifold, modelWithCornersSelf, ENat, Top.top, Bundle.RiemannianBundle, TangentSpace, IsContMDiffRiemannianBundle, WithTop, Bundle.instNormedAddCommGroupOfRiemannianBundleOfIsTopologicalAddGroupOfContinuousConstSMulReal and 10 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
112 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hilbert_smith_conjecture.variants.riemannian {X : Type*} [EMetricSpace X]
    [ConnectedSpace X] [ChartedSpace (EuclideanSpace ℝ (Fin n)) X] [IsManifold (𝓡 n) ∞ X]
    [Bundle.RiemannianBundle (fun x : X ↦ TangentSpace (𝓡 n) x)]
    [IsContMDiffRiemannianBundle (𝓡 n) ∞ (EuclideanSpace ℝ (Fin n))
      (fun x : X ↦ TangentSpace (𝓡 n) x)]
    [IsRiemannianManifold (𝓡 n) X]
    [IsTopologicalGroup G] [LocallyCompactSpace G]
    [MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X]
    (hiso : ∀ g : G, Isometry (g • · : X → X)) :
    AdmitsLieGroupStructure G := by
…
```
Mentions: AdmitsLieGroupStructure
theorem Hilbert5.hilbert_smith_conjecture.variants.dimension_three Pardon's theorem (2013): the Hilbert–Smith conjecture holds for connected 3-manifolds, see [Pardon 2013] sorry line 119 source
Source documentation
Pardon's theorem (2013): the Hilbert–Smith conjecture holds for connected 3-manifolds, see [Pardon 2013].

depends on sorry what this means
Statement uses: Group, TopologicalSpace, T2Space, ConnectedSpace, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, ENNReal, IsTopologicalGroup, LocallyCompactSpace, MulAction, ContinuousSMul, FaithfulSMul, AdmitsLieGroupStructure
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hilbert_smith_conjecture.variants.dimension_three {X : Type*}
    [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 := by
…
```
Mentions: AdmitsLieGroupStructure
theorem Hilbert5.hilbert_smith_padic_formulation p-adic formulation: the p-adic integers ℤ_[p] cannot act continuously and faithfully on a connected finite-dimensional topological manifold sorry line 132 source
Source documentation
p-adic formulation: the p-adic integers ℤ_[p] cannot act continuously and faithfully on a connected finite-dimensional topological manifold. This is equivalent to hilbert_smith_conjecture by the Gleason–Yamabe theorem together with Newman's theorem on periodic transformations of manifolds.

depends on sorry what this means
Statement uses: Nat, TopologicalSpace, T2Space, ConnectedSpace, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, Fact, Nat.Prime, AddAction, PadicInt, ContinuousVAdd, Not, FaithfulVAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
43 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hilbert_smith_padic_formulation {X : Type*}
    [TopologicalSpace X] [T2Space X] [ConnectedSpace X]
    [ChartedSpace (EuclideanSpace ℝ (Fin n)) X]
    (p : ℕ) [Fact p.Prime] [AddAction ℤ_[p] X] [ContinuousVAdd ℤ_[p] X] :
    ¬ FaithfulVAdd ℤ_[p] X := by
…
```
theorem Hilbert5.hilbert_fifth_problem Documented declaration: hilbert_fifth_problem sorry line 146 source
Source documentation
Hilbert's fifth problem (Gleason–Montgomery–Zippin, 1952): every Hausdorff topological group modeled on a finite-dimensional Euclidean space is continuously isomorphic to a real-analytic Lie group.
The input ChartedSpace supplies only a topological atlas. The compatible analytic atlas and analytic group operations belong to the output LieGroupPresentation.

depends on sorry what this means
Statement uses: Group, TopologicalSpace, Nat, IsTopologicalGroup, T2Space, ChartedSpace, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal, Nonempty, LieGroupPresentation
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hilbert_fifth_problem
    [IsTopologicalGroup G] [T2Space G]
    [ChartedSpace (EuclideanSpace ℝ (Fin n)) G] :
    Nonempty (LieGroupPresentation G n) := by
…
```
Mentions: LieGroupPresentation