Linearization of compact group actions

DefinitionBogolyubov's linearization claim (Scottish Book Problem 183)#✓fully proved

The claim that for every \(n \in \mathbb{N}\) and every compact, connected, locally connected topological group \(G\) acting on \(\mathbb{R}^n\) with continuous action map \(G \times \mathbb{R}^n \to \mathbb{R}^n\), there is a homeomorphism \(\varphi\) of \(\mathbb{R}^n\) such that for every \(g \in G\) there is a linear map \(L_g \colon \mathbb{R}^n \to \mathbb{R}^n\) with \(\varphi(g \cdot x) = L_g(\varphi(x))\) for all \(x \in \mathbb{R}^n\).

This is Bogolyubov's problem in its original form: coordinates can be introduced in which every transformation of the group is linear.

def ScottishBookProblem183Claim : Prop :=
  ∀ (n : ℕ) (G : Type) [Group G] [TopologicalSpace G]
    [CompactSpace G] [ConnectedSpace G] [LocallyConnectedSpace G]
    [IsTopologicalGroup G]
    [MulAction G (Fin n → ℝ)] [ContinuousSMul G (Fin n → ℝ)],
    ∃ φ : (Fin n → ℝ) ≃ₜ (Fin n → ℝ), ∀ g : G,
      ∃ L : (Fin n → ℝ) →ₗ[ℝ] (Fin n → ℝ), ∀ x, φ (g • x) = L (φ x)

Used by 1

Used byBogolyubov's Problem 183 has a negative answer

TheoremBogolyubov's Problem 183 has a negative answer#●ready to prove

Bogolyubov's linearization claim is false: there are \(n \in \mathbb{N}\) and a compact, connected, locally connected topological group \(G\) acting continuously on \(\mathbb{R}^n\) such that for no homeomorphism \(\varphi\) of \(\mathbb{R}^n\) is every map \(\varphi \circ g \circ \varphi^{-1}\), for \(g \in G\), linear.

theorem_wanted bogolyubov183_refuted : ¬ ScottishBookProblem183Claim

Sources

• N. N. Bogolyubov, Problem 183 of the Scottish Book; see the commentary in R. D. Mauldin (ed.), The Scottish Book, 2nd ed. (Birkhäuser, 2015).

Uses 1

Statement usesBogolyubov's linearization claim (Scottish Book Problem 183)

1 fully proved1 ready to prove

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