theorem weakSolvability_of_finiteMinimalNormalKernel
{G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G]
(hfinite :
∀ P : TopologicalEmbeddingProblem G, E P →
P.HasFiniteMinimalNormalKernel → P.HasWeakSolution)
(hreduce :
∀ P : TopologicalEmbeddingProblem G, E P →
∃ P₀ : TopologicalEmbeddingProblem G, E P₀ ∧
P₀.HasFiniteMinimalNormalKernel ∧
(P₀.HasWeakSolution → P.HasWeakSolution)) :
∀ P : TopologicalEmbeddingProblem G, E P → P.HasWeakSolutionWeak solvability reduces to the finite minimal normal kernel case when the reduction step is supplied explicitly.
Show proof
by
intro P hP
rcases hreduce P hP with ⟨P₀, hP₀, hker, hback⟩
exact hback (hfinite P₀ hP₀ hker)Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Kernel and image statements are verified after quotienting by sufficiently small open normal subgroups, where they become ordinary finite group calculations. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem weakLiftingProperty_iff_finiteTargetProblems
{G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G]
{E : EmbeddingProblemPredicate.{u}}
(hreduce :
∀ P : TopologicalEmbeddingProblem G, E P →
∃ P₀ : TopologicalEmbeddingProblem G, E P₀ ∧
P₀.HasFiniteMinimalNormalKernel ∧
(P₀.HasWeakSolution → P.HasWeakSolution))
(hfiniteTargetOfFiniteMinimalKernel :
∀ P : TopologicalEmbeddingProblem G, E P →
P.HasFiniteMinimalNormalKernel → Finite P.A) :
HasWeakLiftingPropertyOver E G ↔
∀ P : TopologicalEmbeddingProblem G, E P → Finite P.A → P.HasWeakSolutionShow proof
by
constructor
· intro h P hP _hfinite
exact h P hP
· intro h P hP
rcases hreduce P hP with ⟨P₀, hP₀, hmin, hback⟩
exact hback (h P₀ hP₀ (hfiniteTargetOfFiniteMinimalKernel P₀ hP₀ hmin))Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. Kernel and image statements are verified after quotienting by sufficiently small open normal subgroups, where they become ordinary finite group calculations. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□