| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| namespace Problem340 | |
| abbrev ClockAmbient := EuclideanSpace ℝ (Fin 2) | |
| abbrev SphereAmbient := EuclideanSpace ℝ (Fin 3) | |
| abbrev StandardClock := Metric.sphere (0 : ClockAmbient) (1 : ℝ) | |
| abbrev StandardSphere2 := Metric.sphere (0 : SphereAmbient) (1 : ℝ) | |
| abbrev CounterexampleManifold (q : ℕ) := | |
| StandardClock × (Fin (q + 1) → StandardSphere2) | |
| abbrev CounterexampleAmbient (q : ℕ) := | |
| ClockAmbient × (Fin (q + 1) → SphereAmbient) | |
| def standardEmbedding (q : ℕ) : | |
| CounterexampleManifold q → CounterexampleAmbient q := | |
| fun x => (x.1.1, fun i => (x.2 i).1) | |
| def IsC1SelfMap (q : ℕ) (f : CounterexampleManifold q → CounterexampleManifold q) : Prop := | |
| Continuous f ∧ | |
| ∀ x : CounterexampleManifold q, | |
| ∃ U : Set (CounterexampleAmbient q), | |
| IsOpen U ∧ standardEmbedding q x ∈ U ∧ | |
| ∃ F : CounterexampleAmbient q → CounterexampleAmbient q, | |
| ContDiffOn ℝ 1 F U ∧ | |
| ∀ y : CounterexampleManifold q, | |
| standardEmbedding q y ∈ U → | |
| standardEmbedding q (f y) = F (standardEmbedding q y) | |
| def continuousMapOf {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] | |
| (f : X → Y) (hf : Continuous f) : ContinuousMap X Y := | |
| ⟨f, hf⟩ | |
| abbrev RealSingularHomology (n : ℕ) (X : Type) [TopologicalSpace X] := | |
| (((AlgebraicTopology.singularHomologyFunctor (ModuleCat ℝ) n).obj | |
| (ModuleCat.of ℝ ℝ)).obj (TopCat.of X)) | |
| def realHomologyMap {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] | |
| (n : ℕ) (f : X → Y) (hf : Continuous f) : | |
| RealSingularHomology n X →ₗ[ℝ] RealSingularHomology n Y := | |
| (((AlgebraicTopology.singularHomologyFunctor (ModuleCat ℝ) n).obj | |
| (ModuleCat.of ℝ ℝ)).map | |
| (TopCat.ofHom (continuousMapOf f hf))).hom | |
| def totalRealHomologySpectralRadius {X : Type} [TopologicalSpace X] | |
| (f : X → X) (hf : Continuous f) : ℝ := | |
| sSup {r : ℝ | | |
| ∃ n : ℕ, ∃ a b : ℝ, | |
| ∃ u v : RealSingularHomology n X, | |
| (u ≠ 0 ∨ v ≠ 0) ∧ | |
| realHomologyMap n f hf u = a • u - b • v ∧ | |
| realHomologyMap n f hf v = b • u + a • v ∧ | |
| r = Real.sqrt (a ^ 2 + b ^ 2)} | |
| end Problem340 | |
| end | |
| end OAI | |