File size: 2,303 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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
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