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d23d6f3 | 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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import Mathlib
namespace OAI
/-! Loop-space homeomorphisms and the degree-shift equivalence. -/
noncomputable section
universe u v w
namespace NearbyLagrangian.BasedHomotopy
open ContinuousMap
open scoped Topology Topology.Homotopy
variable {N M X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
variable {x : X} {y : Y}
theorem genLoop_homotopic_iff_joined (f g : GenLoop N X x) :
GenLoop.Homotopic f g ↔ Joined f g := by
constructor
· rintro ⟨H⟩
refine ⟨{
toFun := fun s => ⟨H.toContinuousMap.curry s,fun t ht =>
(H.prop s t ht).trans (GenLoop.boundary f t ht)⟩
continuous_toFun := H.toContinuousMap.curry.continuous.subtype_mk _
source' := ?_
target' := ?_ }⟩
· ext t; exact H.apply_zero t
· ext t; exact H.apply_one t
· rintro ⟨P⟩
refine ⟨{
toFun := fun z => P z.1 z.2
continuous_toFun := ?_
map_zero_left := ?_
map_one_left := ?_
prop' := ?_ }⟩
· exact (continuous_subtype_val.comp (P.continuous.comp continuous_fst)).eval continuous_snd
· intro t; exact congrArg (fun z : GenLoop N X x => z t) P.source
· intro t; exact congrArg (fun z : GenLoop N X x => z t) P.target
· intro s t ht
exact (GenLoop.boundary (P s) t ht).trans (GenLoop.boundary f t ht).symm
def piEquivOfLoopHomeomorph (e : GenLoop N X x ≃ₜ GenLoop M Y y) :
HomotopyGroup N X x ≃ HomotopyGroup M Y y :=
Quotient.congr e.toEquiv fun f g => by
change GenLoop.Homotopic f g ↔ GenLoop.Homotopic (e f) (e g)
rw [genLoop_homotopic_iff_joined,genLoop_homotopic_iff_joined]
constructor
· intro h; exact h.map e.continuous_toFun
· intro h
have hh : Joined (e.symm (e f)) (e.symm (e g)) := h.map e.symm.continuous_toFun
simpa only [Homeomorph.symm_apply_apply] using hh
def piReindexEquiv (e : N ≃ M) (x : X) :
HomotopyGroup N X x ≃ HomotopyGroup M X x :=
piEquivOfLoopHomeomorph (GenLoop.congr x e)
def flattenPathLoop (f : GenLoop N (Path y y) (Path.refl y)) :
GenLoop (N ⊕ Unit) Y y :=
⟨⟨fun a => f (a ∘ Sum.inl) (a (Sum.inr ())),by
apply Continuous.eval
· exact f.1.continuous.comp (by fun_prop)
· fun_prop⟩,by
rintro a ⟨i | i,hi⟩ <;> change f (a ∘ Sum.inl) (a (Sum.inr ())) = y
· have h : a ∘ Sum.inl ∈ Cube.boundary N := ⟨i,hi⟩
rw [GenLoop.boundary f _ h]
rfl
· cases i
rcases hi with hi | hi
· rw [hi]; exact (f _).source
· rw [hi]; exact (f _).target⟩
def curryPathLoop (g : GenLoop (N ⊕ Unit) Y y) :
GenLoop N (Path y y) (Path.refl y) :=
⟨⟨fun a => {
toFun := fun r => g (Sum.elim a (fun _ => r))
continuous_toFun := g.1.continuous.comp (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)
source' := GenLoop.boundary g _ ⟨Sum.inr (),Or.inl rfl⟩
target' := GenLoop.boundary g _ ⟨Sum.inr (),Or.inr rfl⟩ },by
apply Path.continuous_uncurry_iff.mp
exact g.1.continuous.comp (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)⟩,by
rintro a ⟨i,hi⟩
apply Path.ext
funext r
exact GenLoop.boundary g _ ⟨Sum.inl i,hi⟩⟩
@[simp] lemma curry_flatten (f : GenLoop N (Path y y) (Path.refl y)) :
curryPathLoop (flattenPathLoop f) = f := by
apply Subtype.ext
ext a r
rfl
@[simp] lemma flatten_curry (g : GenLoop (N ⊕ Unit) Y y) :
flattenPathLoop (curryPathLoop g) = g := by
apply Subtype.ext
ext a
change g (Sum.elim (a ∘ Sum.inl) (fun _ => a (Sum.inr ()))) = g a
congr 1
funext i
cases i with
| inl i => rfl
| inr i => cases i; rfl
lemma continuous_flattenPathLoop : Continuous (@flattenPathLoop N Y _ y) := by
apply Continuous.subtype_mk
apply ContinuousMap.continuous_of_continuous_uncurry
change Continuous (fun z : GenLoop N (Path y y) (Path.refl y) × ((N ⊕ Unit) → unitInterval) =>
z.1 (z.2 ∘ Sum.inl) (z.2 (Sum.inr ())))
have h : Continuous (fun z : GenLoop N (Path y y) (Path.refl y) × ((N ⊕ Unit) → unitInterval) =>
z.1 (z.2 ∘ Sum.inl)) :=
(continuous_subtype_val.comp continuous_fst).eval
(continuous_pi (fun i => (continuous_apply (Sum.inl i)).comp continuous_snd))
exact h.eval ((continuous_apply (Sum.inr ())).comp continuous_snd)
lemma continuous_curryPathLoop : Continuous (@curryPathLoop N Y _ y) := by
apply Continuous.subtype_mk
apply ContinuousMap.continuous_of_continuous_uncurry
apply Path.continuous_uncurry_iff.mp
exact (continuous_subtype_val.comp (continuous_fst.comp continuous_fst)).eval
(by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)
def loopSpaceCubeHomeomorph :
GenLoop N (Path y y) (Path.refl y) ≃ₜ GenLoop (N ⊕ Unit) Y y where
toFun := flattenPathLoop
invFun := curryPathLoop
left_inv := curry_flatten
right_inv := flatten_curry
continuous_toFun := continuous_flattenPathLoop
continuous_invFun := continuous_curryPathLoop
def loopPiEquiv (y : Y) :
HomotopyGroup N (Path y y) (Path.refl y) ≃ HomotopyGroup (N ⊕ Unit) Y y :=
piEquivOfLoopHomeomorph loopSpaceCubeHomeomorph
def pi9LoopEquivPi10 (y : Y) :
π_ 9 (Path y y) (Path.refl y) ≃ π_ 10 Y y :=
(loopPiEquiv y).trans (piReindexEquiv
((Equiv.sumCongr (Equiv.refl (Fin 9)) (Equiv.ofUnique Unit (Fin 1))).trans finSumFinEquiv) y)
end NearbyLagrangian.BasedHomotopy
end
end OAI
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