| import Mathlib |
|
|
| namespace OAI |
|
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| /-! Maps of based generalized loops and homotopy groups. -/ |
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| noncomputable section |
| universe u v w |
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|
|
| namespace NearbyLagrangian.BasedHomotopy |
| open scoped Topology Topology.Homotopy |
| variable {N X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] |
| variable {x : X} {y : Y} {z : Z} |
|
|
| def loopMap (e : C(X,Y)) (he : e x = y) (f : GenLoop N X x) : GenLoop N Y y := |
| ⟨e.comp f.1,fun t ht => (congrArg e (GenLoop.boundary f t ht)).trans he⟩ |
|
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| lemma loopMap_homotopic (e : C(X,Y)) (he : e x = y) {f g : GenLoop N X x} |
| (h : GenLoop.Homotopic f g) : GenLoop.Homotopic (loopMap e he f) (loopMap e he g) := by |
| obtain ⟨H⟩ := h |
| exact ⟨H.compContinuousMap e⟩ |
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|
| @[simp] theorem loopMap_const (e : C(X,Y)) (he : e x = y) : |
| loopMap (N:=N) e he GenLoop.const = GenLoop.const := by ext t; exact he |
| |
| @[simp] theorem loopMap_id (f : GenLoop N X x) : loopMap (ContinuousMap.id X) rfl f = f := by |
| ext t; rfl |
| |
| @[simp] theorem loopMap_comp (e : C(X,Y)) (he : e x = y) (d : C(Y,Z)) (hd : d y = z) |
| (f : GenLoop N X x) : loopMap d hd (loopMap e he f) = |
| loopMap (d.comp e) ((congrArg d he).trans hd) f := by |
| ext t; rfl |
| |
| lemma loopMap_transAt [DecidableEq N] (e : C(X,Y)) (he : e x = y) |
| (i : N) (f g : GenLoop N X x) : loopMap e he (GenLoop.transAt i f g) = |
| GenLoop.transAt i (loopMap e he f) (loopMap e he g) := by |
| ext t |
| simp only [loopMap,GenLoop.mk_apply,ContinuousMap.comp_apply,GenLoop.transAt,GenLoop.coe_copy,GenLoop.coe_coe] |
| split_ifs <;> rfl |
| |
| def groupMap [DecidableEq N] [Nonempty N] (e : C(X,Y)) (he : e x = y) : |
| HomotopyGroup N X x →* HomotopyGroup N Y y where |
| toFun := Quotient.map (loopMap e he) (fun _ _ h => loopMap_homotopic e he h) |
| map_one' := by |
| rw [HomotopyGroup.one_def,HomotopyGroup.one_def] |
| change Quotient.mk'' (loopMap e he GenLoop.const) = Quotient.mk'' GenLoop.const |
| rw [loopMap_const] |
| map_mul' a b := Quotient.inductionOn₂ a b fun f g => by |
| simp only [HomotopyGroup.mul_spec (i:=Classical.arbitrary N)] |
| change (⟦loopMap e he (GenLoop.transAt (Classical.arbitrary N) g f)⟧ : HomotopyGroup N Y y) = |
| ((· * ·) : HomotopyGroup N Y y → HomotopyGroup N Y y → HomotopyGroup N Y y) |
| ⟦loopMap e he f⟧ ⟦loopMap e he g⟧ |
| rw [HomotopyGroup.mul_spec (i:=Classical.arbitrary N),loopMap_transAt] |
|
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| @[simp] theorem groupMap_id [DecidableEq N] [Nonempty N] (a : HomotopyGroup N X x) : |
| groupMap (ContinuousMap.id X) rfl a = a := by |
| induction a using Quotient.inductionOn with |
| | h f => change Quotient.mk'' (loopMap _ _ f) = _; rw [loopMap_id]; rfl |
| |
| @[simp] theorem groupMap_comp [DecidableEq N] [Nonempty N] |
| (e : C(X,Y)) (he : e x = y) (d : C(Y,Z)) (hd : d y = z) (a : HomotopyGroup N X x) : |
| groupMap d hd (groupMap e he a) = groupMap (d.comp e) ((congrArg d he).trans hd) a := by |
| induction a using Quotient.inductionOn with |
| | h f => change Quotient.mk'' (loopMap _ _ (loopMap _ _ f)) = _; rw [loopMap_comp]; rfl |
| |
| end NearbyLagrangian.BasedHomotopy |
| |
| end |
| |
| end OAI |
| |
