| import Mathlib |
| import OAI.Algebra.GroupRing.DiskTracks |
| import OAI.Algebra.GroupRing.HomotopyTracks |
|
|
| namespace OAI |
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|
| noncomputable section |
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|
| namespace TorsionFreeZeroDivisors.DiskTracks |
| open Set Metric Topology GroupRingTopology GroupRingTopology.Attachment |
| open GroupRingTopology.Cellular hiding Disk Sphere boundary |
| open scoped unitInterval |
| variable {Y : Type} [TopologicalSpace Y] {y : Y} {p : Path y y} |
| {c h : Path p (Path.refl y)} |
| def squareHomotopy (H : c.Homotopy h) : (square c).Homotopy (square h) where |
| toFun z := square (H.eval z.1) z.2 |
| continuous_toFun := (Path.continuous_uncurry_iff.mpr H.continuous).comp |
| ((continuous_fst.prodMk (continuous_diskTime.comp continuous_snd)).prodMk |
| (Cayley.continuous_edgeTime.comp (continuous_diskTail.comp continuous_snd))) |
| map_zero_left d := by |
| change H (0,diskTime d) _=c _ _ |
| rw [H.apply_zero];rfl |
| map_one_left d := by |
| change H (1,diskTime d) _=h _ _ |
| rw [H.apply_one];rfl |
| lemma squareHomotopy_boundary (H : c.Homotopy h) (t : unitInterval) (z : Sphere 2) : |
| squareHomotopy H (t,boundary 2 z)=LoopSphere.map p z := square_boundary (H.eval t) z |
|
|
| def extension (F : C(Disk 2,Y)) (hF : ∀z,F (boundary 2 z)=LoopSphere.map p z) |
| (h : Path p (Path.refl y)) (H : (physical F hF).Homotopy h) : F.Homotopy (.const _ y) := by |
| have e := (squareHomotopy H).trans (contract h) |
| exact { e with |
| map_zero_left := fun d=>(e.apply_zero d).trans (ContinuousMap.congr_fun (square_physical F hF) d) } |
| lemma extension_boundary (F : C(Disk 2,Y)) (hF : ∀z,F (boundary 2 z)=LoopSphere.map p z) |
| (h : Path p (Path.refl y)) (H : (physical F hF).Homotopy h) |
| (t : unitInterval) (z : Sphere 2) : |
| extension F hF h H (t,boundary 2 z)= |
| if ht : (t:ℝ)≤1/2 then LoopSphere.map p z |
| else LoopSphere.map (h ⟨2*t-1,unitInterval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 ht).le,t.2.2⟩⟩) z := by |
| change ((squareHomotopy H).trans (contract h)) (t,boundary 2 z)=_ |
| rw [ContinuousMap.Homotopy.trans_apply] |
| split_ifs |
| · exact squareHomotopy_boundary H _ z |
| · exact contract_boundary h _ z |
| end TorsionFreeZeroDivisors.DiskTracks |
| namespace TorsionFreeZeroDivisors.HomotopyTracks |
| open GroupRingTopology GroupRingTopology.Attachment |
| open scoped unitInterval |
| variable {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] |
| {x : X} {y : Y} {f : C(X,Y)} (hx : f x=y) |
| lemma natural_cast (p : Path x x) : LoopSphere.map (mapped hx p)=f.comp (LoopSphere.map p) := by |
| subst y |
| exact LoopSphere.natural p f |
| variable (H : f.Homotopy (.const _ y)) (hH : ∀t,H (t,x)=y) |
| lemma track_boundary (p : Path x x) (t : unitInterval) (z : Sphere 2) : |
| LoopSphere.map ((track hx H hH p) t) z=H (t,LoopSphere.map p z) := by |
| let k : C(X,Y) := ⟨fun u=>H (t,u),H.continuous.comp (continuous_const.prodMk continuous_id)⟩ |
| exact ContinuousMap.congr_fun (natural_cast (f:=k) (hH t) p) z |
| end TorsionFreeZeroDivisors.HomotopyTracks |
|
|
| end |
|
|
| end OAI |
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