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ae00908 | 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 | import Mathlib
import OAI.Algebra.GroupRing.DiskTracks
import OAI.Algebra.GroupRing.HomotopyTracks
namespace OAI
noncomputable section
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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