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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 65 66 67 68 69 70 71 72 73 74 75 | import Mathlib
import OAI.Algebra.GroupRing.AbsorbTracks
namespace OAI
section
namespace TorsionFreeZeroDivisors.AbsorbCells
open Set GroupRingTopology GroupRingTopology.Attachment
open scoped unitInterval
noncomputable section
variable {G S Q Y : Type} [Group G] [TopologicalSpace G] [DiscreteTopology G]
[TopologicalSpace Y] [SimplyConnectedSpace Y]
(s : S→G) (w : Q→FreeGroup S) (hw : ∀i,FreeGroup.lift s (w i)=1)
(L : CellModule.Module G S→+CellModule.Module G Q)
(hL : ∀g i,L (CellModule.translate g (CellModule.fox (FreeGroup.lift s) (w i)))=Finsupp.single (g,i) 1)
(f : C(IndexedCells.Cover s w hw,Y)) (y : Y)
(hv : ∀g,f (IndexedCells.inclusion s w hw (Cayley.vertex s g))=y)
def graphHomotopy : (AbsorbTracks.graph s w hw f).Homotopy (.const _ y) :=
(ContinuousMap.Homotopy.refl _).trans (AbsorbTracks.final s w hw L f y hv)
def diskHomotopy (e : G×Q) : (AbsorbTracks.cellMap s w hw f e.1 e.2).Homotopy (.const _ y) :=
DiskTracks.extension (AbsorbTracks.cellMap s w hw f e.1 e.2)
(AbsorbTracks.cellMap_boundary s w hw f y hv e.1 e.2)
(AbsorbTracks.finalTrack s w hw L f y hv e.1 e.2)
(AbsorbTracks.homotopic s w hw L hL f y hv e.1 e.2).some
lemma diskHomotopy_boundary (e : G×Q) (t : unitInterval) (z : Sphere 2) :
diskHomotopy s w hw L hL f y hv e (t,boundary 2 z)=
graphHomotopy s w hw L f y hv (t,equivariantFamily (G:=G) (IndexedCells.attaching s w hw) e z) := by
change DiskTracks.extension _ _ _ _ (t,boundary 2 z)=_
erw [DiskTracks.extension_boundary]
change _=((ContinuousMap.Homotopy.refl _).trans (AbsorbTracks.final s w hw L f y hv)) _
rw [ContinuousMap.Homotopy.trans_apply,IndexedCells.attaching_translated]
split_ifs
· exact ContinuousMap.congr_fun (HomotopyTracks.natural_cast (f:=AbsorbTracks.graph s w hw f) (hv e.1) (IndexedCells.translated s w hw e.1 e.2)) z
· exact HomotopyTracks.track_boundary (f:=AbsorbTracks.graph s w hw f) (hv e.1) (AbsorbTracks.final s w hw L f y hv)
(fun t=>AbsorbTracks.final_vertex s w hw L f y hv t e.1) (IndexedCells.translated s w hw e.1 e.2) _ z
def graphPaths : C(Cayley.Graph s,C(unitInterval,Y)) :=
((graphHomotopy s w hw L f y hv).toContinuousMap.comp ⟨Prod.swap,continuous_swap⟩).curry
def cellPaths (e : G×Q) : C(Disk 2,C(unitInterval,Y)) :=
((diskHomotopy s w hw L hL f y hv e).toContinuousMap.comp ⟨Prod.swap,continuous_swap⟩).curry
lemma paths_boundary (e : G×Q) (z : Sphere 2) : cellPaths s w hw L hL f y hv e (boundary 2 z)=
graphPaths s w hw L f y hv (equivariantFamily (G:=G) (IndexedCells.attaching s w hw) e z) := by
ext t
exact diskHomotopy_boundary s w hw L hL f y hv e t z
def paths : C(IndexedCells.Cover s w hw,C(unitInterval,Y)) :=
lift (equivariantFamily (G:=G) (IndexedCells.attaching s w hw))
(graphPaths s w hw L f y hv) (cellPaths s w hw L hL f y hv) (paths_boundary s w hw L hL f y hv)
lemma paths_zero (z : IndexedCells.Cover s w hw) : paths s w hw L hL f y hv z 0=f z := by
cases z with
| inl z => exact (graphHomotopy s w hw L f y hv).apply_zero z
| inr z =>
change diskHomotopy s w hw L hL f y hv z.1 (0,interior 2 z.2)=_
rw [ContinuousMap.Homotopy.apply_zero]
change f (cell _ z.1 (interior 2 z.2))=f (Sum.inr z)
rw [cell_interior]
lemma paths_one (z : IndexedCells.Cover s w hw) : paths s w hw L hL f y hv z 1=y := by
cases z with
| inl z=>exact (graphHomotopy s w hw L f y hv).apply_one z
| inr z=>exact (diskHomotopy s w hw L hL f y hv z.1).apply_one (interior 2 z.2)
def homotopy : f.Homotopy (.const _ y) where
toFun z:=paths s w hw L hL f y hv z.2 z.1
continuous_toFun:=(paths s w hw L hL f y hv).uncurry.continuous.comp continuous_swap
map_zero_left:=paths_zero s w hw L hL f y hv
map_one_left:=paths_one s w hw L hL f y hv
end
end TorsionFreeZeroDivisors.AbsorbCells
end
end OAI
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