| import Mathlib |
| import OAI.Algebra.GroupRing.AbsorbTracks |
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| namespace OAI |
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| 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 |
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| 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 |
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| end |
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| end OAI |
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