| import Mathlib |
| import OAI.Algebra.GroupRing.PartialLift |
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| namespace OAI |
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| section |
| namespace TorsionFreeZeroDivisors.CellHEP |
| open Set Metric Topology GroupRingTopology GroupRingTopology.Cellular |
| open scoped unitInterval |
| noncomputable section |
| variable {n : ℕ} |
| abbrev Cylinder (n : ℕ) := unitInterval × Disk n |
| def denom (q : Cylinder n) : ℝ := max (2-q.1.1) (2*‖q.2.1‖) |
| lemma denom_pos (q : Cylinder n) : 0<denom q := |
| lt_of_lt_of_le (by linarith [q.1.2.2] : 0<2-q.1.1) (le_max_left _ _) |
| lemma denom_le_two (q : Cylinder n) : denom q ≤ 2 := by |
| have h := mem_closedBall_zero_iff.mp q.2.2 |
| exact max_le (by linarith [q.1.2.1]) (by linarith) |
| def scale (q : Cylinder n) : ℝ := 2/denom q |
| lemma scale_nonneg (q : Cylinder n) : 0 ≤ scale q := div_nonneg (by norm_num) (denom_pos q).le |
| lemma scale_ge_one (q : Cylinder n) : 1 ≤ scale q := by |
| rw [scale,le_div_iff₀ (denom_pos q)] |
| simpa using denom_le_two q |
| lemma scale_time (q : Cylinder n) : scale q*(2-q.1.1) ≤ 2 := by |
| dsimp [scale] |
| rw [div_mul_eq_mul_div,div_le_iff₀ (denom_pos q)] |
| have h : 2-q.1.1 ≤ denom q := le_max_left _ _ |
| nlinarith |
| lemma scale_norm (q : Cylinder n) : scale q*‖q.2.1‖ ≤ 1 := by |
| dsimp [scale] |
| rw [div_mul_eq_mul_div,div_le_iff₀ (denom_pos q)] |
| simpa only [one_mul,denom] using le_max_right (2-q.1.1) (2*‖q.2.1‖) |
| def retract (q : Cylinder n) : Cylinder n := |
| (⟨2-scale q*(2-q.1.1),by |
| constructor |
| · linarith [scale_time q] |
| · have h := mul_le_mul_of_nonneg_right (scale_ge_one q) (by linarith [q.1.2.2] : 0 ≤ 2-q.1.1) |
| linarith [q.1.2.2]⟩, |
| ⟨scale q • q.2.1,by |
| rw [mem_closedBall_zero_iff,norm_smul,Real.norm_eq_abs,abs_of_nonneg (scale_nonneg q)] |
| exact scale_norm q⟩) |
| lemma continuous_retract : Continuous (@retract n) := by |
| have hd : Continuous (@denom n) := by unfold denom;fun_prop |
| have hs : Continuous (@scale n) := continuous_const.div hd (fun q=>(denom_pos q).ne') |
| exact (Continuous.subtype_mk ((continuous_const.sub (hs.mul |
| (continuous_const.sub (continuous_subtype_val.comp continuous_fst))))) _).prodMk |
| (Continuous.subtype_mk (hs.smul (continuous_subtype_val.comp continuous_snd)) _) |
| def Rim (n : ℕ) : Set (Cylinder n) := {q | q.1=0 ∨ q.2.1 ∈ sphere 0 1} |
| lemma retract_mem (q : Cylinder n) : retract q∈Rim n := by |
| rcases le_total (2*‖q.2.1‖) (2-q.1.1) with h|h |
| · left |
| apply Subtype.ext |
| change 2-scale q*(2-q.1.1)=0 |
| have he : denom q=2-q.1.1 := max_eq_left h |
| rw [scale,←he,div_mul_cancel₀ _ (denom_pos q).ne'] |
| ring |
| · right |
| rw [mem_sphere_zero_iff_norm] |
| change ‖scale q • q.2.1‖=1 |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (scale_nonneg q)] |
| have he : denom q=2*‖q.2.1‖ := max_eq_right h |
| rw [scale,div_mul_eq_mul_div] |
| change (2*‖q.2.1‖)/denom q=1 |
| rw [←he,div_self (denom_pos q).ne'] |
| lemma retract_eq (q : Cylinder n) (h : q∈Rim n) : retract q=q := by |
| have hm : denom q=2 := by |
| apply le_antisymm (denom_le_two q) |
| rcases h with h|h |
| · have ht : q.1.1=0 := congrArg Subtype.val h |
| simp [denom,ht] |
| · have hn : ‖q.2.1‖=1 := mem_sphere_zero_iff_norm.mp h |
| simp [denom,hn] |
| have hs : scale q=1 := by simp [scale,hm] |
| apply Prod.ext |
| · apply Subtype.ext |
| change 2-scale q*(2-q.1.1)=q.1.1 |
| rw [hs];ring |
| · apply Subtype.ext |
| change scale q • q.2.1=q.2.1 |
| rw [hs,one_smul] |
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| def retraction : C(Cylinder n,Rim n) := ⟨fun q=>⟨retract q,retract_mem q⟩, |
| continuous_retract.subtype_mk _⟩ |
| variable {Y : Type*} [TopologicalSpace Y] |
| def face (n : ℕ) : Bool → Set (Rim n) |
| | false => {z | z.1.1=0} |
| | true => {z | z.1.2.1∈sphere 0 1} |
| lemma face_closed (b : Bool) : IsClosed (face n b) := by |
| cases b |
| · exact isClosed_eq (continuous_fst.comp continuous_subtype_val) continuous_const |
| · exact isClosed_sphere.preimage (continuous_subtype_val.comp |
| (continuous_snd.comp continuous_subtype_val)) |
| lemma face_cover : ⋃b,face n b=univ := by |
| apply eq_univ_of_forall |
| rintro ⟨q,hq⟩ |
| rcases hq with hq|hq |
| · exact mem_iUnion.mpr ⟨false,hq⟩ |
| · exact mem_iUnion.mpr ⟨true,hq⟩ |
| variable (f : C(Disk n,Y)) (H : C(unitInterval × Sphere n,Y)) |
| (h : ∀x,H (0,x)=f (boundary n x)) |
| def faceMap : ∀ b : Bool,C(face n b,Y) |
| | false => ⟨fun z=>f z.1.1.2, |
| f.continuous.comp (continuous_snd.comp (continuous_subtype_val.comp continuous_subtype_val))⟩ |
| | true => ⟨fun z=>H (z.1.1.1,⟨z.1.1.2.1,z.2⟩),H.continuous.comp |
| ((continuous_fst.comp (continuous_subtype_val.comp continuous_subtype_val)).prodMk |
| ((continuous_subtype_val.comp (continuous_snd.comp |
| (continuous_subtype_val.comp continuous_subtype_val))).subtype_mk _))⟩ |
| include h in |
| lemma faceMap_compat (b c : Bool) (z : Rim n) (hb : z∈face n b) (hc : z∈face n c) : |
| faceMap f H b ⟨z,hb⟩=faceMap f H c ⟨z,hc⟩ := by |
| have cross (hb : z∈face n false) (hc : z∈face n true) : |
| faceMap f H false ⟨z,hb⟩=faceMap f H true ⟨z,hc⟩ := by |
| change f z.1.2=H (z.1.1,⟨z.1.2.1,hc⟩) |
| change z.1.1=0 at hb |
| rw [hb,h] |
| rfl |
| cases b <;> cases c |
| · rfl |
| · exact cross hb hc |
| · exact (cross hc hb).symm |
| · rfl |
| def rimMap : C(Rim n,Y) := ⟨Set.liftCover (face n) (fun b=>faceMap f H b) |
| (faceMap_compat f H h) face_cover, |
| continuous_liftCover _ _ _ _ face_closed⟩ |
| lemma rimMap_face (b : Bool) (z : face n b) : |
| rimMap f H h z.1=faceMap f H b z := by |
| apply Set.liftCover_coe (S:=face n) (f:=fun b=>faceMap f H b) |
| (hf:=faceMap_compat f H h) (hS:=face_cover) |
| / |
| def extension : C(Cylinder n,Y) := (rimMap f H h).comp retraction |
| lemma extension_zero (x : Disk n) : extension f H h (0,x)=f x := by |
| have hr : retract (0,x)=(0,x) := retract_eq _ (Or.inl rfl) |
| change rimMap f H h ⟨retract (0,x),retract_mem (0,x)⟩=f x |
| simp only [hr] |
| exact rimMap_face f H h false ⟨⟨(0,x),Or.inl rfl⟩,rfl⟩ |
| lemma extension_boundary (t : unitInterval) (x : Sphere n) : |
| extension f H h (t,boundary n x)=H (t,x) := by |
| have hr : retract (t,boundary n x)=(t,boundary n x) := retract_eq _ (Or.inr x.2) |
| change rimMap f H h ⟨retract (t,boundary n x),retract_mem (t,boundary n x)⟩=H (t,x) |
| simp only [hr] |
| exact rimMap_face f H h true ⟨⟨(t,boundary n x),Or.inr x.2⟩,x.2⟩ |
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| end |
| end TorsionFreeZeroDivisors.CellHEP |
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| end |
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| end OAI |
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