| import OAI.GroupTheory.UniversalGroup.Milnor |
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| namespace OAI |
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| noncomputable section |
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| open Set Metric Topology BigOperators |
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| open Set Metric Topology |
| namespace UniversalFInfinity.Cellular |
| universe u |
| variable {X : Type u} [TopologicalSpace X] [T2Space X] |
| variable {C : Set X} [CWComplex C] |
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| omit [T2Space X] in |
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| lemma closedCell_subset_finite_openCells (n : ℕ) (i : RelCWComplex.cell C n) : |
| ∃ I : Finset (Σ m, RelCWComplex.cell C m), |
| CWComplex.closedCell n i ⊆ ⋃ p ∈ I, CWComplex.openCell p.1 p.2 := by |
| classical |
| obtain ⟨J,hJ⟩ := CWComplex.cellFrontier_subset_finite_openCell n i |
| refine ⟨insert ⟨n,i⟩ ((Finset.range n).sigma J),?_⟩ |
| intro x hx |
| rw [← CWComplex.cellFrontier_union_openCell_eq_closedCell] at hx |
| rcases hx with hx | hx |
| swap |
| · exact Set.mem_iUnion.mpr ⟨⟨n,i⟩,Set.mem_iUnion.mpr ⟨Finset.mem_insert_self _ _,hx⟩⟩ |
| · have hxx := hJ hx |
| simp only [Set.mem_iUnion] at hxx |
| obtain ⟨m,hm,j,hj,hx⟩ := hxx |
| exact Set.mem_iUnion.mpr ⟨⟨m,j⟩,Set.mem_iUnion.mpr |
| ⟨Finset.mem_insert_of_mem (by simp only [Finset.mem_sigma,Finset.mem_range]; exact ⟨hm,hj⟩),hx⟩⟩ |
| |
| lemma cellSelection_closed {I : Type u} (c : I → Σ n, RelCWComplex.cell C n) |
| (hc : Function.Injective c) (x : I → X) |
| (hx : ∀ i, x i ∈ CWComplex.openCell (c i).1 (c i).2) (S : Set X) |
| (hS : S ⊆ Set.range x) : IsClosed S := by |
| apply (CWComplex.closed C S ?_).mpr |
| · intro n j |
| obtain ⟨J,hJ⟩ := closedCell_subset_finite_openCells n j |
| have hfinite : (S ∩ CWComplex.closedCell n j).Finite := by |
| have hcpre : (c ⁻¹' (J : Set (Σ n, RelCWComplex.cell C n))).Finite := |
| J.finite_toSet.preimage hc.injOn |
| apply (hcpre.image x).subset |
| rintro y ⟨hyS,hycell⟩ |
| obtain ⟨i,rfl⟩ := hS hyS |
| have hxx := hJ hycell |
| simp only [Set.mem_iUnion] at hxx |
| obtain ⟨p,hpJ,hxp⟩ := hxx |
| have heq : c i = p := by |
| by_contra hne |
| exact (CWComplex.disjoint_openCell_of_ne hne).notMem_of_mem_left (hx i) hxp |
| exact ⟨i,by change c i ∈ J; rw [heq]; exact hpJ,rfl⟩ |
| exact hfinite.isClosed |
| · rintro y hy |
| obtain ⟨i,rfl⟩ := hS hy |
| exact CWComplex.openCell_subset_complex (c i).1 (c i).2 (hx i) |
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| theorem compact_finitely_many_cells {K : Set X} (hK : IsCompact K) : |
| {p : Σ n, RelCWComplex.cell C n | (K ∩ CWComplex.openCell p.1 p.2).Nonempty}.Finite := by |
| classical |
| let I := {p : Σ n, RelCWComplex.cell C n | (K ∩ CWComplex.openCell p.1 p.2).Nonempty} |
| let x : I → X := fun i => i.2.choose |
| have hx (i : I) : x i ∈ CWComplex.openCell i.1.1 i.1.2 := i.2.choose_spec.2 |
| have hxK (i : I) : x i ∈ K := i.2.choose_spec.1 |
| have hinj : Function.Injective x := by |
| intro i j hij |
| apply Subtype.ext |
| by_contra hne |
| exact (CWComplex.disjoint_openCell_of_ne hne).notMem_of_mem_left (hx i) (hij ▸ hx j) |
| have hall : ∀ S ⊆ Set.range x, IsClosed S := |
| cellSelection_closed (C := C) Subtype.val Subtype.val_injective x hx |
| have hcompact : IsCompact (Set.range x) := |
| hK.of_isClosed_subset (hall _ subset_rfl) (Set.range_subset_iff.mpr hxK) |
| have hdiscrete : IsDiscrete (Set.range x) := by |
| apply isDiscrete_iff_forall_mem_exists_isClosed.mpr |
| intro S hS |
| exact ⟨S,hall S hS,Set.inter_eq_left.mpr hS⟩ |
| have hrange := hcompact.finite hdiscrete |
| have : Finite I := (Set.finite_range_iff hinj).mp hrange |
| exact Set.toFinite I |
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| theorem compact_bounded_skeleton {K : Set X} (hK : IsCompact K) (hKC : K ⊆ C) : |
| ∃ n : ℕ, K ⊆ CWComplex.skeletonLT C n := by |
| classical |
| let S := {p : Σ n, RelCWComplex.cell C n | (K ∩ CWComplex.openCell p.1 p.2).Nonempty} |
| have hS : S.Finite := compact_finitely_many_cells hK |
| let I := hS.toFinset |
| refine ⟨I.sup (fun p => p.1+1),?_⟩ |
| intro x hx |
| have hxC := hKC hx |
| rw [← CWComplex.iUnion_openCell_eq_complex (C := C)] at hxC |
| simp only [Set.mem_iUnion] at hxC |
| obtain ⟨m,j,hj⟩ := hxC |
| have hmem : (⟨m,j⟩ : Σ n, RelCWComplex.cell C n) ∈ I := by |
| exact hS.mem_toFinset.mpr ⟨x,hx,hj⟩ |
| have hm : m+1 ≤ I.sup (fun p => p.1+1) := Finset.le_sup (f := fun p => p.1+1) hmem |
| apply CWComplex.skeletonLT_mono (C := C) |
| (m := (m : ℕ∞)+1) (n := ((I.sup (fun p => p.1+1) : ℕ) : ℕ∞)) (by exact_mod_cast hm) |
| exact CWComplex.closedCell_subset_skeletonLT m j (CWComplex.openCell_subset_closedCell m j hj) |
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| theorem compactMap_bounded_skeleton {A : Type*} [TopologicalSpace A] [CompactSpace A] |
| (f : C(A,X)) (hf : ∀ a, f a ∈ C) : |
| ∃ n : ℕ, ∀ a, f a ∈ CWComplex.skeletonLT C n := by |
| obtain ⟨n,hn⟩ := compact_bounded_skeleton (C := C) (isCompact_range f.continuous) |
| (Set.range_subset_iff.mpr hf) |
| exact ⟨n,fun a => hn (Set.mem_range_self a)⟩ |
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| end UniversalFInfinity.Cellular |
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|
| namespace UniversalFInfinity.Simplicial |
| open Set Metric Topology BigOperators |
| open scoped Classical |
| universe u |
| variable {V : Type u} (K : PreAbstractSimplicialComplex V) |
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| lemma compact_bounded_vertices {A : Set (Space K)} (hA : IsCompact A) : |
| ∃ S : Finset V, ∀ x ∈ A, x.1.support ⊆ S := by |
| let T := {p : Σ n, Topology.RelCWComplex.cell (Set.univ : Set (Space K)) n | |
| (A ∩ CWComplex.openCell p.1 p.2).Nonempty} |
| have hT : T.Finite := Cellular.compact_finitely_many_cells hA |
| let S := hT.toFinset.biUnion (fun p => p.2.1.1) |
| refine ⟨S,?_⟩ |
| intro x hx |
| let s : Face K := ⟨x.1.support,x.2.2.2⟩ |
| let c := cellOfFace K s |
| have hc : (⟨s.1.card-1,c⟩ : Σ n, Topology.RelCWComplex.cell (Set.univ : Set (Space K)) n) ∈ hT.toFinset := by |
| rw [hT.mem_toFinset] |
| refine ⟨x,hx,?_⟩ |
| change x ∈ characteristicPE K c '' ball 0 1 |
| rw [characteristicPE_open] |
| rfl |
| exact Finset.subset_biUnion_of_mem (fun p => p.2.1.1) hc |
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| end UniversalFInfinity.Simplicial |
|
|
| open Set Metric Topology |
| namespace UniversalFInfinity.SphereTopology |
| abbrev Disk (n : ℕ) := closedBall (0 : Fin n → ℝ) 1 |
| abbrev Sphere (n : ℕ) := sphere (0 : Fin n → ℝ) 1 |
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| def boundary (n : ℕ) : C(Sphere n,Disk n) := |
| ⟨fun x => ⟨x.1,sphere_subset_closedBall x.2⟩,continuous_subtype_val.subtype_mk _⟩ |
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| lemma sphere_norm {n : ℕ} (x : Sphere n) : ‖x.1‖ = 1 := by |
| simpa only [mem_sphere,dist_zero_right] using x.2 |
| lemma disk_norm {n : ℕ} (x : Disk n) : ‖x.1‖ ≤ 1 := by |
| simpa only [mem_closedBall,dist_zero_right] using x.2 |
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| def cone (n : ℕ) : C(unitInterval × Sphere n,Disk n) := by |
| have hm (p : unitInterval × Sphere n) : (p.1:ℝ) • p.2.1 ∈ closedBall (0 : Fin n → ℝ) 1 := by |
| simp only [mem_closedBall,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_of_nonneg p.1.2.1,sphere_norm,mul_one] |
| exact p.1.2.2 |
| exact ⟨fun p => ⟨(p.1:ℝ) • p.2.1,hm p⟩, |
| ((continuous_subtype_val.comp continuous_fst).smul |
| (continuous_subtype_val.comp continuous_snd)).subtype_mk hm⟩ |
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| lemma cone_norm {n : ℕ} (p : unitInterval × Sphere n) : ‖(cone n p).1‖ = (p.1:ℝ) := by |
| simp only [cone,ContinuousMap.coe_mk,norm_smul,Real.norm_eq_abs,abs_of_nonneg p.1.2.1,sphere_norm,mul_one] |
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| lemma cone_surjective (n : ℕ) [Nonempty (Sphere n)] : Function.Surjective (cone n) := by |
| intro x |
| by_cases hx : x.1 = 0 |
| · obtain ⟨s⟩ := ‹Nonempty (Sphere n)› |
| refine ⟨(0,s),Subtype.ext ?_⟩ |
| simp only [cone,ContinuousMap.coe_mk,Set.Icc.coe_zero,zero_smul,hx] |
| · let s : Sphere n := ⟨‖x.1‖⁻¹ • x.1,by |
| simp only [mem_sphere,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm, |
| inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)]⟩ |
| refine ⟨(⟨‖x.1‖,norm_nonneg _,disk_norm x⟩,s),Subtype.ext ?_⟩ |
| change ‖x.1‖ • (‖x.1‖⁻¹ • x.1) = x.1 |
| rw [smul_smul,mul_inv_cancel₀ (norm_ne_zero_iff.mpr hx),one_smul] |
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| lemma cone_quotient (n : ℕ) [Nonempty (Sphere n)] : IsQuotientMap (cone n) := |
| IsQuotientMap.of_surjective_continuous (cone_surjective n) (cone n).continuous |
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| variable {Y : Type*} [TopologicalSpace Y] {n : ℕ} |
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| def Extends (f : C(Sphere n,Y)) : Prop := ∃ F : C(Disk n,Y), F.comp (boundary n) = f |
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| lemma extends_of_nullhomotopic [Nonempty (Sphere n)] {y : Y} {f : C(Sphere n,Y)} |
| (H : (ContinuousMap.const (Sphere n) y).Homotopy f) : Extends f := by |
| have hf : Function.FactorsThrough H.toContinuousMap (cone n) := by |
| intro p q hpq |
| have ht : p.1 = q.1 := Subtype.ext ((cone_norm p).symm.trans ((congrArg (fun z : Disk n => ‖z.1‖) hpq).trans (cone_norm q))) |
| by_cases hzero : (p.1:ℝ) = 0 |
| · have hp : p.1 = 0 := Subtype.ext hzero |
| have hq : q.1 = 0 := ht.symm.trans hp |
| change H (p.1,p.2) = H (q.1,q.2) |
| rw [hp,hq,H.apply_zero,H.apply_zero] |
| rfl |
| · have hs : p.2 = q.2 := by |
| apply Subtype.ext |
| have heq := congrArg (fun z : Disk n => z.1) hpq |
| change (p.1:ℝ) • p.2.1 = (q.1:ℝ) • q.2.1 at heq |
| rw [← ht] at heq |
| exact (smul_right_injective _ hzero) heq |
| exact congrArg H.toContinuousMap (Prod.ext ht hs) |
| let F : C(Disk n,Y) := (cone_quotient n).lift H.toContinuousMap hf |
| refine ⟨F,?_⟩ |
| ext s |
| have hh := ContinuousMap.congr_fun ((cone_quotient n).lift_comp H.toContinuousMap hf) (1,s) |
| have hcone : cone n (1,s) = boundary n s := by apply Subtype.ext; simp [cone,boundary] |
| simpa only [ContinuousMap.comp_apply,hcone,ContinuousMap.Homotopy.coe_toContinuousMap, |
| ContinuousMap.Homotopy.apply_one] using hh |
| |
| lemma nullhomotopic_of_extends {f : C(Sphere n,Y)} (hf : Extends f) : |
| ∃ y : Y, Nonempty ((ContinuousMap.const (Sphere n) y).Homotopy f) := by |
| obtain ⟨F,hF⟩ := hf |
| let z : Disk n := ⟨0,by simp⟩ |
| refine ⟨F z,⟨{ |
| toFun := fun p => F (cone n p) |
| continuous_toFun := F.continuous.comp (cone n).continuous |
| map_zero_left := ?_ |
| map_one_left := ?_ }⟩⟩ |
| · intro s |
| exact congrArg F (Subtype.ext (by simp [cone,z])) |
| · intro s |
| have hs : cone n (1,s) = boundary n s := by apply Subtype.ext; simp [cone,boundary] |
| rw [hs] |
| exact ContinuousMap.congr_fun hF s |
| |
| lemma extends_of_homotopy [Nonempty (Sphere n)] {f g : C(Sphere n,Y)} |
| (H : f.Homotopy g) (hg : Extends g) : Extends f := by |
| obtain ⟨y,⟨J⟩⟩ := nullhomotopic_of_extends hg |
| exact extends_of_nullhomotopic (J.trans H.symm) |
|
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| lemma sphere_nonempty (n : ℕ) : Nonempty (Sphere (n+1)) := |
| (NormedSpace.sphere_nonempty.mpr (by norm_num : (0:ℝ) ≤ 1)).coe_sort |
|
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| end UniversalFInfinity.SphereTopology |
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| end |
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| end OAI |
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