| import OAI.GroupTheory.UniversalGroup.CellularLifting |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Set Metric Topology BigOperators |
| open Set Metric Topology |
|
|
| namespace UniversalFInfinity.Milnor |
|
|
| section |
| open Set Metric Topology BigOperators UniversalFInfinity.Simplicial |
| open scoped Classical unitInterval |
| universe u v |
| variable {G : Type u} |
| local instance (priority := 2000) (α : Type*) : DecidableEq (ℕ × α) := Classical.decEq (ℕ × α) |
|
|
| def boundedInclusion (N : ℕ) : C(Simplicial.Space (bounded G N),Space G) := |
| ⟨Simplicial.inclusion (bounded G N) (complex G) (fun _ h => bounded_le G N h), |
| Simplicial.continuous_inclusion _ _ _⟩ |
|
|
| def boundedContraction (N : ℕ) (g : G) : |
| (boundedInclusion (G := G) N).Homotopy (ContinuousMap.const _ (vertex (N,g))) := by |
| let H := Simplicial.straightHomotopy (bounded G N) (complex G) id (fun _ => (N,g)) |
| (bounded_id_face N) (bounded_const_face N g) (bounded_contiguous N g) |
| exact H.cast (ContinuousMap.ext (fun x => bounded_map_id N x)) |
| (ContinuousMap.ext (fun x => map_const_vertex (bounded G N) (N,g) (bounded_const_face N g) x)) |
|
|
| lemma compactMap_bounded {A : Type v} [TopologicalSpace A] [CompactSpace A] (f : C(A,Space G)) : |
| ∃ N : ℕ, ∃ F : C(A,Simplicial.Space (bounded G N)), (boundedInclusion N).comp F = f := by |
| obtain ⟨S,hS⟩ := Simplicial.compact_bounded_vertices (complex G) (isCompact_range f.continuous) |
| let N := S.sup Prod.fst + 1 |
| have hN (a : A) (v : ℕ × G) (hv : v ∈ (f a).1.support) : v.1 < N := by |
| exact Nat.lt_succ_of_le (Finset.le_sup (f := Prod.fst) (hS (f a) (mem_range_self a) hv)) |
| let F : A → Simplicial.Space (bounded G N) := fun a => |
| ⟨(f a).1,(f a).2.1,(f a).2.2.1,(f a).2.2.2,hN a⟩ |
| have hc : Continuous F := |
| (Simplicial.inclusion_closedEmbedding (bounded G N) (complex G) |
| (fun _ h => bounded_le G N h)).isEmbedding.continuous_iff.mpr f.continuous |
| exact ⟨N,⟨F,hc⟩,rfl⟩ |
|
|
| lemma compactMap_nullhomotopic {A : Type v} [TopologicalSpace A] [CompactSpace A] |
| (g : G) (f : C(A,Space G)) : ∃ y, Nonempty (f.Homotopy (ContinuousMap.const _ y)) := by |
| obtain ⟨N,F,hF⟩ := compactMap_bounded f |
| exact ⟨vertex (N,g),⟨((boundedContraction N g).compContinuousMap F).cast hF rfl⟩⟩ |
|
|
| lemma sphereFillable (g : G) : Cellular.SphereFillable (Space G) := by |
| intro n f |
| cases n with |
| | zero => |
| refine ⟨ContinuousMap.const _ (vertex (0,g)),?_⟩ |
| intro x |
| have hx : False := by |
| have he := x.2 |
| simp only [Metric.mem_sphere,Subsingleton.elim x.1 0,dist_self] at he |
| exact zero_ne_one he |
| exact hx.elim |
| | succ n => |
| let := SphereTopology.sphere_nonempty n |
| obtain ⟨y,⟨H⟩⟩ := compactMap_nullhomotopic g f |
| obtain ⟨F,hF⟩ := SphereTopology.extends_of_nullhomotopic H.symm |
| exact ⟨F,fun x => ContinuousMap.congr_fun hF x⟩ |
|
|
| lemma contractible [Nonempty G] : ContractibleSpace (Space G) := by |
| let g : G := Classical.choice ‹Nonempty G› |
| let : Nonempty (Space G) := ⟨vertex (0,g)⟩ |
| exact Cellular.contractible_of_sphereFillable (sphereFillable g) |
|
|
| end |
|
|
| section |
| open Set Metric Topology BigOperators UniversalFInfinity.Simplicial |
| open scoped Classical |
| universe u v |
| variable {G : Type u} [Group G] |
| local instance (priority := 2000) (α : Type*) : DecidableEq (ℕ × α) := Classical.decEq (ℕ × α) |
|
|
| def translate (g : G) (s : Finset (ℕ × G)) : Finset (ℕ × G) := s.image (fun v => (v.1,g*v.2)) |
|
|
| lemma translate_one (s : Finset (ℕ × G)) : translate 1 s = s := by |
| simp only [translate,one_mul,Prod.mk.eta,Finset.image_id'] |
| lemma translate_mul (g h : G) (s : Finset (ℕ × G)) : translate g (translate h s) = translate (g*h) s := by |
| simp only [translate,Finset.image_image,Function.comp_def,mul_assoc] |
| lemma translate_mono (g : G) {s t : Finset (ℕ × G)} (h : s ⊆ t) : translate g s ⊆ translate g t := |
| Finset.image_subset_image h |
| lemma translate_valid (g : G) {s : Finset (ℕ × G)} (hs : s ∈ complex G) : translate g s ∈ complex G := |
| map_face (g * ·) id Function.injective_id ⟨s,hs⟩ |
| |
| def orbitSpan (S : Finset (Finset (ℕ × G))) : PreAbstractSimplicialComplex (ℕ × G) where |
| faces := {t | t ∈ complex G ∧ ∃ g : G, ∃ s ∈ S, t ⊆ translate g s} |
| isRelLowerSet_faces := by |
| intro t ht |
| refine ⟨ht.1.1,?_⟩ |
| intro u hut hu |
| obtain ⟨g,s,hs,hts⟩ := ht.2 |
| exact ⟨(complex G).isRelLowerSet_faces.mem_of_le ht.1 hut hu,g,s,hs,hut.trans hts⟩ |
| |
| lemma orbitSpan_le (S : Finset (Finset (ℕ × G))) : orbitSpan S ≤ complex G := fun _ h => h.1 |
| lemma orbitSpan_mono {S T : Finset (Finset (ℕ × G))} (h : S ⊆ T) : orbitSpan S ≤ orbitSpan T := by |
| intro t ht |
| obtain ⟨g,s,hs,hts⟩ := ht.2 |
| exact ⟨ht.1,g,s,h hs,hts⟩ |
| lemma orbitSpan_contains (S : Finset (Finset (ℕ × G))) {t : Finset (ℕ × G)} |
| (ht : t ∈ complex G) (h : ∃ s ∈ S, t ⊆ s) : t ∈ orbitSpan S := by |
| obtain ⟨s,hs,hts⟩ := h |
| exact ⟨ht,1,s,hs,by rwa [translate_one]⟩ |
| lemma orbitSpan_invariant (S : Finset (Finset (ℕ × G))) (g : G) |
| {t : Finset (ℕ × G)} (ht : t ∈ orbitSpan S) : translate g t ∈ orbitSpan S := by |
| obtain ⟨h,s,hs,hts⟩ := ht.2 |
| exact ⟨translate_valid g ht.1,g*h,s,hs,by rw [←translate_mul]; exact translate_mono g hts⟩ |
| |
| lemma orbitSpan_bounded (S : Finset (Finset (ℕ × G))) : |
| orbitSpan S ≤ bounded G ((S.biUnion id).sup Prod.fst + 1) := by |
| intro t ht |
| refine ⟨ht.1,?_⟩ |
| intro v hv |
| obtain ⟨g,s,hs,hts⟩ := ht.2 |
| obtain ⟨w,hw,rfl⟩ := Finset.mem_image.mp (hts hv) |
| have hh : w.1 ≤ (S.biUnion id).sup Prod.fst := Finset.le_sup (f := Prod.fst) (Finset.mem_biUnion.mpr ⟨s,hs,hw⟩) |
| exact Nat.lt_succ_of_le hh |
| |
| def Controlled (K : PreAbstractSimplicialComplex (ℕ × G)) : Prop := |
| ∃ S : Finset (Finset (ℕ × G)), K ≤ orbitSpan S |
| |
| lemma controlled_orbitSpan (S : Finset (Finset (ℕ × G))) : Controlled (orbitSpan S) := ⟨S,le_rfl⟩ |
| lemma controlled_bounded {K : PreAbstractSimplicialComplex (ℕ × G)} (hK : Controlled K) : |
| ∃ N, K ≤ bounded G N := by |
| obtain ⟨S,hS⟩ := hK |
| exact ⟨_,hS.trans (orbitSpan_bounded S)⟩ |
| |
| lemma orbitSpan_finite_representatives (S : Finset (Finset (ℕ × G))) : |
| ∃ T : Finset (Finset (ℕ × G)), ∀ t ∈ orbitSpan S, ∃ s ∈ T, ∃ g : G, t = translate g s := by |
| refine ⟨S.biUnion Finset.powerset,?_⟩ |
| intro t ht |
| obtain ⟨g,s,hs,hts⟩ := ht.2 |
| have hh : translate g⁻¹ t ⊆ s := by |
| have hh := translate_mono g⁻¹ hts |
| simpa only [translate_mul,inv_mul_cancel,translate_one] using hh |
| refine ⟨translate g⁻¹ t,Finset.mem_biUnion.mpr ⟨s,hs,Finset.mem_powerset.mpr hh⟩,g,?_⟩ |
| rw [translate_mul,mul_inv_cancel,translate_one] |
| |
| end |
| section |
| open Set Metric Topology BigOperators UniversalFInfinity.Simplicial |
| open scoped Classical |
| universe u v |
| variable {G : Type u} {H : Type v} [Group G] [Group H] |
| local instance (priority := 2000) (α : Type*) : DecidableEq (ℕ × α) := Classical.decEq (ℕ × α) |
| |
| def vertexMap (φ : G →* H) (α : ℕ → ℕ) : ℕ × G → ℕ × H := fun v => (α v.1,φ v.2) |
| def faceImage (φ : G →* H) (α : ℕ → ℕ) (s : Finset (ℕ × G)) : Finset (ℕ × H) := s.image (vertexMap φ α) |
| def imageShapes (φ : G →* H) (α : ℕ → ℕ) (S : Finset (Finset (ℕ × G))) : Finset (Finset (ℕ × H)) := S.image (faceImage φ α) |
| |
| lemma faceImage_translate (φ : G →* H) (α : ℕ → ℕ) (a : G) (s : Finset (ℕ × G)) : |
| faceImage φ α (translate a s) = translate (φ a) (faceImage φ α s) := by |
| simp only [faceImage,translate,Finset.image_image,vertexMap,Function.comp_def,map_mul] |
| |
| lemma faceImage_valid (φ : G →* H) (α : ℕ → ℕ) (hα : Function.Injective α) |
| {s : Finset (ℕ × G)} (hs : s ∈ complex G) : faceImage φ α s ∈ complex H := |
| map_face φ α hα ⟨s,hs⟩ |
| |
| lemma faceImage_controlled (φ : G →* H) (α : ℕ → ℕ) (hα : Function.Injective α) |
| (S : Finset (Finset (ℕ × G))) (s : Face (orbitSpan S)) : |
| faceImage φ α s.1 ∈ orbitSpan (imageShapes φ α S) := by |
| obtain ⟨a,t,ht,hst⟩ := s.2.2 |
| refine ⟨faceImage_valid φ α hα s.2.1,φ a,faceImage φ α t, |
| Finset.mem_image.mpr ⟨t,ht,rfl⟩,?_⟩ |
| rw [← faceImage_translate] |
| exact Finset.image_subset_image hst |
| |
| lemma unionFace_valid (φ : G →* H) (α β : ℕ → ℕ) |
| (hα : Function.Injective α) (hβ : Function.Injective β) |
| {s : Finset (ℕ × G)} (hs : s ∈ complex G) |
| (hc : ∀ v ∈ s, ∀ w ∈ s, α v.1 = β w.1 → v.1 = w.1) : |
| faceImage φ α s ∪ faceImage φ β s ∈ complex H := by |
| refine ⟨(faceImage_valid φ α hα hs).1.mono Finset.subset_union_left,?_⟩ |
| intro x hx y hy hxy |
| rcases Finset.mem_union.mp hx with hx | hx <;> rcases Finset.mem_union.mp hy with hy | hy |
| · exact (faceImage_valid φ α hα hs).2 hx hy hxy |
| · obtain ⟨v,hv,rfl⟩ := Finset.mem_image.mp hx |
| obtain ⟨w,hw,rfl⟩ := Finset.mem_image.mp hy |
| have hvw : v = w := hs.2 hv hw (hc v hv w hw hxy) |
| subst w |
| apply Prod.ext |
| · exact hxy |
| · rfl |
| · obtain ⟨v,hv,rfl⟩ := Finset.mem_image.mp hx |
| obtain ⟨w,hw,rfl⟩ := Finset.mem_image.mp hy |
| have hvw : w = v := hs.2 hw hv (hc w hw v hv hxy.symm) |
| subst w |
| apply Prod.ext |
| · exact hxy |
| · rfl |
| · exact (faceImage_valid φ β hβ hs).2 hx hy hxy |
| |
| def unionShapes (φ : G →* H) (α β : ℕ → ℕ) (S : Finset (Finset (ℕ × G))) : |
| Finset (Finset (ℕ × H)) := S.image (fun s => faceImage φ α s ∪ faceImage φ β s) |
| |
| lemma translate_union (a : H) (s t : Finset (ℕ × H)) : translate a (s ∪ t) = translate a s ∪ translate a t := |
| Finset.image_union _ _ |
| |
| lemma unionFace_controlled (φ : G →* H) (α β : ℕ → ℕ) |
| (hα : Function.Injective α) (hβ : Function.Injective β) |
| (S : Finset (Finset (ℕ × G))) (s : Face (orbitSpan S)) |
| (hc : ∀ v ∈ s.1, ∀ w ∈ s.1, α v.1 = β w.1 → v.1 = w.1) : |
| faceImage φ α s.1 ∪ faceImage φ β s.1 ∈ orbitSpan (unionShapes φ α β S) := by |
| obtain ⟨a,t,ht,hst⟩ := s.2.2 |
| refine ⟨unionFace_valid φ α β hα hβ s.2.1 hc,φ a,faceImage φ α t ∪ faceImage φ β t, |
| Finset.mem_image.mpr ⟨t,ht,rfl⟩,?_⟩ |
| rw [translate_union,←faceImage_translate,←faceImage_translate] |
| exact Finset.union_subset_union (Finset.image_subset_image hst) (Finset.image_subset_image hst) |
| |
| def fresh (N : ℕ) (i : ℕ) : ℕ := if i = 0 then 0 else N+i |
| lemma fresh_zero (N : ℕ) : fresh N 0 = 0 := by simp [fresh] |
| lemma fresh_injective (N : ℕ) : Function.Injective (fresh N) := by |
| intro i j h |
| unfold fresh at h |
| split_ifs at h with hi hj hj |
| · exact hi.trans hj.symm |
| · exfalso; omega |
| · exfalso; omega |
| · omega |
| |
| lemma fresh_cross (N : ℕ) (α : ℕ → ℕ) (hα : Function.Injective α) (hzero : α 0 = 0) |
| {i j : ℕ} (hi : α i < N) (h : α i = fresh N j) : i = j := by |
| by_cases hj : j = 0 |
| · subst j |
| apply hα |
| simpa only [fresh_zero,hzero] using h |
| · have hh : N ≤ fresh N j := by simp only [fresh,ite_eq_right hj]; exact Nat.le_add_right _ _ |
| omega |
| end |
| open Set Metric Topology BigOperators UniversalFInfinity.Simplicial |
| open scoped Classical unitInterval |
| universe u v |
| variable {G : Type u} {H : Type v} [Group G] [Group H] |
| local instance (priority := 2000) (α : Type*) : DecidableEq (ℕ × α) := Classical.decEq (ℕ × α) |
| |
| def shapeBound (S : Finset (Finset (ℕ × G))) (α : ℕ → ℕ) : ℕ := |
| (S.biUnion id).sup (fun v => α v.1) + 1 |
| lemma orbit_below_bound (S : Finset (Finset (ℕ × G))) (α : ℕ → ℕ) |
| (s : Face (orbitSpan S)) {v : ℕ × G} (hv : v ∈ s.1) : α v.1 < shapeBound S α := by |
| obtain ⟨a,t,ht,hst⟩ := s.2.2 |
| obtain ⟨w,hw,rfl⟩ := Finset.mem_image.mp (hst hv) |
| have hh : α w.1 ≤ (S.biUnion id).sup (fun v => α v.1) := |
| Finset.le_sup (f := fun v : ℕ × G => α v.1) (Finset.mem_biUnion.mpr ⟨t,ht,hw⟩) |
| exact Nat.lt_succ_of_le hh |
| |
| def relayerFresh (S : Finset (Finset (ℕ × G))) (α β : ℕ → ℕ) : ℕ → ℕ := |
| fresh (max (shapeBound S α) (shapeBound S β)) |
| def relayerTarget (φ : G →* H) (S : Finset (Finset (ℕ × G))) (α β : ℕ → ℕ) := |
| unionShapes φ α (relayerFresh S α β) S ∪ unionShapes φ β (relayerFresh S α β) S |
| |
| lemma relayerLeftValid (φ : G →* H) (S : Finset (Finset (ℕ × G))) |
| (α β : ℕ → ℕ) (hα : Function.Injective α) (hα0 : α 0 = 0) (s : Face (orbitSpan S)) : |
| faceImage φ α s.1 ∪ faceImage φ (relayerFresh S α β) s.1 ∈ orbitSpan (relayerTarget φ S α β) := by |
| apply orbitSpan_mono Finset.subset_union_left |
| apply unionFace_controlled φ α (relayerFresh S α β) hα (fresh_injective _) S s |
| intro v hv w _ h |
| exact fresh_cross _ α hα hα0 ((orbit_below_bound S α s hv).trans_le (le_max_left _ _)) h |
| |
| lemma relayerRightValid (φ : G →* H) (S : Finset (Finset (ℕ × G))) |
| (α β : ℕ → ℕ) (hβ : Function.Injective β) (hβ0 : β 0 = 0) (s : Face (orbitSpan S)) : |
| faceImage φ β s.1 ∪ faceImage φ (relayerFresh S α β) s.1 ∈ orbitSpan (relayerTarget φ S α β) := by |
| apply orbitSpan_mono Finset.subset_union_right |
| apply unionFace_controlled φ β (relayerFresh S α β) hβ (fresh_injective _) S s |
| intro v hv w _ h |
| exact fresh_cross _ β hβ hβ0 ((orbit_below_bound S β s hv).trans_le (le_max_right _ _)) h |
| |
| variable (φ : G →* H) (S : Finset (Finset (ℕ × G))) (α β : ℕ → ℕ) |
| variable (hα : Function.Injective α) (hβ : Function.Injective β) (hα0 : α 0 = 0) (hβ0 : β 0 = 0) |
| include hα hα0 in |
| lemma relayerLeftMapValid (s : Face (orbitSpan S)) : |
| s.1.image (vertexMap φ α) ∈ orbitSpan (relayerTarget φ S α β) := |
| (orbitSpan _).isRelLowerSet_faces.mem_of_le (relayerLeftValid φ S α β hα hα0 s) |
| Finset.subset_union_left (s.2.1.1.image _) |
| include hβ hβ0 in |
| lemma relayerRightMapValid (s : Face (orbitSpan S)) : |
| s.1.image (vertexMap φ β) ∈ orbitSpan (relayerTarget φ S α β) := |
| (orbitSpan _).isRelLowerSet_faces.mem_of_le (relayerRightValid φ S α β hβ hβ0 s) |
| Finset.subset_union_left (s.2.1.1.image _) |
| include hα hα0 in |
| lemma relayerFreshMapValid (s : Face (orbitSpan S)) : |
| s.1.image (vertexMap φ (relayerFresh S α β)) ∈ orbitSpan (relayerTarget φ S α β) := |
| (orbitSpan _).isRelLowerSet_faces.mem_of_le (relayerLeftValid φ S α β hα hα0 s) |
| Finset.subset_union_right (s.2.1.1.image _) |
| |
| def relayerLeftMap := mapC (orbitSpan S) (orbitSpan (relayerTarget φ S α β)) |
| (vertexMap φ α) (relayerLeftMapValid φ S α β hα hα0) |
| def relayerRightMap := mapC (orbitSpan S) (orbitSpan (relayerTarget φ S α β)) |
| (vertexMap φ β) (relayerRightMapValid φ S α β hβ hβ0) |
| def relayerFreshMap := mapC (orbitSpan S) (orbitSpan (relayerTarget φ S α β)) |
| (vertexMap φ (relayerFresh S α β)) (relayerFreshMapValid φ S α β hα hα0) |
| |
| def relayerLeftHomotopy : (relayerLeftMap φ S α β hα hα0).Homotopy (relayerFreshMap φ S α β hα hα0) := |
| straightHomotopy _ _ _ _ _ _ (relayerLeftValid φ S α β hα hα0) |
| def relayerRightHomotopy : (relayerRightMap φ S α β hβ hβ0).Homotopy (relayerFreshMap φ S α β hα hα0) := |
| straightHomotopy _ _ _ _ _ _ (relayerRightValid φ S α β hβ hβ0) |
| |
| def relayerHomotopy : (relayerLeftMap φ S α β hα hα0).Homotopy (relayerRightMap φ S α β hβ hβ0) := |
| (relayerLeftHomotopy φ S α β hα hα0).trans (relayerRightHomotopy φ S α β hα hβ hα0 hβ0).symm |
| |
| lemma relayerLeftPure (x : Simplicial.Space (orbitSpan S)) (g : G) (hx : x.1 = Finsupp.single (0,g) 1) : |
| relayerLeftMap φ S α β hα hα0 x = relayerFreshMap φ S α β hα hα0 x := by |
| apply Subtype.ext |
| change x.1.mapDomain (vertexMap φ α) = x.1.mapDomain (vertexMap φ (relayerFresh S α β)) |
| rw [hx,Finsupp.mapDomain_single,Finsupp.mapDomain_single] |
| congr 1 |
| exact Prod.ext (hα0.trans (fresh_zero (max (shapeBound S α) (shapeBound S β))).symm) rfl |
| lemma relayerRightPure (x : Simplicial.Space (orbitSpan S)) (g : G) (hx : x.1 = Finsupp.single (0,g) 1) : |
| relayerRightMap φ S α β hβ hβ0 x = relayerFreshMap φ S α β hα hα0 x := by |
| apply Subtype.ext |
| change x.1.mapDomain (vertexMap φ β) = x.1.mapDomain (vertexMap φ (relayerFresh S α β)) |
| rw [hx,Finsupp.mapDomain_single,Finsupp.mapDomain_single] |
| congr 1 |
| exact Prod.ext (hβ0.trans (fresh_zero (max (shapeBound S α) (shapeBound S β))).symm) rfl |
| |
| lemma relayerLeftHomotopy_fixed (x : Simplicial.Space (orbitSpan S)) (g : G) |
| (hx : x.1 = Finsupp.single (0,g) 1) (t : I) : |
| relayerLeftHomotopy φ S α β hα hα0 (t,x) = relayerLeftMap φ S α β hα hα0 x := |
| straightHomotopy_fixed _ _ _ _ _ _ (relayerLeftValid φ S α β hα hα0) x |
| (relayerLeftPure φ S α β hα hα0 x g hx) t |
| lemma relayerRightHomotopy_fixed (x : Simplicial.Space (orbitSpan S)) (g : G) |
| (hx : x.1 = Finsupp.single (0,g) 1) (t : I) : |
| relayerRightHomotopy φ S α β hα hβ hα0 hβ0 (t,x) = relayerRightMap φ S α β hβ hβ0 x := |
| straightHomotopy_fixed _ _ _ _ _ _ (relayerRightValid φ S α β hβ hβ0) x |
| (relayerRightPure φ S α β hα hβ hα0 hβ0 x g hx) t |
| |
| theorem relayerHomotopy_fixed (x : Simplicial.Space (orbitSpan S)) (g : G) |
| (hx : x.1 = Finsupp.single (0,g) 1) (t : I) : |
| relayerHomotopy φ S α β hα hβ hα0 hβ0 (t,x) = relayerLeftMap φ S α β hα hα0 x := by |
| rw [relayerHomotopy,ContinuousMap.Homotopy.trans_apply] |
| split |
| · exact relayerLeftHomotopy_fixed φ S α β hα hα0 x g hx _ |
| · rw [ContinuousMap.Homotopy.symm_apply] |
| exact (relayerRightHomotopy_fixed φ S α β hα hβ hα0 hβ0 x g hx _).trans |
| ((relayerRightPure φ S α β hα hβ hα0 hβ0 x g hx).trans |
| (relayerLeftPure φ S α β hα hα0 x g hx).symm) |
| end UniversalFInfinity.Milnor |
| |
| end |
| |
| end OAI |
| |
