openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /AlternatingStability.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonAlternatingStabilization | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonAlternatingConfigurations | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.StabilizationComposition | |
| import OAI.GroupTheory.SimpleAmenable.Homology.PermutationAugmentation | |
| namespace OAI | |
| section | |
| section | |
| open Classical CategoryTheory CategoryTheory.Limits Representation Rep Finsupp | |
| namespace SimpleAmenable | |
| attribute [local instance 1200] Rep.hV2 | |
| namespace PolygonPlacement.Configuration | |
| variable {a m p n : ℕ} | |
| lemma stabilizeAlternating_smul_standard (g : polygonAlternatingGroup a n) : | |
| PolygonTracks.stabilizeAlternating a p n g • standard a p n=standard a p n := by | |
| apply ext | |
| intro i | |
| apply PolygonPlacement.ext | |
| intro x | |
| exact PolygonTracks.stabilize_left g.val (i,x) | |
| noncomputable def alternatingRestrict (a p n : ℕ) (hn : 32 ≤ n) | |
| (g : MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)) : | |
| polygonAlternatingGroup a n := by | |
| have hg : g.val.val∈PolygonTracks.bankFixer a p n := by | |
| have hh := g.property | |
| exact (show g.val ∈ (PolygonTracks.bankFixer a p n).comap | |
| (polygonAlternatingGroup a (p+n)).subtype from | |
| (alternating_stabilizer_standard a p n) ▸ hh) | |
| refine ⟨PolygonTracks.restrict ⟨g.val.val,hg⟩,?_⟩ | |
| rw [← PolygonTracks.stabilize_alternating_comap a p n hn] | |
| change PolygonTracks.stabilize a p n (PolygonTracks.restrict _)∈polygonAlternatingGroup a (p+n) | |
| rw [PolygonTracks.stabilize_restrict] | |
| exact g.val.property | |
| noncomputable def alternatingStabilizerEquiv (a p n : ℕ) (hn : 32 ≤ n) : | |
| polygonAlternatingGroup a n ≃* | |
| MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n) where | |
| toFun g := ⟨PolygonTracks.stabilizeAlternating a p n g,stabilizeAlternating_smul_standard g⟩ | |
| invFun := alternatingRestrict a p n hn | |
| left_inv g := by | |
| apply Subtype.ext | |
| change PolygonTracks.restrict ⟨PolygonTracks.stabilize a p n g.val,_⟩=g.val | |
| exact PolygonTracks.restrict_stabilize g.val | |
| right_inv g := by | |
| apply Subtype.ext | |
| apply Subtype.ext | |
| change PolygonTracks.stabilize a p n (PolygonTracks.restrict _) = g.val.val | |
| exact PolygonTracks.stabilize_restrict _ | |
| map_mul' g h := Subtype.ext ((PolygonTracks.stabilizeAlternating a p n).map_mul g h) | |
| lemma alternatingStabilizerEquiv_subtype (a p n : ℕ) (hn : 32 ≤ n) : | |
| (MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)).subtype.comp | |
| (alternatingStabilizerEquiv a p n hn).toMonoidHom=PolygonTracks.stabilizeAlternating a p n := rfl | |
| lemma alternatingStabilizer_perfect (a p n : ℕ) (hn : 32 ≤ n) : | |
| Group.IsPerfect (MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n)) := by | |
| have := polygonAlternatingGroup_perfect a n (by omega) | |
| exact Group.IsPerfect.ofSurjective (f:=(alternatingStabilizerEquiv a p n hn).toMonoidHom) | |
| (alternatingStabilizerEquiv a p n hn).surjective | |
| noncomputable abbrev altColumn (a m p : ℕ) := | |
| Rep.res (polygonAlternatingGroup a m).subtype (column a m p) | |
| noncomputable abbrev altFaceHom (a m p : ℕ) (i : Fin (p+1)) : | |
| altColumn a m (p+1) ⟶ altColumn a m p := | |
| (Rep.resFunctor (polygonAlternatingGroup a m).subtype).map (faceHom a m p i) | |
| noncomputable abbrev altBoundary (a m p : ℕ) : altColumn a m (p+1) ⟶ altColumn a m p := | |
| (Rep.resFunctor (polygonAlternatingGroup a m).subtype).map (boundary a m p) | |
| lemma altBoundary_square (a m p : ℕ) : altBoundary a m (p+1) ≫ altBoundary a m p=0 := by | |
| rw [← Functor.map_comp, boundary_square, Functor.map_zero] | |
| lemma altBoundary_exact (a m p : ℕ) (hm : 33 ≤ m) (hp : p ≤ 2) : | |
| (ShortComplex.mk (altBoundary a m (p+1)) (altBoundary a m p) (altBoundary_square a m p)).Exact := by | |
| exact (Rep.res_map_exact (polygonAlternatingGroup a m).subtype | |
| (ShortComplex.mk (boundary a m (p+1)) (boundary a m p) (boundary_square a m p))).mpr | |
| (boundary_exact a m p hm hp) | |
| lemma altBoundary_zero_epi (a m : ℕ) (hm : 0 < m) : Epi (altBoundary a m 0) := by | |
| rw [Rep.epi_iff_surjective] | |
| exact (Rep.epi_iff_surjective (boundary a m 0)).mp (boundary_zero_epi a m hm) | |
| lemma altFace_H0_eq (a m p : ℕ) (hm : 3*p+1 < m) (i j : Fin (p+1)) : | |
| (groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altFaceHom a m p i)= | |
| (groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altFaceHom a m p j) := by | |
| let f := standardLE a m p (by omega : p ≤ m) | |
| exact TransitiveInduction.map_equivariant_H0_eq (G:=polygonAlternatingGroup a m) | |
| (face i) (face j) (fun g x => face_smul i g.val x) (fun g x => face_smul j g.val x) | |
| f (fun g => alternating_transitive f g hm) | |
| lemma altFace_H0_isIso (a m p : ℕ) (hm : 3*(p+1)+1 < m) (i : Fin (p+1)) : | |
| IsIso ((groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altFaceHom a m p i)) := by | |
| let f := standardLE a m (p+1) (by omega : p+1 ≤ m) | |
| exact TransitiveInduction.isIso_map_equivariant_H0 (G:=polygonAlternatingGroup a m) | |
| (face i) (fun g x => face_smul i g.val x) f | |
| (fun g => alternating_transitive f g hm) (fun g => alternating_transitive _ g (by omega)) | |
| lemma altBoundary_two_H0_isIso (a m : ℕ) (hm : 10 < m) : | |
| IsIso ((groupHomology.functor ℤ (polygonAlternatingGroup a m) 0).map (altBoundary a m 2)) := by | |
| have he : altBoundary a m 2=altFaceHom a m 2 0-altFaceHom a m 2 1+altFaceHom a m 2 2 := by | |
| change (Rep.resFunctor _).map (boundary a m 2)=_ | |
| rw [boundary_three, Functor.map_add, Functor.map_sub] | |
| rw [he, Functor.map_add, Functor.map_sub, | |
| altFace_H0_eq a m 2 (by omega) 1 0, altFace_H0_eq a m 2 (by omega) 2 0, | |
| sub_self,zero_add] | |
| exact altFace_H0_isIso a m 2 hm 0 | |
| lemma altColumn_H1_isZero (a p n : ℕ) (hn : 32 ≤ n) (hm : 3*p+1 < p+n) : | |
| IsZero (groupHomology (altColumn a (p+n) p) 1) := by | |
| have := alternatingStabilizer_perfect a p n hn | |
| exact (TrivialHomology.H1_isZero_of_perfect | |
| (G:=MulAction.stabilizer (polygonAlternatingGroup a (p+n)) (standard a p n))).of_iso | |
| (alternatingOrbitHomologyIso a p n hm 1) | |
| lemma altBoundary_zero_H2_epi (a m : ℕ) (hm : 34 ≤ m) : | |
| Epi ((groupHomology.functor ℤ (polygonAlternatingGroup a m) 2).map (altBoundary a m 0)) := by | |
| have := altBoundary_zero_epi a m (by omega) | |
| have : IsIso (BoundedHomologyEdge.Hmap 0 (altBoundary a m 2)) := | |
| altBoundary_two_H0_isIso a m (by omega) | |
| have hz : IsZero (groupHomology (altColumn a m 2) 1) := by | |
| have he : m=2+(m-2) := by omega | |
| rw [he] | |
| exact altColumn_H1_isZero a 2 (m-2) (by omega) (by omega) | |
| exact BoundedHomologyEdge.chain_H2_epi (altBoundary a m 0) (altBoundary a m 1) | |
| (altBoundary a m 2) (altBoundary_square a m 0) (altBoundary_square a m 1) | |
| (altBoundary_exact a m 0 (by omega) (by omega)) | |
| (altBoundary_exact a m 1 (by omega) (by omega)) hz | |
| end PolygonPlacement.Configuration | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |