| import OAI.RingTheory.DirectFiniteness.BalanceObstruction |
| |
| namespace OAI |
|
|
| universe u_1 |
|
|
| noncomputable section |
| open scoped Classical |
| open scoped BigOperators |
| open Equiv Equiv.Perm |
| namespace KaplanskyCounterexample.Planar |
| open Equiv Equiv.Perm |
| variable {α : Type u_1} [Fintype α] |
|
|
| lemma sameCycle_after_swap {p : Perm α} {a b x y : α} (hab : p.SameCycle a b) |
| (hxy : (swap a b * p).SameCycle x y) : p.SameCycle x y := by |
| apply setoid_sameCycle_of_step (SameCycle.setoid p) (swap a b * p) _ hxy |
| intro z |
| exact (SameCycle.apply_right SameCycle.rfl).trans |
| (setoid_swap_related (SameCycle.setoid p) hab (p z)) |
|
|
| lemma sign_eq_cycles (p : Perm α) : |
| sign p = (-1 : ℤˣ) ^ (Fintype.card α - Nat.card (Cycles p)) := by |
| classical |
| obtain ⟨l, hl, hlen, hmem⟩ := exists_cycle_factorization p |
| have hs := sign_prod_list_swap (l := l.map fun ab => swap ab.1 ab.2) (by |
| intro g hg |
| obtain ⟨ab, hab, rfl⟩ := List.mem_map.mp hg |
| exact ⟨ab.1, ab.2, (hmem ab hab).1, rfl⟩) |
| rw [hl, List.length_map] at hs |
| have hn : Fintype.card α - Nat.card (Cycles p) = l.length := by omega |
| rwa [hn] |
|
|
| lemma not_sameCycle_swap_of_sameCycle (p : Perm α) (a b : α) |
| (hne : a ≠ b) (hab : p.SameCycle a b) : ¬ (swap a b * p).SameCycle a b := by |
| classical |
| intro hq |
| let q := swap a b * p |
| have hrev : swap a b * q = p := by simp [q, ← mul_assoc] |
| have heq : SameCycle.setoid p = SameCycle.setoid q := by |
| ext x y |
| constructor |
| · intro h |
| apply sameCycle_after_swap hq |
| rwa [hrev] |
| · exact sameCycle_after_swap hab |
| have hc : Nat.card (Cycles p) = Nat.card (Cycles q) := by rw [Cycles, heq] |
| have hs : sign p = sign q := by rw [sign_eq_cycles, sign_eq_cycles, hc] |
| have hsign : (-1 : ℤˣ) = 1 := by |
| have := hs |
| change sign p = sign (swap a b * p) at this |
| rw [Perm.sign_mul, sign_swap hne] at this |
| exact mul_right_cancel (b := sign p) (by simpa using this.symm) |
| norm_num at hsign |
|
|
| lemma swap_mul_cycles_card_split (p : Perm α) (a b : α) |
| (hne : a ≠ b) (hab : p.SameCycle a b) : |
| Nat.card (Cycles (swap a b * p)) = Nat.card (Cycles p) + 1 := by |
| have h := swap_mul_cycles_card (swap a b * p) a b |
| (not_sameCycle_swap_of_sameCycle p a b hne hab) |
| simpa [← mul_assoc] using h.symm |
|
|
| end KaplanskyCounterexample.Planar |
|
|
| end |
|
|
| end OAI |
|
|
