| import OAI.RingTheory.DirectFiniteness.PlanarDiagrams |
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| namespace OAI |
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| universe u_1 u_2 u_3 u_4 |
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| noncomputable section |
| open scoped Classical |
| open scoped BigOperators |
| open Equiv Equiv.Perm |
| namespace KaplanskyCounterexample.Planar |
| open Equiv Equiv.Perm |
| attribute [local simp] SameCycle.rfl |
| universe u |
| variable {α : Type u} {V : Type u_1} {F : Type u_2} {A : Type u_3} {B : Type u_4} [Fintype α] [AddCommGroup A] [AddCommGroup B] |
| variable {R : Ribbon α} {Inc : V → F → Prop} {ι : V → F → (A →+ B)} |
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| theorem WeakBalanced.not_genusZero (D : WeakBalanced R Inc ι) |
| (hinj : ∀ v f, Inc v f → Function.Injective (ι v f)) |
| (hind : ∀ v f g, Inc v f → Inc v g → f ≠ g → |
| ∀ a b, ι v f a + ι v g b = 0 → a = 0) |
| (hgirth : 12 ≤ (incidenceGraph Inc).egirth) : ¬ R.GenusZero := by |
| classical |
| intro hR |
| let C : PlanarDiagram.{u} Inc ι := ⟨α, inferInstance, R, D, hR⟩ |
| obtain ⟨M, hmin, hvmin⟩ := C.exists_minimal |
| have hnz : ∀ E : PlanarDiagram.{u} Inc ι, E.edges = M.edges → |
| ∀ a, E.balanced.weight a ≠ 0 := by |
| intro E hE a ha |
| obtain ⟨β, iβ, S, hcard, hS, hD⟩ := E.balanced.exists_smaller E.genusZero a ha |
| let E' : PlanarDiagram.{u} Inc ι := ⟨β, iβ, S, hD.some, hS⟩ |
| have h := hmin E' |
| change M.edges ≤ Fintype.card β at h |
| change Fintype.card E.Edge = M.edges at hE |
| omega |
| have hparallel : ∀ a b : M.Edge, M.ribbon.black.SameCycle a b → |
| M.ribbon.white.SameCycle a b → a = b := by |
| intro a b hb hw |
| by_contra hab |
| let E : PlanarDiagram.{u} Inc ι := |
| ⟨M.Edge, inferInstance, M.ribbon, M.balanced.transfer a b hb hw, M.genusZero⟩ |
| exact hnz E rfl a (transferWeight_zero M.balanced.weight a b hab) |
| let D' := M.balanced.strong (hnz M rfl) |
| have hred : D'.Reduced := by |
| constructor |
| · intro a ha hl |
| have hw : ¬ M.ribbon.white.SameCycle a (M.ribbon.black a) := by |
| intro h |
| exact ha (hparallel a _ SameCycle.rfl.apply_right h).symm |
| let E : PlanarDiagram.{u} Inc ι := |
| ⟨M.Edge, inferInstance, M.ribbon.mergeWhite a, |
| M.balanced.mergeWhite a hw hl, M.ribbon.genusZero_mergeWhite M.genusZero a hw⟩ |
| have h := hvmin E rfl |
| have hn : ¬ M.ribbon.white.SameCycle a (M.ribbon.face a) := by |
| simpa [Ribbon.face] using hw |
| have hc := swap_mul_cycles_card M.ribbon.white a (M.ribbon.face a) hn |
| change Nat.card (Cycles M.ribbon.black) + Nat.card (Cycles M.ribbon.white) ≤ |
| Nat.card (Cycles M.ribbon.black) + |
| Nat.card (Cycles (swap a (M.ribbon.face a) * M.ribbon.white)) at h |
| omega |
| · intro a ha hl |
| have hb : ¬ M.ribbon.black.SameCycle a (M.ribbon.white a) := by |
| intro h |
| exact ha (hparallel a _ h SameCycle.rfl.apply_right).symm |
| let E : PlanarDiagram.{u} Inc ι := |
| ⟨M.Edge, inferInstance, M.ribbon.mergeBlack a, |
| M.balanced.mergeBlack a hb hl, M.ribbon.genusZero_mergeBlack M.genusZero a hb⟩ |
| have h := hvmin E rfl |
| have hn : ¬ M.ribbon.black.SameCycle a (M.ribbon.black (M.ribbon.white a)) := by |
| simpa using hb |
| have hc := swap_mul_cycles_card M.ribbon.black a (M.ribbon.black (M.ribbon.white a)) hn |
| change Nat.card (Cycles M.ribbon.black) + Nat.card (Cycles M.ribbon.white) ≤ |
| Nat.card (Cycles (swap a (M.ribbon.black (M.ribbon.white a)) * M.ribbon.black)) + |
| Nat.card (Cycles M.ribbon.white) at h |
| omega |
| exact D'.not_genusZero hred hinj hind hgirth M.genusZero |
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| end KaplanskyCounterexample.Planar |
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| end |
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| end OAI |
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