| import OAI.Geometry.SingularHomology.HomologyCycle |
| import OAI.Geometry.SingularHomology.HomologyZero |
|
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| namespace OAI |
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| noncomputable section |
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|
| open CategoryTheory CategoryTheory.Limits Simplicial |
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| namespace Problem340 |
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| def realSingularZeroClass (X : Type) [TopologicalSpace X] : |
| (realSingularChainComplex X).X 0 →ₗ[ℝ] RealSingularHomology 0 X := |
| (((realSingularChainComplex X).liftCycles (𝟙 _) 0 (by simp) (by simp)) ≫ |
| (realSingularChainComplex X).homologyπ 0).hom |
|
|
| theorem realSingularZeroClass_eq_moduleHomologyClass |
| (X : Type) [TopologicalSpace X] (z : (realSingularChainComplex X).X 0) : |
| realSingularZeroClass X z = |
| moduleHomologyClass ((realSingularChainComplex X).sc 0) z (by |
| change ((realSingularChainComplex X).d 0 ((ComplexShape.down ℕ).next 0)).hom z = 0 |
| rw [ChainComplex.next_nat_zero, HomologicalComplex.shape _ 0 0 (by simp)] |
| rfl) := by |
| let C := realSingularChainComplex X |
| erw [moduleHomologyClass_eq_pi] |
| change (C.homologyπ 0).hom ((C.liftCycles (𝟙 _) 0 (by simp) (by simp)).hom z) = _ |
| apply congrArg (C.homologyπ 0).hom |
| apply (ModuleCat.mono_iff_injective (C.iCycles 0)).mp inferInstance |
| have h := congrArg (fun k : C.X 0 ⟶ C.X 0 => k.hom z) |
| (C.liftCycles_i (𝟙 _) 0 (by simp) (by simp)) |
| change (C.iCycles 0).hom _ = (C.iCycles 0).hom _ |
| exact h.trans (((C.sc 0).moduleCatCyclesIso_inv_iCycles_apply |
| ⟨z, by |
| change (C.d 0 ((ComplexShape.down ℕ).next 0)).hom z = 0 |
| rw [ChainComplex.next_nat_zero, HomologicalComplex.shape _ 0 0 (by simp)] |
| rfl⟩).symm) |
|
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| def realSingularAugmentation (X : Type) [TopologicalSpace X] : |
| (realSingularChainComplex X).X 0 →ₗ[ℝ] ℝ := |
| ((TopCat.of X).singularHomology₀ε (ModuleCat.of ℝ ℝ)).hom.comp |
| (realSingularZeroClass X) |
|
|
| theorem realSingularAugmentation_eq_sum (X : Type) [TopologicalSpace X] : |
| realSingularAugmentation X = |
| (Sigma.desc (fun _ : (TopCat.toSSet.obj (TopCat.of X)) _⦋0⦌ => |
| 𝟙 (ModuleCat.of ℝ ℝ))).hom := by |
| let K := TopCat.toSSet.obj (TopCat.of X) |
| let R := ModuleCat.of ℝ ℝ |
| let C := K.chainComplex R |
| change (((C.liftCycles (𝟙 _) 0 (by simp) (by simp)) ≫ |
| C.homologyπ 0) ≫ K.homology₀ε R).hom = _ |
| apply congrArg (fun k : C.X 0 ⟶ R => k.hom) |
| apply SSet.chainComplex_hom_ext |
| intro x |
| rw [Category.assoc, HomologicalComplex.comp_liftCycles_assoc] |
| change C.liftCycles (K.ιChainComplex x) 0 _ _ ≫ |
| C.homologyπ 0 ≫ K.homology₀ε R = _ |
| rw [SSet.liftCycles_ιChainComplex_homologyπ_homology₀ε] |
| simp only [SSet.ιChainComplex, Sigma.ι_comp_desc] |
| rfl |
|
|
| theorem realSingularZeroClass_eq_zero_iff |
| (X : Type) [TopologicalSpace X] (z : (realSingularChainComplex X).X 0) : |
| realSingularZeroClass X z = 0 ↔ |
| ∃ w : (realSingularChainComplex X).X 1, |
| ((realSingularChainComplex X).d 1 0).hom w = z := by |
| rw [realSingularZeroClass_eq_moduleHomologyClass] |
| have h := moduleHomologyClass_eq_zero_iff ((realSingularChainComplex X).sc 0) |
| (show ((realSingularChainComplex X).sc 0).X₂ from z) (by |
| change ((realSingularChainComplex X).d 0 ((ComplexShape.down ℕ).next 0)).hom z = 0 |
| rw [ChainComplex.next_nat_zero, HomologicalComplex.shape _ 0 0 (by simp)] |
| rfl) |
| change _ ↔ ∃ w : (realSingularChainComplex X).X ((ComplexShape.down ℕ).prev 0), |
| ((realSingularChainComplex X).d ((ComplexShape.down ℕ).prev 0) 0).hom w = z at h |
| rw [ChainComplex.prev ℕ 0] at h |
| exact h |
|
|
| theorem realSingularAugmentation_eq_zero_iff |
| (X : Type) [TopologicalSpace X] [PathConnectedSpace X] |
| (z : (realSingularChainComplex X).X 0) : |
| realSingularAugmentation X z = 0 ↔ |
| ∃ w : (realSingularChainComplex X).X 1, |
| ((realSingularChainComplex X).d 1 0).hom w = z := by |
| change (realSingularHomologyZeroEquiv X) (realSingularZeroClass X z) = 0 ↔ _ |
| rw [map_eq_zero_iff _ (realSingularHomologyZeroEquiv X).injective, |
| realSingularZeroClass_eq_zero_iff] |
|
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| @[simp] theorem realSingularAugmentation_boundary |
| (X : Type) [TopologicalSpace X] (w : (realSingularChainComplex X).X 1) : |
| realSingularAugmentation X (((realSingularChainComplex X).d 1 0).hom w) = 0 := by |
| have h := (realSingularZeroClass_eq_zero_iff X _).mpr ⟨w, rfl⟩ |
| simp only [realSingularAugmentation, LinearMap.comp_apply, h, map_zero] |
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|
| @[simp] theorem realSingularAugmentation_generator |
| (X : Type) [TopologicalSpace X] |
| (σ : (TopCat.toSSet.obj (TopCat.of X)) _⦋0⦌) (r : ℝ) : |
| realSingularAugmentation X |
| (((TopCat.toSSet.obj (TopCat.of X)).ιChainComplex (R := ModuleCat.of ℝ ℝ) σ).hom r) = r := by |
| rw [realSingularAugmentation_eq_sum] |
| exact congrArg (fun k : ModuleCat.of ℝ ℝ ⟶ ModuleCat.of ℝ ℝ => k.hom r) |
| (Sigma.ι_comp_desc (fun _ : (TopCat.toSSet.obj (TopCat.of X)) _⦋0⦌ => |
| 𝟙 (ModuleCat.of ℝ ℝ)) σ) |
|
|
| theorem realSingularAugmentation_naturality |
| {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] |
| (f : X → Y) (hf : Continuous f) : |
| (realSingularAugmentation Y).comp ((realSingularChainMap f hf).f 0).hom = |
| realSingularAugmentation X := by |
| rw [realSingularAugmentation_eq_sum, realSingularAugmentation_eq_sum] |
| let K := TopCat.toSSet.obj (TopCat.of X) |
| let L := TopCat.toSSet.obj (TopCat.of Y) |
| let R := ModuleCat.of ℝ ℝ |
| let φ : K ⟶ L := TopCat.toSSet.map (TopCat.ofHom (continuousMapOf f hf)) |
| change ((SSet.chainComplexMap φ R).f 0 ≫ |
| Sigma.desc (fun _ : L _⦋0⦌ => 𝟙 R)).hom = _ |
| apply congrArg (fun k : (K.chainComplex R).X 0 ⟶ R => k.hom) |
| apply SSet.chainComplex_hom_ext |
| intro σ |
| rw [← Category.assoc, SSet.ι_chainComplexMap_f] |
| simp only [SSet.ιChainComplex, Sigma.ι_comp_desc] |
| rfl |
|
|
| def realSingularAugmentedBoundary (X : Type) [TopologicalSpace X] : |
| (realSingularChainComplex X).X 1 →ₗ[ℝ] LinearMap.ker (realSingularAugmentation X) := |
| ((realSingularChainComplex X).d 1 0).hom.codRestrict |
| (LinearMap.ker (realSingularAugmentation X)) |
| (fun w => realSingularAugmentation_boundary X w) |
|
|
| theorem realSingularAugmentedBoundary_surjective |
| (X : Type) [TopologicalSpace X] [PathConnectedSpace X] : |
| Function.Surjective (realSingularAugmentedBoundary X) := by |
| intro z |
| obtain ⟨w, hw⟩ := (realSingularAugmentation_eq_zero_iff X z.val).mp z.property |
| exact ⟨w, Subtype.ext hw⟩ |
|
|
| def realSingularZeroFill (X : Type) [TopologicalSpace X] [PathConnectedSpace X] : |
| LinearMap.ker (realSingularAugmentation X) →ₗ[ℝ] (realSingularChainComplex X).X 1 := |
| Classical.choose ((realSingularAugmentedBoundary X).exists_rightInverse_of_surjective |
| (LinearMap.range_eq_top.mpr (realSingularAugmentedBoundary_surjective X))) |
|
|
| theorem realSingularZeroFill_boundary |
| (X : Type) [TopologicalSpace X] [PathConnectedSpace X] |
| (z : LinearMap.ker (realSingularAugmentation X)) : |
| ((realSingularChainComplex X).d 1 0).hom (realSingularZeroFill X z) = z.val := by |
| have h := LinearMap.congr_fun |
| (Classical.choose_spec ((realSingularAugmentedBoundary X).exists_rightInverse_of_surjective |
| (LinearMap.range_eq_top.mpr (realSingularAugmentedBoundary_surjective X)))) z |
| exact congrArg Subtype.val h |
|
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| end Problem340 |
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| end |
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| end OAI |
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