| import Mathlib | |
| import OAI.Geometry.Borsuk.SphereGeometry.SphereExtension | |
| import OAI.Geometry.Borsuk.MatrixGeometry.SymmetricCoordinates | |
| namespace OAI | |
| namespace BorsukNine | |
| open scoped RealInnerProductSpace ContDiff | |
| open scoped Manifold | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal Pointwise Matrix | |
| section OddSphereMap | |
| open scoped MatrixOrder Matrix.Norms.L2Operator | |
| variable {ν ι : Type*} [Fintype ν] [LinearOrder ν] [Nonempty ν] [Fintype ι] | |
| local notation "E" => EuclideanSpace ℝ ν | |
| noncomputable def sphereExtension (f : E → ι → ℝ) | |
| (x : EuclideanSpace ℝ (UpperTriangle ν)) : EuclideanSpace ℝ ι := | |
| normalizedSignedExtension f (symmetricCoordinates x) | |
| lemma Admissible.sphereExtension_norm {f : E → ι → ℝ} (hf : Admissible f) | |
| {x : EuclideanSpace ℝ (UpperTriangle ν)} (hx : ‖x‖ = 1) : ‖sphereExtension f x‖ = 1 := by | |
| apply hf.normalizedSignedExtension_norm (symmetricCoordinates_selfAdjoint x) | |
| intro hh | |
| have hn : x = 0 := symmetricCoordinates_injective (hh.trans (map_zero _).symm) | |
| simp [hn] at hx | |
| omit [Nonempty ν] in | |
| lemma sphereExtension_odd (f : E → ι → ℝ) (x : EuclideanSpace ℝ (UpperTriangle ν)) : | |
| sphereExtension f (-x) = -sphereExtension f x := by | |
| simp only [sphereExtension, map_neg, normalizedSignedExtension_odd] | |
| lemma Admissible.sphereExtension_continuousOn {f : E → ι → ℝ} (hf : Admissible f) : | |
| ContinuousOn (sphereExtension f) {x : EuclideanSpace ℝ (UpperTriangle ν) | ‖x‖ = 1} := by | |
| apply hf.normalizedSignedExtension_continuousOn.comp | |
| (symmetricCoordinates.continuous_of_finiteDimensional.continuousOn) (fun x hx => ?_) | |
| refine ⟨symmetricCoordinates_selfAdjoint x, ?_⟩ | |
| intro hh | |
| have hn : x = 0 := symmetricCoordinates_injective (hh.trans (map_zero _).symm) | |
| simp [hn] at hx | |
| end OddSphereMap | |
| end BorsukNine | |
| namespace BorsukNine.SmithTheory | |
| variable {K P Q : Type*} [Field K] [AddCommGroup P] [Module K P] | |
| [AddCommGroup Q] [Module K Q] | |
| lemma exists_factor_of_ker_le (N : P →ₗ[K] P) (φ : P →ₗ[K] Q) | |
| (h : LinearMap.ker N ≤ LinearMap.ker φ) : ∃ ψ : P →ₗ[K] Q, ψ.comp N = φ := by | |
| let ψ₀ : LinearMap.range N →ₗ[K] Q := | |
| (N.ker.liftQ φ h).comp N.quotKerEquivRange.symm.toLinearMap | |
| obtain ⟨ψ, hψ⟩ := ψ₀.exists_extend | |
| refine ⟨ψ, ?_⟩ | |
| ext x | |
| have he := LinearMap.congr_fun hψ ⟨N x, LinearMap.mem_range_self N x⟩ | |
| simpa only [LinearMap.comp_apply, Submodule.subtype_apply, ψ₀, | |
| LinearEquiv.coe_toLinearMap, LinearMap.quotKerEquivRange_symm_apply_image, | |
| Submodule.mkQ_apply, Submodule.liftQ_apply] using he | |
| lemma exists_factor_of_vanish_norm (N : P →ₗ[K] P) | |
| (hexact : LinearMap.ker N ≤ LinearMap.range N) (φ : P →ₗ[K] Q) | |
| (hφ : φ.comp N = 0) : ∃ ψ : P →ₗ[K] Q, ψ.comp N = φ := by | |
| apply exists_factor_of_ker_le N φ | |
| intro x hx | |
| obtain ⟨y, rfl⟩ := hexact hx | |
| exact LinearMap.congr_fun hφ y | |
| lemma tower_augmentation_zero (d N : P →ₗ[K] P) | |
| (hd : d.comp d = 0) (hcomm : d.comp N = N.comp d) | |
| (hexact : LinearMap.ker N ≤ LinearMap.range N) | |
| (n : ℕ) (φ : P →ₗ[K] Q) (hφN : φ.comp N = 0) (hφd : φ.comp d = 0) | |
| (c : ℕ → P) (htower : ∀ i < n, d (c (i + 1)) = N (c i)) | |
| (htop : N (c n) = 0) : φ (c 0) = 0 := by | |
| induction n generalizing φ c with | |
| | zero => | |
| obtain ⟨ψ, hψ⟩ := exists_factor_of_vanish_norm N hexact φ hφN | |
| rw [← hψ, LinearMap.comp_apply, htop, map_zero] | |
| | succ n ih => | |
| obtain ⟨ψ, hψ⟩ := exists_factor_of_vanish_norm N hexact φ hφN | |
| have hN : (ψ.comp d).comp N = 0 := by | |
| rw [LinearMap.comp_assoc, hcomm, ← LinearMap.comp_assoc, hψ, hφd] | |
| have hd' : (ψ.comp d).comp d = 0 := by | |
| rw [LinearMap.comp_assoc, hd, LinearMap.comp_zero] | |
| have hh := ih (ψ.comp d) hN hd' (fun i => c (i + 1)) | |
| (fun i hi => htower (i + 1) (Nat.add_lt_add_right hi 1)) htop | |
| rw [← hψ, LinearMap.comp_apply, ← htower 0 (Nat.zero_lt_succ n)] | |
| exact hh | |
| end BorsukNine.SmithTheory | |
| end OAI | |