| import Mathlib.Analysis.Complex.Basic |
| import Mathlib.Topology.Algebra.Monoid |
| import Mathlib.Topology.Compactification.OnePoint.Basic |
| import Mathlib.Topology.Homotopy.Basic |
| import Mathlib.Topology.VectorBundle.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| /-! Gluing, homotopy, and operations on vector bundle sections. -/ |
|
|
| section |
| open Bundle Topology |
| namespace Kaplansky |
|
|
| section |
| variable {B V : Type*} [TopologicalSpace B] |
| [NormedAddCommGroup V] [NormedSpace ℂ V] |
| {E : B → Type*} [∀ b, AddCommGroup (E b)] [∀ b, Module ℂ (E b)] |
| [∀ b, TopologicalSpace (E b)] [TopologicalSpace (TotalSpace V E)] |
| [FiberBundle V E] [VectorBundle ℂ V E] |
|
|
| /-- Addition over a varying base, with the actual total-space topology. -/ |
| theorem continuous_bundle_add_over {X : Type*} [TopologicalSpace X] |
| (b : X → B) (hb : Continuous b) (v w : ∀ x, E (b x)) |
| (hv : Continuous (fun x => TotalSpace.mk' V (b x) (v x))) |
| (hw : Continuous (fun x => TotalSpace.mk' V (b x) (w x))) : |
| Continuous (fun x => TotalSpace.mk' V (b x) (v x + w x)) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x |
| rw [FiberBundle.continuousAt_totalSpace] |
| refine ⟨hb.continuousAt, ?_⟩ |
| have hv' := (FiberBundle.continuousAt_totalSpace V _ (x₀ := x)).mp hv.continuousAt |>.2 |
| have hw' := (FiberBundle.continuousAt_totalSpace V _ (x₀ := x)).mp hw.continuousAt |>.2 |
| apply (hv'.add hw').congr_of_eventuallyEq |
| let e := trivializationAt V E (b x) |
| have hn := hb.continuousAt (e.open_baseSet.mem_nhds (mem_baseSet_trivializationAt V E (b x))) |
| filter_upwards [hn] with y hy |
| exact (e.linear ℂ hy).map_add (v y) (w y) |
|
|
| /-- Scalar multiplication over a varying base. -/ |
| theorem continuous_bundle_smul_over {X : Type*} [TopologicalSpace X] |
| (b : X → B) (hb : Continuous b) (c : X → ℂ) (hc : Continuous c) |
| (v : ∀ x, E (b x)) |
| (hv : Continuous (fun x => TotalSpace.mk' V (b x) (v x))) : |
| Continuous (fun x => TotalSpace.mk' V (b x) (c x • v x)) := by |
| apply continuous_iff_continuousAt.mpr |
| intro x |
| rw [FiberBundle.continuousAt_totalSpace] |
| refine ⟨hb.continuousAt, ?_⟩ |
| have hv' := (FiberBundle.continuousAt_totalSpace V _ (x₀ := x)).mp hv.continuousAt |>.2 |
| apply (hc.continuousAt.smul hv').congr_of_eventuallyEq |
| let e := trivializationAt V E (b x) |
| have hn := hb.continuousAt (e.open_baseSet.mem_nhds (mem_baseSet_trivializationAt V E (b x))) |
| filter_upwards [hn] with y hy |
| exact (e.linear ℂ hy).map_smul (c y) (v y) |
|
|
| /-- The genuine global affine homotopy of two continuous bundle sections. |
| It permits intermediate degenerate zeros. -/ |
| def bundleSectionHomotopy (s t : ∀ b, E b) |
| (hs : Continuous (fun b => TotalSpace.mk' V b (s b))) |
| (ht : Continuous (fun b => TotalSpace.mk' V b (t b))) : |
| ContinuousMap.Homotopy ⟨_, hs⟩ ⟨_, ht⟩ where |
| toFun z := ⟨z.2, (1 - (z.1.val : ℂ)) • s z.2 + (z.1.val : ℂ) • t z.2⟩ |
| continuous_toFun := by |
| apply continuous_bundle_add_over Prod.snd continuous_snd |
| · apply continuous_bundle_smul_over Prod.snd continuous_snd |
| · exact continuous_const.sub (Complex.continuous_ofReal.comp (continuous_subtype_val.comp continuous_fst)) |
| · exact hs.comp continuous_snd |
| · apply continuous_bundle_smul_over Prod.snd continuous_snd |
| · exact Complex.continuous_ofReal.comp (continuous_subtype_val.comp continuous_fst) |
| · exact ht.comp continuous_snd |
| map_zero_left b := by simp |
| map_one_left b := by simp |
|
|
| /-- The map of a section into the one-point compactification of its actual total space. -/ |
| def compactifiedSection (s : ∀ b, E b) |
| (hs : Continuous (fun b => TotalSpace.mk' V b (s b))) : |
| C(B, OnePoint (TotalSpace V E)) := |
| ⟨fun b => (TotalSpace.mk' V b (s b) : OnePoint (TotalSpace V E)), |
| OnePoint.continuous_coe.comp hs⟩ |
|
|
| /-- Global section independence at the level of homotopy, without a |
| regularity or finiteness assumption on any intermediate zero set. -/ |
| def compactifiedSectionHomotopy (s t : ∀ b, E b) |
| (hs : Continuous (fun b => TotalSpace.mk' V b (s b))) |
| (ht : Continuous (fun b => TotalSpace.mk' V b (t b))) : |
| ContinuousMap.Homotopy (compactifiedSection s hs) (compactifiedSection t ht) := |
| (ContinuousMap.Homotopy.refl ⟨_, OnePoint.continuous_coe⟩).comp |
| (bundleSectionHomotopy s t hs ht) |
|
|
| end |
|
|
| variable {𝕜 Y F I : Type*} [NontriviallyNormedField 𝕜] [TopologicalSpace Y] [NormedAddCommGroup F] [NormedSpace 𝕜 F] |
|
|
| /-- Gluing compatible local coordinate functions produces a continuous |
| section of the ordinary, potentially nontrivial vector bundle. -/ |
| theorem glue_core_section (H : VectorBundleCore 𝕜 Y F I) |
| (L : I → Y → F) (hL : ∀ a, ContinuousOn (L a) (H.baseSet a)) |
| (hcompat : ∀ a b y, y ∈ H.baseSet a → y ∈ H.baseSet b → |
| H.coordChange a b y (L a y) = L b y) : |
| ∃ s : ∀ y, H.Fiber y, |
| Continuous (fun y => TotalSpace.mk' F y (s y)) ∧ |
| ∀ a y, y ∈ H.baseSet a → (H.localTriv a ⟨y,s y⟩).2 = L a y := by |
| let s : ∀ y, H.Fiber y := fun y => L (H.indexAt y) y |
| have he (a : I) (y : Y) (hy : y ∈ H.baseSet a) : |
| (H.localTriv a ⟨y,s y⟩).2 = L a y := |
| hcompat (H.indexAt y) a y (H.mem_baseSet_at y) hy |
| refine ⟨s, ?_, he⟩ |
| rw [continuous_iff_continuousAt] |
| intro y |
| rw [FiberBundle.continuousAt_section] |
| have hn := (H.isOpen_baseSet (H.indexAt y)).mem_nhds (H.mem_baseSet_at y) |
| have hc := (hL (H.indexAt y) y (H.mem_baseSet_at y)).continuousAt hn |
| apply hc.congr_of_eventuallyEq |
| filter_upwards [hn] with z hz |
| exact he (H.indexAt y) z hz |
|
|
| end Kaplansky |
|
|
| end |
|
|
| open Bundle Topology |
| namespace Kaplansky |
| variable {𝕜 Y V : Type*} [NontriviallyNormedField 𝕜] [TopologicalSpace Y] |
| variable [NormedAddCommGroup V] [NormedSpace 𝕜 V] |
| variable {E : Y → Type*} [∀ y, AddCommGroup (E y)] [∀ y, Module 𝕜 (E y)] |
| variable [∀ y, TopologicalSpace (E y)] [TopologicalSpace (TotalSpace V E)] |
| variable [FiberBundle V E] [VectorBundle 𝕜 V E] |
|
|
| include 𝕜 in |
| /-- Finite sums of continuous local sections in an ordinary vector bundle. -/ |
| theorem continuousOn_bundle_sum {ι : Type*} (u : Finset ι) (v : ι → ∀ y, E y) {s : Set Y} |
| (hv : ∀ i ∈ u, ContinuousOn (fun y => TotalSpace.mk' V y (v i y)) s) : |
| ContinuousOn (fun y => TotalSpace.mk' V y (∑ i ∈ u, v i y)) s := by |
| intro y hy |
| rw [FiberBundle.continuousWithinAt_section] |
| have hc : ContinuousWithinAt |
| (fun z => ∑ i ∈ u, ((trivializationAt V E y) ⟨z,v i z⟩).2) s y := |
| tendsto_finsetSum u (fun i hi => |
| ((FiberBundle.continuousWithinAt_section V).mp (hv i hi y hy))) |
| apply hc.congr_of_eventuallyEq |
| · |
| have hn := (trivializationAt V E y).open_baseSet.mem_nhds |
| (mem_baseSet_trivializationAt V E y) |
| filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hn] with z hz |
| exact (map_sum (((trivializationAt V E y).linear 𝕜 hz).mk' _) (fun i => v i z) u) |
| · exact map_sum (((trivializationAt V E y).linear 𝕜 (mem_baseSet_trivializationAt V E y)).mk' _) (fun i => v i y) u |
|
|
| /-- Scalar multiplication of continuous local sections. -/ |
| theorem continuousOn_bundle_smul (f : Y → 𝕜) (v : ∀ y, E y) {s : Set Y} |
| (hf : ContinuousOn f s) (hv : ContinuousOn (fun y => TotalSpace.mk' V y (v y)) s) : |
| ContinuousOn (fun y => TotalSpace.mk' V y (f y • v y)) s := by |
| intro y hy |
| rw [FiberBundle.continuousWithinAt_section] |
| have hc := (hf y hy).smul ((FiberBundle.continuousWithinAt_section V).mp (hv y hy)) |
| apply hc.congr_of_eventuallyEq |
| · |
| have hn := (trivializationAt V E y).open_baseSet.mem_nhds |
| (mem_baseSet_trivializationAt V E y) |
| filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds hn] with z hz |
| exact (((trivializationAt V E y).linear 𝕜 hz).map_smul (f z) (v z)) |
| · exact ((trivializationAt V E y).linear 𝕜 (mem_baseSet_trivializationAt V E y)).map_smul (f y) (v y) |
|
|
| end Kaplansky |
|
|
| end |
|
|
| end OAI |
|
|
