Standard constructions on fiber bundles

This file contains several standard constructions on fiber bundles:

• Bundle.Trivial.fiberBundle 𝕜 B F: the trivial fiber bundle with model fiber F over the base B

• FiberBundle.prod: for fiber bundles E₁ and E₂ over a common base, a fiber bundle structure on their fiberwise product E₁ ×ᵇ E₂ (the notation stands for fun x ↦ E₁ x × E₂ x).

• FiberBundle.pullback: for a fiber bundle E over B, a fiber bundle structure on its pullback f *ᵖ E by a map f : B' → B (the notation is a type synonym for E ∘ f).

Tags

fiber bundle, fibre bundle, fiberwise product, pullback

The trivial bundle

instance Bundle.Trivial.topologicalSpace (B : Type u_1) (F : Type u_2) [t₁ : TopologicalSpace B] [t₂ : TopologicalSpace F] : TopologicalSpace (Bundle.TotalSpace F (Bundle.Trivial B F))

Equations

• Bundle.Trivial.topologicalSpace B F = TopologicalSpace.induced Bundle.TotalSpace.proj t₁ ⊓ TopologicalSpace.induced (Bundle.TotalSpace.trivialSnd B F) t₂

theorem Bundle.Trivial.inducing_toProd (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] : Inducing ⇑(Bundle.TotalSpace.toProd B F)

def Bundle.Trivial.homeomorphProd (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] : Bundle.TotalSpace F (Bundle.Trivial B F) ≃ₜ B × F

Homeomorphism between the total space of the trivial bundle and the Cartesian product.

Equations

• Bundle.Trivial.homeomorphProd B F = Equiv.toHomeomorphOfInducing (Bundle.TotalSpace.toProd B F) ⋯

def Bundle.Trivial.trivialization (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] : Trivialization F Bundle.TotalSpace.proj

Local trivialization for trivial bundle.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Bundle.Trivial.trivialization_source (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] : (Bundle.Trivial.trivialization B F).source = Set.univ

@[simp]

theorem Bundle.Trivial.trivialization_target (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] : (Bundle.Trivial.trivialization B F).target = Set.univ

instance Bundle.Trivial.fiberBundle (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] : FiberBundle F (Bundle.Trivial B F)

Fiber bundle instance on the trivial bundle.

Equations

• One or more equations did not get rendered due to their size.

theorem Bundle.Trivial.eq_trivialization (B : Type u_1) (F : Type u_2) [TopologicalSpace B] [TopologicalSpace F] (e : Trivialization F Bundle.TotalSpace.proj) [i : MemTrivializationAtlas e] : e = Bundle.Trivial.trivialization B F

Fibrewise product of two bundles

instance FiberBundle.Prod.topologicalSpace {B : Type u_1} (F₁ : Type u_2) (E₁ : B → Type u_3) (F₂ : Type u_4) (E₂ : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] : TopologicalSpace (Bundle.TotalSpace (F₁ × F₂) fun (x : B) => E₁ x × E₂ x)

Equip the total space of the fiberwise product of two fiber bundles E₁, E₂ with the induced topology from the diagonal embedding into TotalSpace F₁ E₁ × TotalSpace F₂ E₂.

Equations

• One or more equations did not get rendered due to their size.

theorem FiberBundle.Prod.inducing_diag {B : Type u_1} (F₁ : Type u_2) (E₁ : B → Type u_3) (F₂ : Type u_4) (E₂ : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] : Inducing fun (p : Bundle.TotalSpace (F₁ × F₂) fun (x : B) => E₁ x × E₂ x) => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })

The diagonal map from the total space of the fiberwise product of two fiber bundles E₁, E₂ into TotalSpace F₁ E₁ × TotalSpace F₂ E₂ is Inducing.

def Trivialization.Prod.toFun' {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) : (Bundle.TotalSpace (F₁ × F₂) fun (x : B) => E₁ x × E₂ x) → B × F₁ × F₂

Given trivializations e₁, e₂ for fiber bundles E₁, E₂ over a base B, the forward function for the construction Trivialization.prod, the induced trivialization for the fiberwise product of E₁ and E₂.

Equations

• Trivialization.Prod.toFun' e₁ e₂ p = (p.proj, (↑e₁ { proj := p.proj, snd := p.snd.1 }).2, (↑e₂ { proj := p.proj, snd := p.snd.2 }).2)

theorem Trivialization.Prod.continuous_to_fun {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} : ContinuousOn (Trivialization.Prod.toFun' e₁ e₂) (Bundle.TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))

noncomputable def Trivialization.Prod.invFun' {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] (p : B × F₁ × F₂) : Bundle.TotalSpace (F₁ × F₂) fun (x : B) => E₁ x × E₂ x

Given trivializations e₁, e₂ for fiber bundles E₁, E₂ over a base B, the inverse function for the construction Trivialization.prod, the induced trivialization for the fiberwise product of E₁ and E₂.

Equations

• Trivialization.Prod.invFun' e₁ e₂ p = { proj := p.1, snd := (Trivialization.symm e₁ p.1 p.2.1, Trivialization.symm e₂ p.1 p.2.2) }

theorem Trivialization.Prod.left_inv {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] {x : Bundle.TotalSpace (F₁ × F₂) fun (x : B) => E₁ x × E₂ x} (h : x ∈ Bundle.TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) : Trivialization.Prod.invFun' e₁ e₂ (Trivialization.Prod.toFun' e₁ e₂ x) = x

theorem Trivialization.Prod.right_inv {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] {x : B × F₁ × F₂} (h : x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ Set.univ) : Trivialization.Prod.toFun' e₁ e₂ (Trivialization.Prod.invFun' e₁ e₂ x) = x

theorem Trivialization.Prod.continuous_inv_fun {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] : ContinuousOn (Trivialization.Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ Set.univ)

noncomputable def Trivialization.prod {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] : Trivialization (F₁ × F₂) Bundle.TotalSpace.proj

Given trivializations e₁, e₂ for bundle types E₁, E₂ over a base B, the induced trivialization for the fiberwise product of E₁ and E₂, whose base set is e₁.baseSet ∩ e₂.baseSet.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Trivialization.baseSet_prod {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] : (Trivialization.prod e₁ e₂).baseSet = e₁.baseSet ∩ e₂.baseSet

theorem Trivialization.prod_symm_apply {B : Type u_1} [TopologicalSpace B] {F₁ : Type u_2} [TopologicalSpace F₁] {E₁ : B → Type u_3} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_4} [TopologicalSpace F₂] {E₂ : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] (e₁ : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ : Trivialization F₂ Bundle.TotalSpace.proj) [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] (x : B) (w₁ : F₁) (w₂ : F₂) : ↑(PartialEquiv.symm (Trivialization.prod e₁ e₂).toPartialEquiv) (x, w₁, w₂) = { proj := x, snd := (Trivialization.symm e₁ x w₁, Trivialization.symm e₂ x w₂) }

noncomputable instance FiberBundle.prod {B : Type u_1} [TopologicalSpace B] (F₁ : Type u_2) [TopologicalSpace F₁] (E₁ : B → Type u_3) [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] (F₂ : Type u_4) [TopologicalSpace F₂] (E₂ : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₁ E₁] [FiberBundle F₂ E₂] : FiberBundle (F₁ × F₂) fun (x : B) => E₁ x × E₂ x

The product of two fiber bundles is a fiber bundle.

Equations

• One or more equations did not get rendered due to their size.

instance instMemTrivializationAtlasProdInstTopologicalSpaceProdProdTopologicalSpaceInstTopologicalSpaceProdProdProd {B : Type u_1} [TopologicalSpace B] (F₁ : Type u_2) [TopologicalSpace F₁] (E₁ : B → Type u_3) [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] (F₂ : Type u_4) [TopologicalSpace F₂] (E₂ : B → Type u_5) [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → Zero (E₁ x)] [(x : B) → Zero (E₂ x)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₁ E₁] [FiberBundle F₂ E₂] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [MemTrivializationAtlas e₁] [MemTrivializationAtlas e₂] : MemTrivializationAtlas (Trivialization.prod e₁ e₂)

Equations

• ⋯ = ⋯

Pullbacks of fiber bundles

instance instForAllTopologicalSpacePullback {B : Type u} (E : B → Type w₁) {B' : Type w₂} (f : B' → B) [(x : B) → TopologicalSpace (E x)] (x : B') : TopologicalSpace ((f *ᵖ E) x)

Equations

• instForAllTopologicalSpacePullback E f x = Eq.mpr ⋯ inferInstance

theorem pullbackTopology_def {B : Type u_1} (F : Type u_2) (E : B → Type u_3) {B' : Type u_4} (f : B' → B) [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] : pullbackTopology F E f = TopologicalSpace.induced Bundle.TotalSpace.proj inst✝¹ ⊓ TopologicalSpace.induced (Bundle.Pullback.lift f) inst✝

@[irreducible]

def pullbackTopology {B : Type u_1} (F : Type u_2) (E : B → Type u_3) {B' : Type u_4} (f : B' → B) [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] : TopologicalSpace (Bundle.TotalSpace F (f *ᵖ E))

Definition of Pullback.TotalSpace.topologicalSpace, which we make irreducible.

Equations

• @pullbackTopology = wrapped✝.1

instance Pullback.TotalSpace.topologicalSpace {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} (f : B' → B) [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] : TopologicalSpace (Bundle.TotalSpace F (f *ᵖ E))

The topology on the total space of a pullback bundle is the coarsest topology for which both the projections to the base and the map to the original bundle are continuous.

Equations

• Pullback.TotalSpace.topologicalSpace F E f = pullbackTopology F E f

theorem Pullback.continuous_proj {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] (f : B' → B) : Continuous Bundle.TotalSpace.proj

theorem Pullback.continuous_lift {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] (f : B' → B) : Continuous (Bundle.Pullback.lift f)

theorem inducing_pullbackTotalSpaceEmbedding {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] (f : B' → B) : Inducing (Bundle.pullbackTotalSpaceEmbedding f)

theorem Pullback.continuous_totalSpaceMk {B : Type u} (F : Type v) (E : B → Type w₁) {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] [TopologicalSpace F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] {f : B' → B} {x : B'} : Continuous (Bundle.TotalSpace.mk x)

noncomputable def Trivialization.pullback {B : Type u} {F : Type v} {E : B → Type w₁} {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] [TopologicalSpace F] [TopologicalSpace B] [(_b : B) → Zero (E _b)] {K : Type U} [FunLike K B' B] [ContinuousMapClass K B' B] (e : Trivialization F Bundle.TotalSpace.proj) (f : K) : Trivialization F Bundle.TotalSpace.proj

A fiber bundle trivialization can be pulled back to a trivialization on the pullback bundle.

Equations

• One or more equations did not get rendered due to their size.

noncomputable instance FiberBundle.pullback {B : Type u} {F : Type v} {E : B → Type w₁} {B' : Type w₂} [TopologicalSpace B'] [TopologicalSpace (Bundle.TotalSpace F E)] [TopologicalSpace F] [TopologicalSpace B] [(_b : B) → Zero (E _b)] {K : Type U} [FunLike K B' B] [ContinuousMapClass K B' B] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] (f : K) : FiberBundle F (⇑f *ᵖ E)

Equations

• One or more equations did not get rendered due to their size.