Theorems · Definition · algebraic topology

Bundle.Trivialization.continuousAlternatingMap

(𝕜 : Type u_1) →
  (ι : Type u_2) →
    [inst : NontriviallyNormedField 𝕜] →
      [inst_1 : Fintype ι] →
        {B : Type u_3} →
          [inst_2 : TopologicalSpace B] →
            {F₁ : Type u_4} →
              [inst_3 : NormedAddCommGroup F₁] →
                [inst_4 : NormedSpace 𝕜 F₁] →
                  {E₁ : B → Type u_5} →
                    [inst_5 : (x : B) → AddCommGroup (E₁ x)] →
                      [inst_6 : (x : B) → Module 𝕜 (E₁ x)] →
                        [inst_7 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)] →
                          [inst_8 : (x : B) → TopologicalSpace (E₁ x)] →
                            [inst_9 : FiberBundle F₁ E₁] →
                              [inst_10 : VectorBundle 𝕜 F₁ E₁] →
                                {F₂ : Type u_6} →
                                  [inst_11 : NormedAddCommGroup F₂] →
                                    [inst_12 : NormedSpace 𝕜 F₂] →
                                      {E₂ : B → Type u_7} →
                                        [inst_13 : (x : B) → AddCommGroup (E₂ x)] →
                                          [inst_14 : (x : B) → Module 𝕜 (E₂ x)] →
                                            [inst_15 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)] →
                                              [inst_16 : (x : B) → TopologicalSpace (E₂ x)] →
                                                [inst_17 : FiberBundle F₂ E₂] →
                                                  [inst_18 : VectorBundle 𝕜 F₂ E₂] →
                                                    [inst_19 : ∀ (x : B), IsTopologicalAddGroup (E₂ x)] →
                                                      [inst_20 : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] →
                                                        (e₁ : Bundle.Trivialization F₁ Bundle.TotalSpace.proj) →
                                                          (e₂ : Bundle.Trivialization F₂ Bundle.TotalSpace.proj) →
                                                            [he₁ : MemTrivializationAtlas e₁] →
                                                              [he₂ : MemTrivializationAtlas e₂] →
                                                                Bundle.Trivialization (F₁ [⋀^ι]→L[𝕜] F₂)
                                                                  Bundle.TotalSpace.proj

Given trivializations e₁, e₂ in the atlas for vector bundles E₁, E₂ over a base B, the induced trivialization for the continuous σ-semilinear maps from E₁ to E₂, whose base set is e₁.baseSet ∩ e₂.baseSet.

Defined in

Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Cited by

3 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldFintypeTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleIsTopologicalAddGroupContinuousSMulMemTrivializationAtlasMemTrivializationAtlas