Theorems · Definition · algebraic topology

Bundle.Pretrivialization.continuousAlternatingMap

(𝕜 : Type u_1) →
  (ι : Type u_2) →
    [inst : NontriviallyNormedField 𝕜] →
      {B : Type u_3} →
        [inst_1 : TopologicalSpace B] →
          {F₁ : Type u_4} →
            [inst_2 : NormedAddCommGroup F₁] →
              [inst_3 : NormedSpace 𝕜 F₁] →
                {E₁ : B → Type u_5} →
                  [inst_4 : (x : B) → AddCommGroup (E₁ x)] →
                    [inst_5 : (x : B) → Module 𝕜 (E₁ x)] →
                      [inst_6 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)] →
                        {F₂ : Type u_6} →
                          [inst_7 : NormedAddCommGroup F₂] →
                            [inst_8 : NormedSpace 𝕜 F₂] →
                              {E₂ : B → Type u_7} →
                                [inst_9 : (x : B) → AddCommGroup (E₂ x)] →
                                  [inst_10 : (x : B) → Module 𝕜 (E₂ x)] →
                                    [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)] →
                                      [inst_12 : (x : B) → TopologicalSpace (E₁ x)] →
                                        [FiberBundle F₁ E₁] →
                                          [inst_14 : (x : B) → TopologicalSpace (E₂ x)] →
                                            [FiberBundle F₂ E₂] →
                                              (e₁ : Bundle.Trivialization F₁ Bundle.TotalSpace.proj) →
                                                (e₂ : Bundle.Trivialization F₂ Bundle.TotalSpace.proj) →
                                                  [Bundle.Trivialization.IsLinear 𝕜 e₁] →
                                                    [Bundle.Trivialization.IsLinear 𝕜 e₂] →
                                                      Bundle.Pretrivialization (F₁ [⋀^ι]→L[𝕜] F₂) Bundle.TotalSpace.proj

Given trivializations e₁, e₂ for vector bundles E₁, E₂ over a base B, Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂ is the induced pretrivialization for the continuous σ-semilinear maps from E₁ to E₂. That is, the map which will later become a trivialization, after the bundle of continuous semilinear maps is equipped with the right topological vector bundle structure.

Defined in

Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Cited by

5 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceFiberBundleTopologicalSpaceFiberBundleBundle.Trivialization.IsLinearBundle.Trivialization.IsLinear