Theorems · Definition · algebraic topology

Bundle.Pretrivialization.continuousAlternatingMapCoordChange

(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)] →
                                      (e₁ e₁' : Bundle.Trivialization F₁ Bundle.TotalSpace.proj) →
                                        (e₂ e₂' : Bundle.Trivialization F₂ Bundle.TotalSpace.proj) →
                                          [Bundle.Trivialization.IsLinear 𝕜 e₁] →
                                            [Bundle.Trivialization.IsLinear 𝕜 e₁'] →
                                              [Bundle.Trivialization.IsLinear 𝕜 e₂] →
                                                [Bundle.Trivialization.IsLinear 𝕜 e₂'] →
                                                  B → F₁ [⋀^ι]→L[𝕜] F₂ →L[𝕜] F₁ [⋀^ι]→L[𝕜] F₂

Assume eᵢ and eᵢ' are trivializations of the bundles Eᵢ over base B with fiber Fᵢ (i ∈ {1,2}), then Pretrivialization.continuousAlternatingMapCoordChange 𝕜 ι e₁ e₁' e₂ e₂' is the coordinate change function between the two induced (pre)trivializations Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂ and Pretrivialization.continuousAlternatingMap 𝕜 ι e₁' e₂' of the bundle of continuous alternating maps.

Defined in

Mathlib.Topology.VectorBundle.ContinuousAlternatingMap

Cited by

2 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceBundle.Trivialization.IsLinearBundle.Trivialization.IsLinearBundle.Trivialization.IsLinearBundle.Trivialization.IsLinear