nLab external direct sum of vector bundles

Contents

Context

Bundles

bundles

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The external direct sum or external Whitney sum of vector bundles is the operation which from a pair of vector bundles 𝒱\mathscr{V} \,\in\, Vect(X), 𝒲\mathscr{W} \,\in\, Vect ( Y ) Vect(Y) over possibly different base spaces X,YX,\ Y \,\in\, Top forms the direct sum of vector bundles (“Whitney sum”) ()()(-)\bigoplus(-) of their pullback? to the product space X×YX \times Y of their base spaces:

Vect(X)×Vect(Y) oversey Vect(X×Y) (𝒱,𝒲) ((pr X) *𝒱) X×Y((pr Y) *𝒲) \array{ Vect(X) \times Vect(Y) &\oversey{\boxplus}{\longrightarrow}& Vect(X\times Y) \\ \big( \mathscr{V} ,\, \mathscr{W} \big) &\mapsto& \big( (pr_X)^\ast \mathscr{V} \big) \oplus_{X \times Y} \big( (pr_Y)^\ast \mathscr{W} \big) }

Since the Whitney sum :Vect(X×Y)×Vect(X×Y)Vect(X×Y)\oplus \,\colon\, Vect(X \times Y) \times Vect(X \times Y) \longrightarrow Vect(X \times Y) is the Cartesian product in Vect ( X × Y ) Vect(X \times Y) , one may understand the external Whitney sum as the Cartesian product in the Grothendieck construction XVect(X)\int_{X} Vect(X), see the the example there.

References

Discussion in the context of topological K-theory:

Created on May 17, 2023 at 10:05:50. See the history of this page for a list of all contributions to it.