nLab trivial vector bundle




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



Paths and cylinders

Homotopy groups

Basic facts




For XX a suitable space then a vector bundle over XX is called trivial if there is no twist in how the fibers vary over it.


For XX a topological space, then a topological vector bundle EXE \to X over a topological field kk is called trivial if its total space is the product topological space

E=X×k npr 1X E = X \times k^n \overset{pr_1}{\longrightarrow} X

with the topological vector space k nk^n, for some nn \in \mathbb{N}. For n=1n = 1, one also speaks of a trivial line bundle.

An isomorphism of vector bundles over XX of the form

EX× n E \longrightarrow X \times \mathbb{R}^n

is called a trivialization of EE. If EE admits such an isomorphism, then it is called a trivializable vector bundle.

Last revised on February 26, 2020 at 19:00:06. See the history of this page for a list of all contributions to it.