nLab
trivial vector bundle

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

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

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see also algebraic topology

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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.

Definition

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 isomorphis, then it is called a trivializable vector bundle.

Last revised on July 22, 2017 at 09:53:58. See the history of this page for a list of all contributions to it.