#
nLab
trivial vector bundle

Contents
### Context

#### 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

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

## Definition

For $X$ a topological space, then a topological vector bundle $E \to X$ over a topological field $k$ is called *trivial* if its total space is the product topological space

$E = X \times k^n \overset{pr_1}{\longrightarrow} X$

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

An isomorphism of vector bundles over $X$ of the form

$E \longrightarrow X \times \mathbb{R}^n$

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

Last revised on February 26, 2020 at 19:00:06.
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