---
tag: 4RK8T21D27
kind: definition
name: Fiber bundle
msc: [55R10]
defines: CX4M8K61AB
uses: [0CQHFPGJ45, 0X3WGX95FE, 0QT00PB5G2]
sources:
  - url: https://en.wikipedia.org/wiki/Fiber_bundle
---

## Definition

Let $E$, $B$ and $F$ be [[topological spaces|0CQHFPGJ45]], and let $\pi \colon E \to B$ be a [[continuous|0X3WGX95FE]] surjection.
The quadruple $(E, B, \pi, F)$ is a *fiber bundle* if the following holds.
Every point $b \in B$ has an open neighbourhood $U$ and a [[homeomorphism|0QT00PB5G2]]
$$\varphi \colon \pi^{-1}(U) \to U \times F$$
such that $\operatorname{pr}_1 \circ \varphi = \pi$ on $\pi^{-1}(U)$, where $\operatorname{pr}_1 \colon U \times F \to U$ is the first projection.
Such a pair $(U, \varphi)$ is a *local trivialization*.
The spaces $E$, $B$ and $F$ are the total space, the base and the fiber, and $\pi$ is the projection.

## Remarks

For any spaces $B$ and $F$, the product $B \times F$ with $\pi = \operatorname{pr}_1$ is a fiber bundle, the trivial bundle; one trivialization over $U = B$ suffices.
The Möbius band is a fiber bundle over the circle with fiber $[0, 1]$ that is not trivial.

Non-example: let $E = \set{(x, y) \in \R^2 : xy = 0}$ and $\pi(x, y) = x$, so $\pi \colon E \to \R$.
The preimage of $0$ is a line, while the preimage of each $x \neq 0$ is a single point.
A local trivialization around $0$ would make all preimages near $0$ homeomorphic to the same $F$, which is impossible.

Each fiber $\pi^{-1}(b)$ is homeomorphic to $F$.
A *smooth fiber bundle* uses smooth manifolds, a smooth submersion $\pi$ and diffeomorphisms $\varphi$.
The notation $F \to E \to B$ is common.

## Explanations

### L1 human

A fiber bundle is a map onto a base space that looks like the projection of a product over small open sets of the base.