Definition:Fiber Bundle

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Let $M, E, F$ be topological spaces.

Let $\pi: E \to M$ be a continuous surjection.

Let $\UU := \set {U_\alpha \subseteq M: \alpha \in I}$ be an open cover of $M$ with index set $I$.

Let $\pr_{1, \alpha}: U_\alpha \times F \to U_\alpha$ be the first projection on $U_\alpha \times F$.

Let there exist homeomorphisms:

$\chi_\alpha: \map {\pi^{-1} } {U_\alpha} \to U_\alpha \times F$

such that for all $\alpha \in I$:

$\pi {\restriction}_{U_\alpha} = \pr_{1, \alpha} \circ \chi_\alpha$

where $\pi {\restriction}_{U_\alpha}$ is the restriction of $\pi$ to $U_\alpha \in \UU$.

Then the ordered tuple $\struct {E, M, \pi, F}$ is called a fiber bundle over $M$.

Total Space

The topological space $E$ is called the total space of $B$.

Base Space

The topological space $M$ is called the base space of $B$.

Bundle Projection

The continuous surjection $\pi: E \to M$ is called the bundle projection of $B$.

Model Fiber

The topological space $F$ is called the model fiber of $B$.

System of Local Trivializations

The set $\set {\struct {U_\alpha, \chi_\alpha}: \alpha \in I}$ is called a system of local trivializations of $E$ on $M$.

Base Point

A point $m \in M$ is called a base point of $B$.

Also known as

Some sources refer to a fiber bundle just as a bundle.

Some sources use the term twisted product.

By an abuse of language, it is common to say that $E$ is a fiber bundle over $M$.

One also finds the formulation Let $E \overset \pi \to M$ be a fiber bundle in the literature.

Also see

  • Results about fiber bundles can be found here.

Linguistic Note

In British English, the word fibre is used instead of fiber.