Definition:Fiber Bundle

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Definition

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

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

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

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

Let there exist homeomorphisms:

$\chi_\alpha : \pi^{-1} \left({U_\alpha}\right) \to U_\alpha \times F$

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

$\pi {\restriction}_{U_\alpha} = \operatorname{pr}_{1, \alpha} \mathop \circ \chi_\alpha$

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


Then the ordered tuple $\left({E, M, \pi, F}\right)$ 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 $\left\{ {\left({U_\alpha, \chi_\alpha}\right): \alpha \in I}\right\}$ is called a system of local trivializations of $E$ on $M$.


Base Point

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


Remarks

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.


Linguistic Note

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


Also see


Sources