Definition:Fiber Bundle
Definition
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
- Definition:Local Trivialization
- Definition:Transition Mapping
- Definition:Fiber (Relation)
- Definition:Section (Topology)
- Definition:Smooth Fiber Bundle
- Definition:Vector Bundle
- Results about fiber bundles can be found here.
Linguistic Note
In British English, the word fibre is used instead of fiber.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bundle
- 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 10$: Fiber Bundles
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bundle