Fiber bundle - Knowlpedia

A fiber bundle with nonempty model fiber $F$ is a bundle $\pi:E\to B$ , with $F$ a topological space, for which there is an open cover $\{U_i\}$ of $B$ and homeomorphisms

\Phi_i:\pi^{-1}(U_i)\longrightarrow U_i\times F, \qquad \operatorname{pr}_1\circ\Phi_i=\pi|_{\pi^{-1}(U_i)}.

Each product carries the product topology. These maps are local trivializations: they identify the entire bundle over $U_i$ with a product while preserving its projection to the base.

Fibers and smooth bundles

Restricting $\Phi_i$ over a point $b$ gives a homeomorphism $E_b\cong F$ . Such an identification depends on a choice of trivialization. Having homeomorphic fibers alone does not supply local trivializations.

For smooth manifolds and diffeomorphic local product charts, the corresponding notion is a smooth fiber bundle. Forgetting smoothness gives a fiber bundle in the topological sense defined here.

Examples

The projection $B\times F\to B$ has a single global trivialization. A vector bundle is locally a product with a vector space and has additional linear compatibility. A topological principal bundle has a group as its model fiber and an action compatible with its trivializations.

References

Ralph L. Cohen, The Topology of Fiber Bundles, Chapter 1, Definition 1.1. Author-hosted notes. The definition here does not require the base to be connected or pointed.