Fiber bundle

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Definition

Suppose E,B,F are topological spaces. A continuous map p:E \to B is termed a fiber bundle with fiber F if:

  • p is surjective
  • For any point b \in B, the fiber p^{-1}(\{ b \}), given the subspace topology from E, is homeomorphic to F.
  • For any point b \in B, there exists an open subset U of B such that b \in U, and such that there exists a homeomorphism \varphi: U \times F \to p^{-1}(U) (where U \times F is endowed with the product topology) with the property that p \circ \varphi coincides with the projection map from U \times F to U. In other words, the bundle is locally trivial, i.e., locally like a product space.

We use the following terminology:

  • The space F is termed the fiber space or fiber or fiber type.
  • The space E is termed the total space.
  • The space B is termed the base space.

The letters F,E,B are typical in this context.

Particular cases

Product bundle

A special case of a fiber bundle is the trivial fiber bundle, where E = B \times F with p:E \to B being the projection onto the first coordinate. We can also think of this as being globally trivial.

Covering map

When F is a discrete space, then a fiber bundle with fiber F is a covering map. In fact, a covering map can be defined as a fiber bundle with discrete fiber.