Topological Fibration

Abstract

A topological fibration is a continuous map equipped with a lifting property that allows homotopies in the base to be lifted to the total space. Different precise notions refine what is meant by “lifting,” leading to several standard definitions used in algebraic topology. This note is an overview that links to the main variants.

Main notions

• Hurewicz Fibration: a map with the homotopy lifting property for all spaces.
• Serre Fibration: a map with the homotopy lifting property for disks (equivalently CW-complexes).
• Fibre Bundle: a locally trivial map with specified fibre and structure group.

Properties

• Every fibre bundle is a Hurewicz Fibration (via the covering homotopy property).
• Every Hurewicz Fibration is a Serre Fibration.
• The converses do not hold in general: Serre fibrations need not be Hurewicz fibrations, and Hurewicz fibrations need not be locally trivial bundles.

Examples

• Path-space fibration: evaluation at the endpoint, ${\mathrm{ev}}_1:X^I\to X$, is a Hurewicz fibration with fibre $\Omega X$.
• Projection of a product: $\pi :B\times F\to B$ is a trivial fibre bundle and hence a Hurewicz fibration.
• covering spaces: special cases of fibre bundles with discrete fibre.

Related concepts

• Fibration
• Hurewicz Fibration
• Serre Fibration
• Fibre Bundle
• Kan Fibration
• Grothendieck Fibration