Fibrations

A space can be considered as a fibre bundle if there is a sub-space (the fibre) which can be reproduced by a displacement so that any point of the space is on a fibre, and only one. For example the Euclidean space R3 can be considered as a fibre bundle of parallel straight lines, all perpendicular to the same plane.

What is a fibration?

From a mathematical point of view, a fibred space E is defined by a mapping p from E onto the so-called ‘base’, B, any point of a given fibre being mapped onto the same base point. A fibre is therefore the full pre-image of one base point under the mapping p. In a three-dimensional fibred space with one-dimensional fibres, the base is a two-dimensional manifold. In the above simple R3 example, the two-dimensional base space is just the plane orthogonal to the fibres; in this case the base is a sub-space of the whole space, and the fibration is called ‘trivial’. But this latter property is not general, the base may not be embedded in the fibre bundle space, as will be seen below with the Hopf fibration.

Fibration of S2 × R1

When a space is the direct product between two spaces, it can trivially be considered as a fibre bundle. If fibres are straight lines in S2 × R1 then the base is the sphere S2.