Examples
In the following examples a fibration is denoted
- F → E → B,
where the first map is the inclusion of "the" fiber F into the total space E and the second map is the fibration onto the basis B. This is also referred to as a fibration sequence.
- The projection map from a product space is very easily seen to be a fibration.
- Fiber bundles have local trivializations such Cartesian product structures exist locally on B, and this is usually enough to show that a fiber bundle is a fibration. More precisely, if there are local trivializations over a "numerable open cover" of B, the bundle is a fibration. Any open cover of a paracompact space is numerable. For example, any open cover of a metric space has a locally finite refinement, so any bundle over such a space is a fibration. The local triviality also implies the existence of a well-defined fiber (up to homeomorphism), at least on each connected component of B.
- The Hopf fibration S1 → S3 → S2 was historically one of the earliest non-trivial examples of a fibration.
- The Serre fibration SO(2) → SO(3) → S2 comes from the action of the rotation group SO(3) on the 2-sphere S2.
- Over complex projective space, there is a fibration S1 → S2n+1 → CPn.
Read more about this topic: Fibration
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—André Breton (1896–1966)
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—G.C. (Georg Christoph)