In topology, the wordsfiber and fiber space appeared for the first time in a paper by Seifert in 1932, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space of a fiber space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space is given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Hopf, Feldbau, Whitney, Steenrod, Ehresmann, Serre, and others.
Formal definition
A triple where and are differentiable manifolds and is a surjective submersion, is called a fibered manifold. E is called the total space, B is called the base.
Examples
Every differentiable fiber bundle is a fibered manifold.
Every differentiable covering space is a fibered manifold with discrete fiber.
In general, a fibered manifold needs not to be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by taking the trivial bundle and deleting two points in two different fibers over the base manifold.The result is a new fibered manifold where all the fibers except two are connected.
Properties
Any surjective submersion is open: for each open, the set is open in.
Let be an -dimensional manifold. A fibered manifold admits fiber charts. We say that a chart on is a fiber chart, or is adapted to the surjective submersion if there exists a chart on such that and where The above fiber chart condition may be equivalently expressed by where is the projection onto the first coordinates. The chart is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart are usually denoted by where, , the coordinates of the corresponding chart on are then denoted, with the obvious convention, by where. Conversely, if a surjection admits a fibered atlas, then is a fibered manifold.
Let be a fibered manifold and any manifold. Then an open covering of together with maps called trivialization maps, such that is a local trivializationwith respect to. A fibered manifold together with a manifold is a fiber bundle with typical fiber if it admits a local trivialization with respect to. The atlas is then called a bundle atlas.
