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In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres of some dimension n.[1] Similarly, in a disk bundle, the fibers are disks . From a topological perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the Alexander trick, which implies

An example of a sphere bundle is the torus, which is orientable and has fibers over an base space. The non-orientable Klein bottle also has fibers over an base space, but has a twist that produces a reversal of orientation as one follows the loop around the base space.[1]

A circle bundle is a special case of a sphere bundle.

Orientation of a sphere bundleEdit

A sphere bundle that is a product space is orientable, as is any sphere bundle over a simply connected space.[1]

If E be a real vector bundle on a space X and if E is given an orientation, then a sphere bundle formed from E, Sph(E), inherits the orientation of E.

Spherical fibrationEdit

A spherical fibration, a generalization of the concept of a sphere bundle, is a fibration whose fibers are homotopy equivalent to spheres. For example, the fibration

 

has fibers homotopy equivalent to Sn.[2]

See alsoEdit

NotesEdit

  1. ^ a b c Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018.
  2. ^ Since, writing   for the one-point compactification of  , the homotopy fiber of   is  .

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Further readingEdit

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