Sphere bundle


In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres of some dimension n. 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.
A circle bundle is a special case of a sphere bundle.

Orientation of a sphere bundle

A sphere bundle that is a product space is orientable, as is any sphere bundle over a simply connected space.
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, inherits the orientation of E.

Spherical fibration

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.