Sphere bundle

In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres formula_1 of some dimension "n". Similarly, in a disk bundle, the fibers are disks formula_2. From a topological perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the Alexander trick, which implies formula_3
An example of a sphere bundle is the torus, which is orientable and has formula_4 fibers over an formula_4 base space. The non-orientable Klein bottle also has formula_4 fibers over an formula_4 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("E"), 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 "S""n".