In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres
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
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. 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.