Sphere bundle

In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres ${\displaystyle S^{n}}$ of some dimension n.^[1] Similarly, in a disk bundle, the fibers are disks ${\displaystyle D^{n}}$. From a topological perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the Alexander trick, which implies ${\displaystyle \operatorname {BTop} (D^{n+1})\simeq \operatorname {BTop} (S^{n}).}$

An example of a sphere bundle is the torus, which is orientable and has ${\displaystyle S^{1}}$ fibers over an ${\displaystyle S^{1}}$ base space. The non-orientable Klein bottle also has ${\displaystyle S^{1}}$ fibers over an ${\displaystyle S^{1}}$ 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.