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.

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

${\displaystyle \operatorname {BTop} (\mathbb {R} ^{n})\to \operatorname {BTop} (S^{n})}$

has fibers homotopy equivalent to Sⁿ.²