Sphere bundle

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.

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.