A sphere bundle that is a product space is orientable, as is any sphere bundle over a simply connected space.3
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.4
Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018. 9780521795401 ↩
Since, writing X + {\displaystyle X^{+}} for the one-point compactification of X {\displaystyle X} , the homotopy fiber of BTop ( X ) → BTop ( X + ) {\displaystyle \operatorname {BTop} (X)\to \operatorname {BTop} (X^{+})} is Top ( X + ) / Top ( X ) ≃ X + {\displaystyle \operatorname {Top} (X^{+})/\operatorname {Top} (X)\simeq X^{+}} . /wiki/Alexandroff_extension ↩