Sphere bundle

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

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

Orientation of a sphere bundle edit

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.

Spherical fibration edit

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

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

has fibers homotopy equivalent to Sⁿ.^[2]

Notes edit

^ ^a ^b ^c Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018.
^ Since, writing $X^{+}$ for the one-point compactification of $X$ , the homotopy fiber of $\operatorname {BTop} (X)\to \operatorname {BTop} (X^{+})$ is $\operatorname {Top} (X^{+})/\operatorname {Top} (X)\simeq X^{+}$ .