In the
Mathematics field of
topology, a
sphere bundle is a
fiber bundle in which the fibers are
of some dimension
n.
Similarly, in a disk bundle, the fibers are disks
. From a topological perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the
Alexander trick, which implies
An example of a sphere bundle is the torus, which is Orientability and has fibers over an base space. The non-orientable Klein bottle also has fibers over an 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.
Spherical fibration
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
S n.
[Since, writing for the one-point compactification of , the homotopy fiber of is .]
See also
Notes
Further reading
External links
