In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres 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 orientable 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 bundleEdit
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.
has fibers homotopy equivalent to Sn.
- Is it true that all sphere bundles are boundaries of disk bundles?
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