Encyclopedia

Sphere bundle

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated June 28, 2022

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. 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.

A circle bundle is a special case of a sphere bundle.

01Orientation 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.

02Spherical 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

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

has fibers homotopy equivalent to Sⁿ.

Watch videos about Sphere bundleExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Sphere bundle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Sphere bundle

01Orientation of a sphere bundle

02Spherical fibration

Sources and credits

Related topics

Smale conjecture