Sphere bundle
In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres of some dimension n.[1] 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.[1]
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.[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
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.[2]
See also
- Smale conjecture
Notes
- ^ a b c Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018.
- ^ Since, writing for the one-point compactification of , the homotopy fiber of is .
References
- Dennis Sullivan, Geometric Topology, the 1970 MIT notes
Further reading
- The Adams conjecture I
- Johannes Ebert, The Adams Conjecture, after Edgar Brown
- Strunk, Florian. On motivic spherical bundles
External links
- Is it true that all sphere bundles are boundaries of disk bundles?
- https://ncatlab.org/nlab/show/spherical+fibration
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