In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S n {\displaystyle S^{n}} of some dimension n. Similarly, in a disk bundle, the fibers are disks D n {\displaystyle 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 BTop ( D n + 1 ) ≃ BTop ( S n ) . {\displaystyle \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 {\displaystyle S^{1}} fibers over an S 1 {\displaystyle S^{1}} base space. The non-orientable Klein bottle also has S 1 {\displaystyle S^{1}} fibers over an S 1 {\displaystyle 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.
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.3
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
BTop ( R n ) → BTop ( S n ) {\displaystyle \operatorname {BTop} (\mathbb {R} ^{n})\to \operatorname {BTop} (S^{n})}has fibers homotopy equivalent to Sn.4
See also
Notes
- 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
References
Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018. 9780521795401 ↩
Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018. 9780521795401 ↩
Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018. 9780521795401 ↩
Since, writing X + {\displaystyle X^{+}} for the one-point compactification of X {\displaystyle X} , the homotopy fiber of BTop ( X ) → BTop ( X + ) {\displaystyle \operatorname {BTop} (X)\to \operatorname {BTop} (X^{+})} is Top ( X + ) / Top ( X ) ≃ X + {\displaystyle \operatorname {Top} (X^{+})/\operatorname {Top} (X)\simeq X^{+}} . /wiki/Alexandroff_extension ↩