Sphere bundle
In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres [math]\displaystyle{ S^n }[/math] of some dimension n.[1] Similarly, in a disk bundle, the fibers are disks [math]\displaystyle{ D^n }[/math]. From a topological perspective, there is no difference between sphere bundles and disk bundles: this is a consequence of the Alexander trick, which implies [math]\displaystyle{ \operatorname{BTop}(D^{n+1}) \simeq \operatorname{BTop}(S^n). }[/math] An example of a sphere bundle is the torus, which is orientable and has [math]\displaystyle{ S^1 }[/math] fibers over an [math]\displaystyle{ S^1 }[/math] base space. The non-orientable Klein bottle also has [math]\displaystyle{ S^1 }[/math] fibers over an [math]\displaystyle{ S^1 }[/math] 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
- [math]\displaystyle{ \operatorname{BTop}(\mathbb{R}^n) \to \operatorname{BTop}(S^n) }[/math]
has fibers homotopy equivalent to Sn.[2]
See also
Notes
- ↑ 1.0 1.1 1.2 Hatcher, Allen (2002) (in en). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. https://books.google.com/books?id=BjKs86kosqgC&dq=sphere+bundle&pg=PA442. Retrieved 28 February 2018.
- ↑ Since, writing [math]\displaystyle{ X^+ }[/math] for the one-point compactification of [math]\displaystyle{ X }[/math], the homotopy fiber of [math]\displaystyle{ \operatorname{BTop}(X) \to \operatorname{BTop}(X^+) }[/math] is [math]\displaystyle{ \operatorname{Top}(X^+)/\operatorname{Top}(X) \simeq X^+ }[/math].
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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