sphere bundle

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.{{cite book|last=Hatcher|first=Allen|authorlink=Allen Hatcher|title=Algebraic Topology|date=2002|publisher=Cambridge University Press|isbn=9780521795401|page=442|url=https://books.google.com/books?id=BjKs86kosqgC&dq=sphere+bundle&pg=PA442|accessdate=28 February 2018|language=en}} 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.

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.

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

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

has fibers homotopy equivalent to Sⁿ.Since, writing $X^+$ for the one-point compactification of $X$ , the homotopy fiber of $\operatorname{BTop}(X) \to \operatorname{BTop}(X^+)$ is $\operatorname{Top}(X^+)/\operatorname{Top}(X) \simeq X^+$ .

Notes

References

Dennis Sullivan, [https://www.maths.ed.ac.uk/~v1ranick/books/gtop.pdf Geometric Topology], the 1970 MIT notes

External links

[https://mathoverflow.net/q/74756 Is it true that all sphere bundles are boundaries of disk bundles?]
https://ncatlab.org/nlab/show/spherical+fibration

Category:Algebraic topology

Category:Fiber bundles