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''. 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 $\operatorname(D^) \simeq \operatorname(S^n).$

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
:$\operatorname(\mathbb^n) \to \operatorname(S^n)$
has fibers homotopy equivalent to ''S''^''n''.Since, writing $X^+$ for the one-point compactification of $X$, the homotopy fiber of $\operatorname(X) \to \operatorname(X^+)$ is $\operatorname(X^+)/\operatorname(X) \simeq X^+$.

See also

* Smale conjecture

Notes

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

{{topology-stub

Algebraic topology
Fiber bundles