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 $\mathrm{BTop}(D^{n+1})\simeq \mathrm{BTop}(S^n).$ 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

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

has fibers homotopy equivalent to Sⁿ.

See also

• Smale conjecture

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