Sphere bundle

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. 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.

Tags