Sphere bundle

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In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S n {\displaystyle S^{n}} {\displaystyle S^{n}} of some dimension n.[1] Similarly, in a disk bundle, the fibers are disks D n {\displaystyle 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}).} {\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}} {\displaystyle S^{1}} fibers over an S 1 {\displaystyle S^{1}} {\displaystyle S^{1}} base space. The non-orientable Klein bottle also has S 1 {\displaystyle S^{1}} {\displaystyle S^{1}} fibers over an S 1 {\displaystyle 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.[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

BTop ⁡ ( R n ) → BTop ⁡ ( S n ) {\displaystyle \operatorname {BTop} (\mathbb {R} ^{n})\to \operatorname {BTop} (S^{n})} {\displaystyle \operatorname {BTop} (\mathbb {R} ^{n})\to \operatorname {BTop} (S^{n})}

has fibers homotopy equivalent to Sn.[2]

See also

Notes

  1. Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. p. 442. ISBN 9780521795401. Retrieved 28 February 2018.
  2. Since, writing X + {\displaystyle X^{+}} {\displaystyle X^{+}} for the one-point compactification of X {\displaystyle X} {\displaystyle X}, the homotopy fiber of BTop ⁡ ( X ) → BTop ⁡ ( X + ) {\displaystyle \operatorname {BTop} (X)\to \operatorname {BTop} (X^{+})} {\displaystyle \operatorname {BTop} (X)\to \operatorname {BTop} (X^{+})} is Top ⁡ ( X + ) / Top ⁡ ( X ) ≃ X + {\displaystyle \operatorname {Top} (X^{+})/\operatorname {Top} (X)\simeq X^{+}} {\displaystyle \operatorname {Top} (X^{+})/\operatorname {Top} (X)\simeq X^{+}}.

References

Further reading