Sphere bundle

In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Topology/2EqW47cU">topology</a>, a sphere bundle is a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a> in which the fibers are <a href="/facts/Sphere/jywYLNM2">spheres</a> 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S^{n}}
 
 of some dimension n. Similarly, in a disk bundle, the fibers are <a href="/facts/Disk_(mathematics)/DgMVr97H">disks</a> 
 
 
 
 
 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 <a href="/facts/Alexander_trick/fTsPL4aT">Alexander trick</a>, 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 <a href="/facts/Orientability/1SEQM2Pt">orientable</a> and has 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
 fibers over an 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
 base space. The non-orientable <a href="/facts/Klein_bottle/LTenG5E2">Klein bottle</a> 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 <a href="/facts/Circle_bundle/vXA6hHCM">circle bundle</a> is a special case of a sphere bundle.

Sphere bundle open-in-new

Sphere bundle