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This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles.
Stable homotopy groups of spheres are used to describe the group Θ n of h-cobordism classes of oriented homotopy n-spheres (for n ≠ 4, this is the group of smooth structures on n-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by exotic spheres). More precisely, there is an ...
meaning that the fiber space S 1 (a circle) is embedded in the total space S 3 (the 3-sphere), and p : S 3 → S 2 (Hopf's map) projects S 3 onto the base space S 2 (the ordinary 2-sphere). The Hopf fibration, like any fiber bundle, has the important property that it is locally a product space.
The Möbius strip can be constructed by a non-trivial gluing of two trivial bundles on open subsets U and V of the circle S 1.When glued trivially (with g UV =1) one obtains the trivial bundle, but with the non-trivial gluing of g UV =1 on one overlap and g UV =-1 on the second overlap, one obtains the non-trivial bundle E, the Möbius strip.
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
The orthonormal frame bundle () of the Möbius strip is a non-trivial principal /-bundle over the circle. In mathematics , a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} .
Non-trivial Z/2Z principal bundle over the circle. There is no obvious way to identify which point corresponds to +1 or -1 in each fibre. This bundle is non-trivial as there is no globally defined section of the projection π.
At each point in the fiber , the vertical fiber is unique. It is the tangent space to the fiber. The horizontal fiber is non-unique. It merely has to be transverse to the vertical fiber. In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle.