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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.
However it is not a trivial fiber bundle, i.e., S 3 is not globally a product of S 2 and S 1 although locally it is indistinguishable from it. This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general.
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.
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.
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.
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
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} .
In general, a fibered manifold need not be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by taking the trivial bundle ( S 1 × R , π 1 , S 1 ) {\displaystyle \left(S^{1}\times \mathbb {R} ,\pi _{1},S^{1}\right)} and deleting two points in two different fibers over the base ...