I-bundle explained

In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be twisted if it is not trivial.

Two simple examples of I-bundles are the annulus and the Möbius band, the only two possible I-bundles over the circle

S ^{1} . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product S ^{1 x} I , and the Möbius band is a non-trivial or twisted bundle. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the Möbius band is a non-orientable manifold.K . That surface has three I-bundles: the trivial bundle K x I and two twisted bundles.

Together with the Seifert fiber spaces, I-bundles are fundamental elementary building blocks for the description of three-dimensional spaces. These observations are simple well known facts on elementary 3-manifolds.

Line bundles are both I-bundles and vector bundles of rank one. When considering I-bundles, one is interested mostly in their topological properties and not their possible vector properties, as one might be for line bundles.

References

• Scott . Peter . G. Peter Scott. The geometries of 3-manifolds . . 15 . 5 . 1983 . 10.1112/blms/15.5.401 . 401–487 . 0705527 . 2027.42/135276 . free .
• Book: Hempel, John . John Hempel. 3-manifolds . 1976 . . Annals of Mathematics Studies. 86. 978-0-8218-6939-0 . en.

External links

• Example of use of I-bundles, nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa.