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 {\displaystyle S^{1}} . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product S 1 × I {\displaystyle S^{1}\times 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. There are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle K {\displaystyle K} . That surface has three I-bundles: the trivial bundle K × I {\displaystyle K\times 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.
See also Khovanov homology
References Scott, Peter (1983). "The geometries of 3-manifolds". Bulletin of the London Mathematical Society. 15 (5): 401–487. doi:10.1112/blms/15.5.401. hdl:2027.42/135276. MR 0705527. Hempel, John (1976). 3-manifolds. Annals of Mathematics Studies. Vol. 86. Princeton University Press. ISBN 978-0-8218-6939-0.
External links Example of use of I-bundles, nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa.
