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[[File:MobiusF.PNG|297px|right|thumb|A Möbius band is a non-orientable I-bundle. The dark line is the base for a set of transversal lines that are [[homeomorphic]] to the fiber and that each touch the edge of the band twice.]] | |||
[[File:Hopf_band_wikipedia.png|right|thumb|An annulus is an orientable I-bundle. This example is embedded in 3-space with an even number of twists|200px]] | |||
[[File:MxS1.PNG|right|thumb|This image represents the twisted I-bundle over the 2-torus, which is also fibered as a Möbius strip times the circle. So, this space is also a [[circle bundle]]|200px]] | |||
In mathematics, an '''I-bundle''' is a [[fiber bundle]] whose fiber is an [[interval (mathematics)|interval]] and whose base is a [[manifold]]. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even [[Line (mathematics)#Ray|ray]]s, can be the fiber. | |||
Two simple examples of '''I-bundles''' are the [[Annulus (mathematics)|annulus]] and the [[Möbius band]], the only two possible '''I-bundles''' over the circle <math>\scriptstyle S^1</math>. The annulus is a trivial or untwisted bundle because it corresponds to the [[Cartesian product]] <math>\scriptstyle S^1\times I</math>, and the Möbius band is a non-trivial or twisted bundle. Both bundles are [[2-manifold]]s, but the annulus is an [[orientable manifold]] while the Möbius band is a [[non-orientable manifold]]. | |||
Curiously, there are only two kinds of '''I-bundles''' when the base manifold is any [[surface]] but the [[Klein bottle]] <math>\scriptstyle K</math>. That surface has three I-bundles: the trivial bundle <math>\scriptstyle K\times I</math> and two twisted bundles. | |||
Together with the [[Seifert fiber space]]s, '''I-bundles''' are fundamental elementary building blocks for the description of three dimensional spaces. These observations are simple well known facts on elementary [[3-manifold]]s. | |||
[[Line bundle]]s are both '''I-bundles''' and [[vector bundle]]s of rank one. When considering '''I-bundles''', one is interested mostly in their [[topological property|topological properties]] and not their possible vector properties, as we might be for [[line bundle]]s. | |||
==References== | |||
* Scott, Peter, "The geometries of 3-manifolds". ''Bulletin of the London Mathematical Society'' 15 (1983), number 5, 401–487. | |||
* Hempel, John, "3-manifolds", ''Annals of Mathematics Studies'', number 86, Princeton University Press (1976). | |||
==External links== | |||
* [http://www.math.lsu.edu/~kasten/LSUTalk.pdf Example of use of I-bundles], nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa. | |||
* [http://www.m-hikari.com/imf-password2008/5-8-2008/casaliIMF5-8-2008.pdf I-bundles over the Klein-Bottle], "elementary" work on the orientable I-bundle over ''K'', by Maria Rita Casali, Dipartimento di Matematica Pura e Applicata, Università di Modena e Reggio Emilia. | |||
[[Category:Fiber bundles]] | |||
[[Category:Geometric topology]] | |||
[[Category:3-manifolds]] | |||
Revision as of 01:26, 1 November 2013
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.
Two simple examples of I-bundles are the annulus and the Möbius band, the only two possible I-bundles over the circle . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product , 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.
Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle . That surface has three I-bundles: the trivial bundle 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 we might be for line bundles.
References
- Scott, Peter, "The geometries of 3-manifolds". Bulletin of the London Mathematical Society 15 (1983), number 5, 401–487.
- Hempel, John, "3-manifolds", Annals of Mathematics Studies, number 86, Princeton University Press (1976).
External links
- Example of use of I-bundles, nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa.
- I-bundles over the Klein-Bottle, "elementary" work on the orientable I-bundle over K, by Maria Rita Casali, Dipartimento di Matematica Pura e Applicata, Università di Modena e Reggio Emilia.