OPTICS algorithm

2013-10-31T23:26:30Z

2602:306:CD4A:D330:582A:1FD:FD8A:2064: /* Basic idea */

[[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]]

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OPTICS algorithm