Lesson plan
I-bundle
to the fiber and that each touch the edge of the band twice. |200px 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\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 . That surface has three I-bundles: the trivial bundle 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.
Article summary
to the fiber and that each touch the edge of the band twice. |200px 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\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 . That surface has three I-bundles: the trivial bundle 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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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