In mathematics, an I-bundle is a

fiber bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...

whose fiber is an interval and whose base is a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

. 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 In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

S^1\times I

, and the Möbius band is a non-trivial or twisted bundle. Both bundles are

2-manifold In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces ari ...

s, but the annulus is an

orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is ori ...

while the Möbius band is a

non-orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is ori ...

. Curiously, there are only two kinds of I-bundles when the base manifold is any

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

but the

Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...

K

. That surface has three I-bundles: the trivial bundle

K\times I

and two twisted bundles. Together with the

Seifert fiber space A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for a ...

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 In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

Line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...

s are both I-bundles and

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...

s of rank one. When considering I-bundles, one is interested mostly in their

topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

and not their possible vector properties, as one might be for

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...

References

* * {{Knottheory-stub

External links

Example of use of I-bundles
nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa. Fiber bundles Geometric topology 3-manifolds

See Also

References

External links