In mathematics, an I-bundle is a

fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in 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 p ...

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

ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (g ...

s, 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 Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...

and the

Möbius band Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...

, 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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

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 solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...

s, but the annulus is an orientable manifold 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 "counterclockwise". A space is ...

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

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...

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-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

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 organisin ...

s 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 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 organisin ...

References

* {{cite journal , last=Scott , first=Peter , authorlink=G. Peter Scott, title=The geometries of 3-manifolds , journal= Bulletin of the London Mathematical Society , volume=15 , issue=5 , year=1983 , doi=10.1112/blms/15.5.401 , pages=401–487 , mr=0705527 , hdl=2027.42/135276 , hdl-access=free * 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. Fiber bundles Geometric topology 3-manifolds