Bundle (mathematics)

In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E→ B with E and B sets. It is no longer true that the preimages [math]\displaystyle{ \pi^{-1}(x) }[/math] must all look alike, unlike fiber bundles where the fibers must all be isomorphic (in the case of vector bundles) and homeomorphic.

Definition

A bundle is a triple $(E, p, B)$ where $E, B$ are sets and $p : E \to B$ is a map.^[1]

$E$ is called the total space
$B$ is the base space of the bundle
$p$ is the projection

This definition of a bundle is quite unrestrictive. For instance, the empty function defines a bundle. Nonetheless it serves well to introduce the basic terminology, and every type of bundle has the basic ingredients of above with restrictions on $E, p, B$ and usually there is additional structure.

For each $b \in B, p -1 (b)$ is the fibre or fiber of the bundle over $b$ .

A bundle $(E*, p*, B*)$ is a subbundle of $(E, p, B)$ if $B* \subset B, E* \subset E$ and $p* = p | E*$ .

A cross section is a map $s : B \to E$ such that $p (s (b)) = b$ for each $b \in B$ , that is, $s (b) \in p -1 (b)$ .

Examples

If $E$ and $B$ are smooth manifolds and $p$ is smooth, surjective and in addition a submersion, then the bundle is a fibered manifold. Here and in the following examples, the smoothness condition may be weakened to continuous or sharpened to analytic, or it could be anything reasonable, like continuously differentiable ( $C 1$ ), in between.
If for each two points $b 1$ and $b 2$ in the base, the corresponding fibers $p -1 (b 1)$ and $p -1 (b 2)$ are homotopy equivalent, then the bundle is a fibration.
If for each two points $b 1$ and $b 2$ in the base, the corresponding fibers $p -1 (b 1)$ and $p -1 (b 2)$ are homeomorphic, and in addition the bundle satisfies certain conditions of local triviality outlined in the pertaining linked articles, then the bundle is a fiber bundle. Usually there is additional structure, e.g. a group structure or a vector space structure, on the fibers besides a topology. Then is required that the homeomorphism is an isomorphism with respect to that structure, and the conditions of local triviality are sharpened accordingly.
A principal bundle is a fiber bundle endowed with a right group action with certain properties. One example of a principal bundle is the frame bundle.
If for each two points $b 1$ and $b 2$ in the base, the corresponding fibers $p -1 (b 1)$ and $p -1 (b 2)$ are vector spaces of the same dimension, then the bundle is a vector bundle if the appropriate conditions of local triviality are satisfied. The tangent bundle is an example of a vector bundle.

Bundle objects

More generally, bundles or bundle objects can be defined in any category: in a category C, a bundle is simply an epimorphism π: E → B. If the category is not concrete, then the notion of a preimage of the map is not necessarily available. Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do; for instance, for a category with pullbacks and a terminal object 1 the points of B can be identified with morphisms p:1→B and the fiber of p is obtained as the pullback of p and π. The category of bundles over B is a subcategory of the slice category (C↓B) of objects over B, while the category of bundles without fixed base object is a subcategory of the comma category (C↓C) which is also the functor category C², the category of morphisms in C.

The category of smooth vector bundles is a bundle object over the category of smooth manifolds in Cat, the category of small categories. The functor taking each manifold to its tangent bundle is an example of a section of this bundle object.

Notes

↑ Husemoller 1994 p 11.

References

Goldblatt, Robert (2006) [1984]. Topoi, the Categorial Analysis of Logic. Dover Publications. ISBN 978-0-486-45026-1. http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=gold010. Retrieved 2009-11-02.
Husemoller, Dale (1994) [1966], Fibre bundles, Graduate Texts in Mathematics, 20, Springer, ISBN 0-387-94087-1
Vassiliev, Victor (2001) [2001], Introduction to Topology, Student Mathematical Library, Amer Mathematical Society, ISBN 0821821628

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[1] Husemoller 1994 p 11.

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