bundle (mathematics)

A bundle is a triple {{math|(E, p, B)}} where {{math|E, B}} are sets and {{math|p : E → B}} is a map.{{harvnb|Husemoller|1994}} p 11.

• {{math|E}} is called the total space
• {{math|B}} is the base space of the bundle
• {{math|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 {{math|E, p, B}} and usually there is additional structure.

For each {{math|b ∈ B, p⁻¹(b)}} is the fibre or fiber of the bundle over {{math|b}}.

A bundle {{math|(E*, p*, B*)}} is a subbundle of {{math|(E, p, B)}} if {{math|B* ⊂ B, E* ⊂ E}} and {{math|p* {{=}} p{{!}}_E*}}.

A cross section is a map {{math|s : B → E}} such that {{math|p(s(b)) {{=}} b}} for each {{math|b ∈ B}}, that is, {{math|s(b) ∈ p⁻¹(b)}}.

• If {{math|E}} and {{math|B}} are smooth manifolds and {{math|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 ({{math|C¹}}), in between.
• If for each two points {{math|b₁}} and {{math|b₂}} in the base, the corresponding fibers {{math|p⁻¹(b₁)}} and {{math|p⁻¹(b₂)}} are homotopy equivalent, then the bundle is a fibration.
• If for each two points {{math|b₁}} and {{math|b₂}} in the base, the corresponding fibers {{math|p⁻¹(b₁)}} and {{math|p⁻¹(b₂)}} 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 {{math|b₁}} and {{math|b₂}} in the base, the corresponding fibers {{math|p⁻¹(b₁)}} and {{math|p⁻¹(b₂)}} 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.

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.

• Fiber bundle
• Fibration
• Fibered manifold

{{reflist}}

{{refbegin}}

• {{cite book

|last=Goldblatt

|first=Robert

|title=Topoi, the Categorial Analysis of Logic

|url=http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=gold010

|accessdate=2009-11-02

|origyear=1984

|year=2006

|publisher=Dover Publications

|isbn=978-0-486-45026-1

}}

• {{citation|last=Husemoller|first=Dale|authorlink=Dale Husemoller |title=Fibre bundles|year=1994|publisher=Springer|origyear=1966|isbn=0-387-94087-1|series=Graduate Texts in Mathematics|volume=20}}
• {{citation|last=Vassiliev|first=Victor|title=Introduction to Topology|year=2001|publisher=Amer Mathematical Society|origyear=2001|isbn=0821821628|series=Student Mathematical Library}}

{{refend}}

{{Topology}}

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Category:Category theory

Category:Fiber bundles