Editing Vector bundle (section)

{{Short description|Mathematical parametrization of vector spaces by another space}}
[[File:Mobius strip illus.svg|thumb|250px|right|The (infinitely extended) [[Möbius strip]] is a [[line bundle]] over the [[N-sphere|1-sphere]] '''S'''<sup>1</sup>. Locally around every point in '''S'''<sup>1</sup>, it [[homeomorphism|looks like]] ''U''&nbsp;×&nbsp;'''R''' (where ''U'' is an open [[Arc (topology)|arc]] including the point), but the total bundle is different from '''S'''<sup>1</sup>&nbsp;×&nbsp;'''R''' (which is a [[Cartesian product|cylinder]] instead).]]

In [[mathematics]], a '''vector bundle''' is a [[topological]] construction that makes precise the idea of a [[Family of sets|family]] of [[vector space]]s parameterized by another [[space (mathematics)|space]] <math>X</math> (for example <math>X</math> could be a [[topological space]], a [[manifold]], or an [[algebraic variety]]): to every point <math>x</math> of the space <math>X</math> we associate (or "attach") a vector space <math>V(x)</math> in such a way that these vector spaces fit together to form another space of the same kind as <math>X</math> (e.g. a topological space, manifold, or algebraic variety), which is then called a '''vector bundle over <math>X</math>'''.

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space <math>V</math> such that <math>V(x)=V</math> [[for all]] <math>x</math> in <math>X</math>: in this case there is a copy of <math>V</math> for each <math>x</math> in <math>X</math> and these copies fit together to form the vector bundle <math>X\times V</math> over <math>X</math>. Such vector bundles are said to be [[Fiber bundle#Trivial bundle|''trivial'']]. A more complicated (and prototypical) class of examples are the [[tangent bundle]]s of [[manifold|smooth (or differentiable) manifolds]]: to every point of such a manifold we attach the [[tangent space]] to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the [[hairy ball theorem]]. In general, a manifold is said to be [[Parallelizable manifold|parallelizable]] if, and only if, its tangent bundle is trivial.

Vector bundles are almost always required to be ''locally trivial'', which means they are examples of [[fiber bundle]]s. Also, the vector spaces are usually required to be over the [[Real number|real]] or [[complex number]]s, in which case the vector bundle is said to be a real or complex vector bundle (respectively). [[Complex vector bundle]]s can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the [[category of topological spaces]].