'''Wiązka wektorowa''' jest pojęciem matematycznym, dotyczącym topologii. Wiązka wektorowa to przestrzenią wektorową w każdym punkcie w taki sposób, że całość tworzy także przestrzeń topologiczną.

Wiązkę wektorową można rozważać również nad rozmaitością różniczkową. Wtedy wymaga się by była ona rozmaitością różniczkową (a nie tylko przestrzenią topologiczną).

== Definicja ==
<math>(E, M, \pi)\,</math> jest wiązką wektorową nad rozmaitością różniczkową <math>M</math> jeśli:
# <math>E</math> jest rozmaitością różniczkową,
# <math>\pi: E \to M</math> jest ciągłą suriekcją (zwaną ''kanoniczną projekcją''),
# każde włókno <math>E_p = \pi^{-1}(p)\,</math> ma strukturę przestrzeni liniowej nad <math>\mathbb{R}</math>,
# dla każdego punktu rozmaitości <math>M</math> istnieją jego otoczenie <math>U \subset M</math> oraz liczba naturalna <math>n</math>, takie że <math>\pi^{-1}(U)\,</math> jest dyfeomorficzny z <math>U \times\mathbb{R}^n</math> za pomocą dyfeomorfizmu <math>\Phi_U : U \times\mathbb{R}^n \to \pi^{-1}(U)</math>, takiego że <math>\pi \circ \Phi_U</math> jest rzutowaniem na pierwszą współrzędną w iloczynie kartezjańskim <math>U \times\mathbb{R}^n</math> .

== Przykłady ==

* Wiązka styczna i wiązka kostyczna są przykładami wiązki wektorowej.
* strukturą różniczkową jest wiązką wektorową zwaną ''trywialną wiązką wektorową''.

In topological construction which 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 every point ''x'' of the space ''X'' we associate (or "attach") a vector space ''V''(''x'') in such a way that these vector spaces fit together to form another space of the same kind as ''X'' (e.g. a topological space, manifold, or algebraic variety), which is then called a '''vector bundle over ''X'''''.

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space ''V'' such that ''V''(''x'') = ''V'' for all ''x'' in ''X'': in this case there is a copy of ''V'' for each ''x'' in ''X'' and these copies fit together to form the vector bundle ''X''&times;''V'' over ''X''. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the 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 (two dimensional) sphere is not trivial by the Hairy ball theorem.

Vector bundles are almost always required to be ''locally trivial'', however, which means they are examples of fiber bundles. Also, the vector spaces are usually vector spaces over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles 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.

== Definition and first consequences ==
===Definition 1===
A '''vector bundle''' is a fiber bundle, where the standard fiber ''V'' is a vector space, with structure group the general linear group of ''V''.

===Definition 2===
A '''real vector bundle''' consists of:
# topological spaces ''X'' (''base space'') and ''E'' (''total space'')
# a continuous map ''π'' : ''E'' → ''X'' (''bundle projection'')
# for every ''x''  in ''X'', the structure of a finite-dimensional real fiber ''π'' <sup>-1</sup>({''x''})
 
where the following compatibility condition is satisfied: for every point in ''X'', there is an open neighborhood ''U'', a natural number ''k'', and a homeomorphism 

:<math>\varphi:  U \times \mathbf{R}^{k} \rightarrow \pi^{-1}(U) </math> 

such that for all ''x'' ∈ ''U'',
 
* ''πφ''(''x'',''v'') = ''x''  for all vectors ''v'' in '''R'''<sup>''k''</sup>, and

* the map ''v'' <math>\mapsto</math> φ(''x'',''v'') is an isomorphism between the vector spaces '''R'''<sup>''k''</sup> and π<sup>&minus;1</sup>({''x''}).

The open neighborhood ''U'' together with the homeomorphism ''φ'' is called a '''local trivialization''' of the vector bundle. The local trivialization shows that ''locally'' the map ''π'' "looks like" the projection of ''U'' &times; '''R'''<sup>''k''</sup> on ''U''.

Every fiber ''π''<sup>&minus;1</sup>({''x''}) is a finite-dimensional real vector space and hence has a dimension ''k''<sub>''x''</sub>. The local trivializations show that the function ''x'' <math>\mapsto</math> ''k''<sub>''x''</sub> is connected component of ''X''. If ''k''<sub>''x''</sub> is equal to a constant ''k'' on all of ''X'', then ''k'' is called the '''rank''' of the vector bundle, and ''E'' is said to be a '''vector bundle of rank ''k'''''. Vector bundles of rank 1 are called line bundles.

The cartesian product ''X'' &times; '''R'''<sup>''k''</sup> , equipped with the projection ''X'' &times; '''R'''<sup>''k''</sup> → ''X'', is called the '''trivial bundle''' of rank ''k'' over ''X''.

== Vector bundle morphisms ==

A '''morphism''' from the vector bundle π<sub>1</sub> : ''E''<sub>1</sub> → ''X''<sub>1</sub> to the vector bundle π<sub>2</sub> : ''E''<sub>2</sub> → ''X''<sub>2</sub> is given by a pair of continuous maps ''f'' : ''E''<sub>1</sub> → ''E''<sub>2</sub> and ''g'' : ''X''<sub>1</sub> → ''X''<sub>2</sub> such that 
* ''g''π<sub>1</sub> = π<sub>2</sub>''f''
<div style="text-align: center;">Image:BundleMorphism-01.png</div>

* for every ''x'' in ''X''<sub>1</sub>, the map π<sub>1</sub><sup>&minus;1</sup>({''x''}) → π<sub>2</sub><sup>&minus;1</sup>({''g''(''x'')}) induced by ''f'' is a linear map between vector spaces.

Note that ''g'' is determined by ''f'' (because π<sub>1</sub> is surjective), and ''f'' is then said to '''cover ''g'''''.

The class of all vector bundles together with bundle morphisms forms a category. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundles, and are also often called '''(vector) bundle homomorphisms'''.

A bundle homomorphism from ''E''<sub>1</sub> to ''E''<sub>2</sub> with an inverse which is also a bundle homomorphism (from ''E''<sub>2</sub> to ''E''<sub>1</sub>) is called a '''(vector) bundle isomorphism''', and then ''E''<sub>1</sub> and ''E''<sub>2</sub> are said to be '''isomorphic''' vector bundles. An isomorphism of a (rank ''k'') vector bundle ''E'' over ''X'' with the trivial bundle (of rank ''k'' over ''X'') is called a '''trivialization''' of ''E'', and ''E'' is then said to be '''trivial''' (or '''trivializable'''). The definition of a vector bundle shows that any vector bundle is '''locally trivial'''.

We can also consider the category of all vector bundles over a fixed base space ''X''. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the commutes:
<div style="text-align: center;">Image:BundleMorphism-02.png</div>

(Note that this category is ''not'' abelian; the kernel of a morphism of vector bundles is in general not a vector bundle in any natural way.)

A vector bundle morphism between vector bundles π<sub>1</sub> : ''E''<sub>1</sub> → ''X''<sub>1</sub> and π<sub>2</sub> : ''E''<sub>2</sub> → ''X''<sub>2</sub> covering a map ''g'' from ''X''<sub>1</sub>  to ''X''<sub>2</sub> can also be viewed as a vector bundle morphism over ''X''<sub>1</sub> from ''E''<sub>1</sub> to the pullback bundle ''g''<sup>*</sup>''E''<sub>2</sub>.

== Sections and locally free sheaves ==
right|thumb|300px|The map associating to each point on a surface a vector normal to it can be thought of  as a section. The surface is the space ''X'', and at each point ''x'' there is a vector in the vector space attached at ''x''.

Given a vector bundle π : ''E'' → ''X'' and an open subset ''U'' of ''X'', we can consider '''sections''' of π on ''U'', i.e. continuous functions ''s'' : ''U'' → ''E'' with π''s'' = id<sub>''U''</sub>. Essentially, a section assigns to every point of ''U'' a vector from the attached vector space, in a continuous manner. 
As an example, sections of the tangent bundle of a differential manifold are nothing but vector fields on that manifold.

Let ''F''(''U'') be the set of all sections on ''U''. ''F''(''U'') always contains at least one element, namely the '''zero section''': the function ''s'' that maps every element ''x'' of ''U'' to the zero element of the vector space π<sup>&minus;1</sup>({''x''}). With the pointwise addition and scalar multiplication of sections, ''F''(''U'') becomes itself a real vector space. The collection of these vector spaces is a sheaf of vector spaces on ''X''.

If ''s'' is an element of ''F''(''U'') and α : ''U'' → '''R''' is a continuous map, then α''s'' (pointwise scalar multiplication) is in ''F''(''U''). We see that ''F''(''U'') is a module over the ring of continuous real-valued functions on ''U''. Furthermore, if O<sub>''X''</sub> denotes the structure sheaf of continuous real-valued functions on ''X'', then ''F'' becomes a sheaf of O<sub>''X''</sub>-modules.

Not every sheaf of O<sub>''X''</sub>-modules arises in this fashion from a vector bundle: only the locally free ones do. (The reason: locally we are looking for sections of a projection ''U'' &times; '''R'''<sup>''k''</sup> → ''U''; these are precisely the continuous functions ''U'' → '''R'''<sup>''k''</sup>, and such a function is an ''k''-tuple of continuous functions ''U'' → '''R'''.)

Even more: the category of real vector bundles on ''X'' is equivalent to the category of locally free and finitely generated sheaves of O<sub>''X''</sub>-modules.
So we can think of the category of real vector bundles on ''X'' as sitting inside the category of sheaves of O<sub>''X''</sub>-modules; this latter category is abelian, so this is where we can compute kernels and cokernels of morphisms of vector bundles.

== Operations on vector bundles ==

Most operations on vector spaces can be extended to vector bundles by performing the vector space operation ''fiberwise''.