Inmathematics, avector bundle is atopological construction that makes precise the idea of afamily ofvector spaces parameterized by anotherspace${\displaystyle X}$ (for example${\displaystyle X}$ could be atopological space, amanifold, or analgebraic variety): to every point${\displaystyle x}$ of the space${\displaystyle X}$ we associate (or "attach") a vector space${\displaystyle V(x)}$ in such a way that these vector spaces fit together to form another space of the same kind as${\displaystyle X}$ (e.g. a topological space, manifold, or algebraic variety), which is then called avector bundle over${\displaystyle X}$.

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space${\displaystyle V}$ such that${\displaystyle V(x)=V}$for all${\displaystyle x}$ in${\displaystyle X}$: in this case there is a copy of${\displaystyle V}$ for each${\displaystyle x}$ in${\displaystyle X}$ and these copies fit together to form the vector bundle${\displaystyle X\times V}$ over${\displaystyle X}$. Such vector bundles are said to betrivial. A more complicated (and prototypical) class of examples are thetangent bundles ofsmooth (or differentiable) manifolds: to every point of such a manifold we attach thetangent 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 thehairy ball theorem. In general, a manifold is said to beparallelizable if, and only if, its tangent bundle is trivial.

Vector bundles are almost always required to belocally trivial, which means they are examples offiber bundles. Also, the vector spaces are usually required to be over thereal orcomplex 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 thecategory of topological spaces.

Areal vector bundle consists of:

1. topological spaces${\displaystyle X}$ (base space) and${\displaystyle E}$ (total space)
2. acontinuoussurjection${\displaystyle \pi :E\to X}$ (bundle projection)
3. for every${\displaystyle x}$ in${\displaystyle X}$, the structure of afinite-dimensionalrealvector space on thefiber${\displaystyle {\pi }^{-1}(\{x\})}$

where the following compatibility condition is satisfied: for every point${\displaystyle p}$ in${\displaystyle X}$, there is anopen neighborhood${\displaystyle U\subseteq X}$ of${\displaystyle p}$, anatural number${\displaystyle k}$, and ahomeomorphism

${\displaystyle \phi :U\times {\mathbb{R}}^k\to {\pi }^{-1}(U)}$

such that for all${\displaystyle x}$ in${\displaystyle U}$,

• ${\displaystyle (\pi \circ \phi )(x,v)=x}$ for allvectors${\displaystyle v}$ in${\displaystyle {\mathbb{R}}^k}$, and
• the map${\displaystyle v\mapsto \phi (x,v)}$ is alinearisomorphism between the vector spaces${\displaystyle {\mathbb{R}}^k}$ and${\displaystyle {\pi }^{-1}(\{x\})}$.

The open neighborhood${\displaystyle U}$ together with the homeomorphism${\displaystyle \phi }$ is called alocal trivialization of the vector bundle. The local trivialization shows thatlocally the map${\displaystyle \pi }$ "looks like" the projection of${\displaystyle U\times {\mathbb{R}}^k}$ on${\displaystyle U}$.

Every fiber${\displaystyle {\pi }^{-1}(\{x\})}$ is a finite-dimensional real vector space and hence has adimension${\displaystyle k_x}$. The local trivializations show that thefunction${\displaystyle x\to k_x}$ islocally constant, and is therefore constant on eachconnected component of${\displaystyle X}$. If${\displaystyle k_x}$ is equal to a constant${\displaystyle k}$ on all of${\displaystyle X}$, then${\displaystyle k}$ is called therank of the vector bundle, and${\displaystyle E}$ is said to be avector bundle of rank${\displaystyle k}$. Often the definition of a vector bundle includes that the rank is well defined, so that${\displaystyle k_x}$ is constant. Vector bundles of rank 1 are calledline bundles, while those of rank 2 are less commonly called plane bundles.

TheCartesian product${\displaystyle X\times {\mathbb{R}}^k}$, equipped with the projection${\displaystyle X\times {\mathbb{R}}^k\to X}$, is called thetrivial bundle of rank${\displaystyle k}$ over${\displaystyle X}$.

Given a vector bundle${\displaystyle E\to X}$ of rank${\displaystyle k}$, and a pair of neighborhoods${\displaystyle U}$ and${\displaystyle V}$ over which the bundle trivializes via

thecomposite function

${\displaystyle {\phi }_U^{-1}\circ {\phi }_V:(U\cap V)\times {\mathbb{R}}^k\to (U\cap V)\times {\mathbb{R}}^k}$

is well-defined on the overlap, and satisfies

${\displaystyle {\phi }_U^{-1}\circ {\phi }_V(x,v)=\left(x,g_{UV}(x)v\right)}$

for some${\displaystyle \text{GL}(k)}$-valued function

${\displaystyle g_{UV}:U\cap V\to \mathrm{GL}(k).}$

These are called thetransition functions (or thecoordinate transformations) of the vector bundle.

Theset of transition functions forms aČech cocycle in the sense that

${\displaystyle g_{UU}(x)=I,g_{UV}(x)g_{VW}(x)g_{WU}(x)=I}$

for all${\displaystyle U,V,W}$ over which the bundle trivializes satisfying${\displaystyle U\cap V\cap W\ne \mathrm{\varnothing }}$. Thus the data${\displaystyle (E,X,\pi ,{\mathbb{R}}^k)}$ defines afiber bundle; the additional data of the${\displaystyle g_{UV}}$ specifies a${\displaystyle \text{GL}(k)}$ structure group in which theaction on the fiber is the standard action of${\displaystyle \text{GL}(k)}$.

Conversely, given a fiber bundle${\displaystyle (E,X,\pi ,{\mathbb{R}}^k)}$ with a${\displaystyle \text{GL}(k)}$ cocycle acting in the standard way on the fiber${\displaystyle {\mathbb{R}}^k}$, there isassociated a vector bundle. This is an example of thefibre bundle construction theorem for vector bundles, and can be taken as an alternative definition of a vector bundle.

One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle${\displaystyle \pi :E\to X}$ over a topological space, a subbundle is simply atopological subspace${\displaystyle F\subset E}$ for which therestriction${\displaystyle {\pi |}_F}$ of${\displaystyle \pi }$ to${\displaystyle F}$ gives${\displaystyle {\pi |}_F:F\to X}$ the structure of a vector bundle also. In this case the fibre${\displaystyle F_x\subset E_x}$ is a vector subspace for every${\displaystyle x\in X}$.

A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, theMöbius band, a non-trivialline bundle over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.

Amorphism from the vector bundle${\displaystyle {\pi }_1:E_1\to X_1}$ to the vector bundle${\displaystyle {\pi }_2:E_2\to X_2}$ is given by a pair of continuous maps${\displaystyle f:E_1\to E_2}$ and${\displaystyle g:X_1\to X_2}$ such that$${\displaystyle g\circ {\pi }_1={\pi }_2\circ f}$$

for every${\displaystyle x}$ in${\displaystyle X_1}$, the map${\displaystyle {\pi }_1^{-1}(\{x\})\to {\pi }_2^{-1}(\{g(x)\})}$induced by${\displaystyle f}$ is alinear map between vector spaces.

Note that${\displaystyle g}$ is determined by${\displaystyle f}$ (because${\displaystyle {\pi }_1}$ is surjective), and${\displaystyle f}$ is then said tocoverg.

The class of all vector bundles together with bundle morphisms forms acategory. 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 abundle map betweenfiber bundles, and are sometimes called(vector) bundle homomorphisms.

A bundle homomorphism from${\displaystyle E_1}$ to${\displaystyle E_2}$ with aninverse which is also a bundle homomorphism (from${\displaystyle E_2}$ to${\displaystyle E_1}$) is called a(vector) bundle isomorphism, and then${\displaystyle E_1}$ and${\displaystyle E_2}$ are said to beisomorphic vector bundles. An isomorphism of a (rank${\displaystyle k}$) vector bundle${\displaystyle E}$ over${\displaystyle X}$ with the trivial bundle (of rank${\displaystyle k}$ over${\displaystyle X}$) is called atrivialization of${\displaystyle E}$, and${\displaystyle E}$ is then said to betrivial (ortrivializable). The definition of a vector bundle shows that any vector bundle islocally trivial.

We can also consider the category of all vector bundles over a fixed base space${\displaystyle X}$. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is theidentity map on${\displaystyle X}$. That is, bundle morphisms for which the following diagramcommutes:

(Note that this category isnotabelian; thekernel of a morphism of vector bundles is in general not a vector bundle in any natural way.)

A vector bundle morphism between vector bundles${\displaystyle {\pi }_1:E_1\to X_1}$ and${\displaystyle {\pi }_2:E_2\to X_2}$ covering a map${\displaystyle g}$ from${\displaystyle X_1}$ to${\displaystyle X_2}$ can also be viewed as a vector bundle morphism over${\displaystyle X_1}$ from${\displaystyle E_1}$ to thepullback bundle${\displaystyle g^{\ast }{E}_2}$.

Given a vector bundleπ:E →X and an open subsetU ofX, we can considersections ofπ onU, i.e. continuous functionss:U →E where the compositeπ ∘ s is such that(π ∘s)(u) =u for allu inU. A section overU is an assignment, to every pointp ofU, a vector from the vector space fibre abovep, in a continuous manner. As an example, a section of the tangent bundle of a differential manifold is the same as avector field on that manifold.

LetF(U) be the set of all sections onU.F(U) always contains at least one element, namely thezero section: the functions that maps every elementx ofU to thezero element of the vector spaceπ⁻¹({x}). With thepointwise addition andscalar multiplication of sections,F(U) becomes itself a real vector space. The collection of these vector spaces is asheaf of vector spaces onX.

Ifs is an element ofF(U) and f :U →R is a continuous map, then their product fs (pointwise scalar multiplication) is inF(U). This shows thatF(U) is amodule over thering of continuousreal-valued functions onU. Furthermore, if O_X denotes the structure sheaf of continuous real-valued functions onX, thenF becomes a sheaf of O_X-modules.

Not every sheaf of O_X-modules arises in this fashion from a vector bundle: only thelocally free ones do. (The reason is that locally we are looking for sections of a projectionU ×R^k →U; these are precisely the continuous functionsU →R^k, and such a function is ak-tuple of continuous functionsU →R.)

Even more: the category of real vector bundles onX isequivalent to the category of locally free andfinitely generated sheaves of O_X-modules.

So we can think of the category of real vector bundles onX as sitting inside the category ofsheaves of O_X-modules; this latter category is abelian, so this is where we can computekernels andcokernels of morphisms of vector bundles.

A rankn vector bundle is trivialif and only if it hasnlinearly independent global sections.

Mostoperations on vector spaces can be extended to vector bundles by performing the vector space operationfiberwise.

For example, ifE is a vector bundle overX, then there is a bundleE* overX, called thedual bundle, whose fiber atx ∈X is thedual vector space (E_x)*. FormallyE* can be defined as the set of pairs (x, φ), wherex ∈X and φ ∈ (E_x)*. The dual bundle is locally trivial because thedual space of the inverse of a local trivialization ofE is a local trivialization ofE*: the key point here is that the operation of taking the dual vector space isfunctorial.

There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundlesE,F onX (over the given field). A few examples follow.

• TheWhitney sum (named forHassler Whitney) ordirect sum bundle ofE andF is a vector bundleE ⊕F overX whose fiber overx is thedirect sumE_x ⊕F_x of the vector spacesE_x andF_x.
• Thetensor product bundleE ⊗F is defined in a similar way, using fiberwisetensor product of vector spaces.
• TheHom-bundle Hom(E,F) is a vector bundle whose fiber atx is the space of linear maps fromE_x toF_x (which is often denoted Hom(E_x,F_x) orL(E_x,F_x)). The Hom-bundle is so-called (and useful) because there is abijection between vector bundle homomorphisms fromE toF overX and sections of Hom(E,F) overX.
• Building on the previous example, given a sections of anendomorphism bundle Hom(E,E) and a functionf:X →R, one can construct aneigenbundle by taking the fiber over a pointx ∈X to be thef(x)-eigenspace of the linear maps(x):E_x →E_x. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case ofs being the zero section andf having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them inE, while everywhere else the fiber is the trivial 0-dimensional vector space.
• Thedual vector bundleE* is the Hom bundle Hom(E,R ×X) of bundle homomorphisms ofE and the trivial bundleR ×X. There is a canonical vector bundle isomorphism Hom(E,F) =E* ⊗F.

Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on thecategory of vector spaces can also be performed on the category of vector bundles in afunctorial manner. This is made precise in the language ofsmooth functors. An operation of a different nature is thepullback bundle construction. Given a vector bundleE →Y and a continuous mapf:X →Y one can "pull back"E to a vector bundlef*E overX. The fiber over a pointx ∈X is essentially just the fiber overf(x) ∈Y. Hence, Whitney summingE ⊕F can be defined as the pullback bundle of the diagonal map fromX toX ×X where the bundle overX ×X isE × F.

Remark: LetX be acompact space. Any vector bundleE overX is a direct summand of a trivial bundle; i.e., there exists a bundleE' such thatE ⊕E' is trivial. This fails ifX is not compact: for example, thetautological line bundle over the infinite real projective space does not have this property.^[1]

Vector bundles are often given more structure. For instance, vector bundles may be equipped with avector bundle metric. Usually this metric is required to bepositive definite, in which case each fibre ofE becomes aEuclidean space. A vector bundle with acomplex structure corresponds to acomplex vector bundle, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings becomplex-linear in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resultingreduction of the structure group of a bundle. Vector bundles over more generaltopological fields may also be used.

If instead of a finite-dimensional vector space, the fiberF is taken to be aBanach space then aBanach bundle is obtained.^[2] Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions

${\displaystyle g_{UV}:U\cap V\to \mathrm{GL}(F)}$

are continuous mappings ofBanach manifolds. In the corresponding theory for C^p bundles, all mappings are required to be C^p.

Vector bundles are specialfiber bundles, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for examplesphere bundles are fibered by spheres.

A vector bundle (E,p,M) issmooth, ifE andM aresmooth manifolds, p:E →M is a smooth map, and the local trivializations arediffeomorphisms. Depending on the required degree ofsmoothness, there are different corresponding notions ofC^p bundles,infinitely differentiableC^∞-bundles andreal analyticC^ω-bundles. In this section we will concentrate onC^∞-bundles. The most important example of aC^∞-vector bundle is thetangent bundle (TM,π_TM,M) of aC^∞-manifoldM.

A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which aresmooth functions on overlaps of trivializing chartsU andV. That is, a vector bundleE is smooth if it admits a covering by trivializing open sets such that for any two such setsU andV, the transition function

${\displaystyle g_{UV}:U\cap V\to \mathrm{GL}(k,\mathbb{R})}$

is a smooth function into thematrix group GL(k,R), which is aLie group.

Similarly, if the transition functions are:

• C^r then the vector bundle is aC^r vector bundle,
• realanalytic then the vector bundle is areal analytic vector bundle (this requires the matrix group to have a real analytic structure),
• holomorphic then the vector bundle is aholomorphic vector bundle (this requires the matrix group to be acomplex Lie group),
• algebraic functions then the vector bundle is analgebraic vector bundle (this requires the matrix group to be analgebraic group).

TheC^∞-vector bundles (E,p,M) have a very important property not shared by more generalC^∞-fibre bundles. Namely, the tangent spaceT_v(E_x) at anyv ∈E_x can be naturally identified with the fibreE_x itself. This identification is obtained through thevertical liftvl_v:E_x →T_v(E_x), defined as

${\displaystyle {\mathrm{vl}}_{v}w[f]:={\frac{d}{dt}|}_{t=0}f(v+tw),f\in C^{\mathrm{\infty }}(E_x).}$

The vertical lift can also be seen as a naturalC^∞-vector bundle isomorphismp*E →VE, where (p*E,p*p,E) is the pull-back bundle of (E,p,M) overE throughp:E →M, andVE := Ker(p_*) ⊂TE is thevertical tangent bundle, a natural vector subbundle of the tangent bundle (TE,π_TE,E) of the total spaceE.

The total spaceE of any smooth vector bundle carries a natural vector fieldV_v := vl_vv, known as thecanonical vector field. More formally,V is a smooth section of (TE,π_TE,E), and it can also be defined as the infinitesimal generator of theLie-group action${\displaystyle (t,v)\mapsto e^{tv}}$ given by the fibrewise scalar multiplication. The canonical vector fieldV characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that whenX is a smooth vector field on a smooth manifoldM andx ∈M such thatX_x = 0, the linear mapping

${\displaystyle C_x(X):T_{x}M\to T_{x}M;C_x(X)Y=({\mathrm{\nabla }}_{Y}X{)}_x}$

does not depend on the choice of the linearcovariant derivative ∇ onM. The canonical vector fieldV onE satisfies the axioms

1. The flow (t,v) → Φ^t_V(v) ofV is globally defined.
2. For eachv ∈V there is a unique lim_t→∞ Φ^t_V(v) ∈V.
3. C_v(V)∘C_v(V) =C_v(V) wheneverV_v = 0.
4. Thezero set ofV is a smoothsubmanifold ofE whosecodimension is equal to the rank ofC_v(V).

Conversely, ifE is any smooth manifold andV is a smooth vector field onE satisfying 1–4, then there is a unique vector bundle structure onE whose canonical vector field isV.

For any smooth vector bundle (E,p,M) the total spaceTE of its tangent bundle (TE,π_TE,E) has a naturalsecondary vector bundle structure (TE,p_*,TM), wherep_* is thepush-forward of the canonical projectionp:E →M. The vector bundle operations in this secondary vector bundle structure are the push-forwards +_*:T(E ×E) →TE and λ_*:TE →TE of the original addition +:E ×E →E and scalar multiplication λ:E →E.

The K-theory group,K(X), of a compactHausdorff topological space is defined as theabelian group generated byisomorphism classes[E] ofcomplex vector bundles under the group operation ofWhitney sum, modulo therelation that, whenever we have anexact sequence$${\displaystyle 0\to A\to B\to C\to 0,}$$then$${\displaystyle [B]=[A]+[C]}$$ intopological K-theory.KO-theory is a version of this construction which considers real vector bundles. K-theory withcompact supports can also be defined, as well as higher K-theory groups.

The famousperiodicity theorem ofRaoul Bott asserts that the K-theory of any spaceX is isomorphic to that of theS²X, the double suspension ofX.

Inalgebraic geometry, one considers the K-theory groups consisting ofcoherent sheaves on aschemeX, as well as the K-theory groups of vector bundles on the scheme with the aboveequivalence relation. The two constructs are naturally isomorphic provided that the underlying scheme issmooth.

• Grassmannian:classifying spaces for vector bundle, among whichprojective spaces forline bundles
• Characteristic class
• Splitting principle
• Stable bundle

• Connection: the notion needed to differentiate sections of vector bundles.
• Gauge theory: the general study of connections on vector bundles and principal bundles and their relations to physics.

• Algebraic vector bundle
• Picard group
• Holomorphic vector bundle

• "Vector bundle",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Why is it useful to study vector bundles ? onMathOverflow
• Why is it useful to classify the vector bundles of a space ?

• Vector bundles

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