 Vector bundle

In mathematics, a vector bundle is a topological construction that 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 × V over X. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of 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 nontrivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if and only if its tangent bundle is trivial.
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 required to be 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.
Contents
Definition and first consequences
A real vector bundle consists of:
 topological spaces X (base space) and E (total space)
 a continuous surjection π : E → X (bundle projection)
 for every x in X, the structure of a finitedimensional real vector space on the fiber π^{−1}({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
such that for all x ∈ U,
 (π∘φ)(x,v) = x for all vectors v in R^{k}, and
 the map v ↦ φ(x,v) is an isomorphism between the vector spaces R^{k} and π^{−1}({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 × R^{k} on U.
Every fiber π^{−1}({x}) is a finitedimensional real vector space and hence has a dimension k_{x}. The local trivializations show that the function x ↦ k_{x} is locally constant, and is therefore constant on each connected component of X. If k_{x} 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. Often the definition of a vector bundle includes that the rank is well defined, so that k_{x} is constant. Vector bundles of rank 1 are called line bundles, while those of rank 2 are less commonly called plane bundles.
The Cartesian product X × R^{k} , equipped with the projection X × R^{k} → X, is called the trivial bundle of rank k over X.
Transition functions
Given a vector bundle E → X of rank k, and a pair of neighborhoods U and V over which the bundle trivializes via
the composite function
is welldefined on the overlap, and satisfies
for some GL(k)valued function
These are called the transition functions (or the coordinate transformations) of the vector bundle.
The set of transition functions forms a Čech cocycle in the sense that
for all U,V,W over which the bundle trivializes. Thus the data (E,X,π,R^{k}) defines a fiber bundle; the additional data of the g_{UV} specifies a GL(k) structure group in which the action on the fiber is the standard action of GL(k).
Conversely, given a fiber bundle (E,X,π,R^{k}) with a GL(k) cocycle acting in the standard way on the fiber R^{k}, there is associated a vector bundle. This is sometimes taken as the definition of a vector bundle.
Vector bundle morphisms
A morphism from the vector bundle π_{1} : E_{1} → X_{1} to the vector bundle π_{2} : E_{2} → X_{2} is given by a pair of continuous maps f : E_{1} → E_{2} and g : X_{1} → X_{2} such that
 g ∘ π_{1} = π_{2} ∘ f
 for every x in X_{1}, the map π_{1}^{−1}({x}) → π_{2}^{−1}({g(x)}) induced by f is a linear map between vector spaces.
Note that g is determined by f (because π_{1} 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_{1} to E_{2} with an inverse which is also a bundle homomorphism (from E_{2} to E_{1}) is called a (vector) bundle isomorphism, and then E_{1} and E_{2} 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 identity map on X. That is, bundle morphisms for which the following diagram commutes:
(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_{1} : E_{1} → X_{1} and π_{2} : E_{2} → X_{2} covering a map g from X_{1} to X_{2} can also be viewed as a vector bundle morphism over X_{1} from E_{1} to the pullback bundle g*E_{2}.
Sections and locally free sheaves
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 where the composite π∘s is such that (π∘s)(u) = u for all u in U. 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^{−1}({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 αs (pointwise scalar multiplication) is in F(U). We see that F(U) is a module over the ring of continuous realvalued functions on U. Furthermore, if O_{X} denotes the structure sheaf of continuous realvalued functions on X, then F becomes a sheaf of O_{X}modules.
Not every sheaf of O_{X}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 × R^{k} → U; these are precisely the continuous functions U → R^{k}, and such a function is an ktuple 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_{X}modules. So we can think of the category of real vector bundles on X as sitting inside the category of sheaves of O_{X}modules; this latter category is abelian, so this is where we can compute kernels and cokernels of morphisms of vector bundles.
Note that a rank n vector bundle is trivial if and only if it has n linearly independent global sections.
Operations on vector bundles
Most operations on vector spaces can be extended to vector bundles by performing the vector space operation fiberwise.
For example, if E is a vector bundle over X, then the there is a bundle E* over X, called the dual bundle, whose fiber at x∈X is the dual vector space (E_{x})*. Formally E* can be defined as the set of pairs (x,φ), where x∈X and φ∈(E_{x})*. The dual bundle is locally trivial because the dual space of the inverse of a local trivialization of E is a local trivialization of E*: the key point here is that the operation of taking the dual vector space is functorial.
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 bundles E, F on X (over the given field). A few examples follow.
 The Whitney sum (named for Hassler Whitney) or direct sum bundle of E and F is a vector bundle over X whose fiber over x is the direct sum of the vector spaces E_{x} and F_{x}.
 The tensor product bundle is defined in a similar way, using fiberwise tensor product of vector spaces.
 The Hombundle Hom(E,F) is a vector bundle whose fiber at x is the space of linear maps from E_{x} to F_{x} (which is often denoted Hom(E_{x},F_{x}) or L(E_{x},F_{x})). The Hombundle is socalled (and useful) because there is a bijection between vector bundle homomorphisms from E to F over X and sections of Hom(E,F) over X.
 The dual vector bundle E^{∗} is the Hom bundle Hom(E,R×X) of bundle homomorphisms of E and the trivial bundle R×X. There is a canonical vector bundle isomorphism
Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the category of vector spaces can also be performed on the category of vector bundles in a functorial manner. This is made precise in the language of smooth functors. An operation of a different nature is the pullback bundle construction. Given a vector bundle E → Y and a continuous map f : X → Y one can "pull back" E to a vector bundle f*E over X. The fiber over a point x ∈ X is essentially just the fiber over f(x) ∈ Y. Hence, Whitney summing can be defined as the pullback bundle of the diagonal map from X to X x X where the bundle over X x X is E x F.
Additional structures and generalizations
Vector bundles are often given more structure. For instance, vector bundles may be equipped with a vector bundle metric. Usually this metric is required to be positive definite, in which case each fibre of E becomes a Euclidean space. A vector bundle with a complex structure corresponds to a complex vector bundle, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be complexlinear in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting reduction of the structure group of a bundle. Vector bundles over more general topological fields may also be used.
If instead of a finitedimensional vector space, if the fiber F is taken to be a Banach space then a Banach bundle is obtained.^{[1]} Specifically, one must require that the local trivializions are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions
are continuous mappings of Banach manifolds. In the corresponding theory for C^{p} bundles, all mappings are required to be C^{p}.
Vector bundles are special fiber 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 example sphere bundles are fibered by spheres.
Smooth vector bundles
A vector bundle (E,p,M) is smooth, if E and M are smooth manifolds, p : E → M is a smooth map, and the local trivializations are diffeomorphisms. Depending on the required degree of smoothness, there are different corresponding notions of C^{p} bundles, infinitely differentiable C^{∞}bundles and real analytic C^{ω}bundles. In this section we will concentrate on C^{∞}bundles. The most important example of a C^{∞}vector bundle is the tangent bundle (TM,π_{TM},M) of a C^{∞}manifold M.
The C^{∞}vector bundles (E,p,M) have a very important property not shared by more general C^{∞}fibre bundles. Namely, the tangent space T_{v}(E_{x}) at any v∈E_{x} can be naturally identified with the fibre E_{x} itself. This identification is obtained through the vertical lift vl_{v}:E_{x}→T_{v}(E_{x}), defined as
The vertical lift can also be seen as a natural C^{∞}vector bundle isomorphism p*E→VE, where (p*E,p*p,E) is the pullback bundle of (E,p,M) over E through p:E→M, and VE:=Ker(p_{*})⊂TE is the vertical tangent bundle, a natural vector subbundle of the tangent bundle (TE,π_{TE},E) of the total space E.
The slit vector bundle E/0, obtained from (E,p,M) by removing the zero section 0⊂E, carries a natural vector field V_{v} := vl_{v}v, known as the canonical 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 the Liegroup action
For any smooth vector bundle (E,p,M) the total space TE of its tangent bundle (TE,π_{TE},E) has a natural secondary vector bundle structure (TE,p_{*},TM), where p_{*} is the pushforward of the canonical projection p:E→M. The vector bundle operations in this secondary vector bundle structure are the pushforwards +_{*}:T(E×E)→TE and λ_{*}:TE→TE of the original addition +:E×E→E and scalar multiplication λ:E→E.
Ktheory
The Ktheory group
 K(X)
of a manifold is defined as the abelian group generated by isomorphism classes [E] of (complex) vector bundles modulo the relation that whenever we have an exact sequence
 0 → A → B → C → 0
then
 [B]=[A]+[C]
in topological Ktheory. KOtheory is a version of this construction which considers real vector bundles. Ktheory with compact supports can also be defined, as well as higher Ktheory groups.
The famous periodicity theorem of Raoul Bott asserts that the Ktheory of any space X is isomorphic to that of the Cartesian product
 X × S^{2},
where S^{2} denotes the 2sphere.
In algebraic geometry, one considers the Ktheory groups consisting of coherent sheaves on a scheme X, as well as the Ktheory groups of vector bundles on the scheme with the above equivalence relation. The two constructs are the same provided that the underlying scheme is smooth.
See also
General notions
 Grassmannian: classifying spaces for vector bundle, among which Projective spaces for line bundles
 Characteristic class
 Splitting principle
Topology and differential geometry
 Fiber bundle: the general topological notion, among which Covering spaces
 Connection (vector bundle): the notion needed to differentiate sections of vector bundles.
 Sheaf (mathematics)
 Topological Ktheory
Algebraic and analytic geometry
 Coherent sheaf, in particular Picard group
 Holomorphic vector bundle
Notes
 ^ Lang, Serge (1995), Differential and Riemannian manifolds, Berlin, New York: SpringerVerlag, ISBN 9780387943381
References
 Hatcher, Allen (2003), Vector Bundles & KTheory (2.0 ed.), http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html
 Lee, John M. (2003), Introduction to Smooth Manifolds, New York: Springer, ISBN 0387954481, http://www.math.washington.edu/~lee/Books/smooth.html see Ch.5
 Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis (3rd ed.), Berlin, New York: SpringerVerlag, ISBN 9783540426271, see section 1.5.
 Abraham, Ralph H.; Marsden, Jerrold E. (1978), Foundations of mechanics, London: BenjaminCummings, ISBN 9780805301021, see section 1.5
External links
Categories: Differential topology
 Algebraic topology
 Vector bundles
 Vectors
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