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 : to every point of the space we associate a vector space in such a way that these vector spaces fit together to form another space of the same kind as , which is then called a vector bundle over .
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space such that for all in : in this case there is a copy of for each in and these copies fit together to form the vector bundle over. Such vector bundles are said to be trivial. A more complicated class of examples are the tangent bundles of smooth 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 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 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. 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

A real vector bundle consists of:

topological spaces and
a continuous surjection
for every in, the structure of a finite-dimensional real vector space on the fiber

where the following compatibility condition is satisfied: for every point in, there is an open neighborhood of, a natural number, and a homeomorphism
such that for all in,

for all vectors in, and
the map is a linear isomorphism between the vector spaces and.

The open neighborhood together with the homeomorphism is called a local trivialization of the vector bundle. The local trivialization shows that locally the map "looks like" the projection of on.
Every fiber is a finite-dimensional real vector space and hence has a dimension. The local trivializations show that the function is locally constant, and is therefore constant on each connected component of. If is equal to a constant on all of, then is called the rank of the vector bundle, and is said to be a vector bundle of rank . Often the definition of a vector bundle includes that the rank is well defined, so that 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, equipped with the projection, is called the trivial bundle of rank over.

Transition functions

Given a vector bundle of rank, and a pair of neighborhoods and over which the bundle trivializes via
the composite function
is well-defined on the overlap, and satisfies
for some -valued function
These are called the transition functions of the vector bundle.
The set of transition functions forms a Čech cocycle in the sense that
for all over which the bundle trivializes satisfying. Thus the data defines a fiber bundle; the additional data of the specifies a structure group in which the action on the fiber is the standard action of.
Conversely, given a fiber bundle with a cocycle acting in the standard way on the fiber, there is associated a vector bundle. This is an example of the fibre bundle construction theorem for vector bundles, and can be taken as an alternative definition of a vector bundle.

Subbundles

One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle over a topological space, a subbundle is simply a topological subspace for which the restriction of to gives the structure of a vector bundle also. In this case the fibre is a vector subspace for every.
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, the Möbius band, a non-trivial line bundle over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.

Vector bundle morphisms

A morphism from the vector bundle to the vector bundle is given by a pair of continuous maps and such that
Note that is determined by , and 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 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 sometimes called bundle homomorphisms.
A bundle homomorphism from to with an inverse which is also a bundle homomorphism is called a bundle isomorphism, and then and are said to be isomorphic vector bundles. An isomorphism of a vector bundle over with the trivial bundle is called a trivialization of, and is then said to be trivial. 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. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the identity map on. That is, bundle morphisms for which the following diagram commutes:
A vector bundle morphism between vector bundles and covering a map from to can also be viewed as a vector bundle morphism over from to the pullback bundle.

Sections and locally free sheaves

Given a vector bundle : E → X and an open subset U of X, we can consider sections of on U, i.e. continuous functions s: U → E where the composite ∘ s is such that for all u in U. A section over U is an assignment, to every point p of U, a vector from the vector space fibre above p, in a continuous manner. As an example, a section of the tangent bundle of a differential manifold is the same as a vector field on that manifold.
Let F be the set of all sections on U. F 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 ⁻¹. With the pointwise addition and scalar multiplication of sections, F 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 and f : U → R is a continuous map, then their product fs is in F. This shows that F is a module over the ring of continuous real-valued functions on U. Furthermore, if O_X denotes the structure sheaf of continuous real-valued 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.
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.
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 there is a bundle E* over X, called the dual bundle, whose fiber at x ∈ X is the dual vector space *. Formally E* can be defined as the set of pairs, where x ∈ X and φ ∈ *. 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, and these extend straightforwardly to pairs of vector bundles E, F on X. A few examples follow.

The Whitney sum or direct sum bundle of E and F is a vector bundle E ⊕ F over X whose fiber over x is the direct sum E_x ⊕ F_x of the vector spaces E_x and F_x.
The tensor product bundle E ⊗ F is defined in a similar way, using fiberwise tensor product of vector spaces.
The Hom-bundle Hom is a vector bundle whose fiber at x is the space of linear maps from E_x to F_x or L). The Hom-bundle is so-called because there is a bijection between vector bundle homomorphisms from E to F over X and sections of Hom over X.
Building on the previous example, given a section s of an endomorphism bundle Hom and a function f: X → R, one can construct an eigenbundle by taking the fiber over a point x ∈ X to be the f-eigenspace of the linear map s: E_x → E_x. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of s being the zero section and f having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in E, while everywhere else the fiber is the trivial 0-dimensional vector space.
The dual vector bundle E* is the Hom bundle Hom of bundle homomorphisms of E and the trivial bundle R × X. There is a canonical vector bundle isomorphism Hom = E* ⊗ F.

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 ∈ Y. Hence, Whitney summing E ⊕ F can be defined as the pullback bundle of the diagonal map from X to X × X where the bundle over X × X is E × F.
Remark: Let X be a compact space. Any vector bundle E over X is a direct summand of a trivial bundle; i.e., there exists a bundle E such that E ⊕ E is trivial. This fails if X is not compact: for example, the tautological line bundle over the infinite real projective space does not have this property.