In mathematics, a vector bundle is a 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ...

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 not trivial by the Hairy ball theorem. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... A failed attempt to comb a hairy ball flat, leaving two tufts at the top and bottom. ...

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. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... The category Top has topological spaces as objects and continuous maps as morphisms. ...

Definition and first consequences

Definition 1

A vector bundle is a generalization of a fiber bundle, where the standard fiber V is a vector space, with structure group the general linear group of V. The "generality" is in the fact that in the case of a fiber bundle, all of its fibers must be vector spaces which are all homeomorphic to each other. In the case of a vector bundle, the fibers are vector spaces which might have different dimensions. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...

Definition 2

A real vector bundle consists of:

1. topological spaces X (base space) and E (total space)
2. a continuous map π : E → X (bundle projection)
3. for every x in X, the structure of a finite-dimensional 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 In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

such that for all x ∈ U,

• πφ(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 finite-dimensional 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. Vector bundles of rank 1 are called line bundles. This article is about functions in mathematics. ... In mathematics, a function f from a topological space A to a set B is called locally constant, iff for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant. ... This article is about mathematics. ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ...

The cartesian product X × R^k , equipped with the projection X × R^k → X, is called the trivial bundle of rank k over X. In mathematics, the Cartesian product is a direct product of sets. ...

Vector bundle morphisms

A morphism from the vector bundle π₁ : E₁ → X₁ to the vector bundle π₂ : E₂ → X₂ is given by a pair of continuous maps f : E₁ → E₂ and g : X₁ → X₂ such that In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

• gπ₁ = π₂f

• for every x in X₁, the map π₁⁻¹({x}) → π₂⁻¹({g(x)}) induced by f is a linear map between vector spaces.

Note that g is determined by f (because π₁ is surjective), and f is then said to cover g. Image File history File links BundleMorphism-01. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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. In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

A bundle homomorphism from E₁ to E₂ with an inverse which is also a bundle homomorphism (from E₂ to E₁) is called a (vector) bundle isomorphism, and then E₁ and E₂ 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: An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

(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.) Image File history File links BundleMorphism-02. ... In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... The word kernel has several meanings in mathematics, some related to each other and some not. ...

A vector bundle morphism between vector bundles π₁ : E₁ → X₁ and π₂ : E₂ → X₂ covering a map g from X₁ to X₂ can also be viewed as a vector bundle morphism over X₁ from E₁ to the pullback bundle g^*E₂. In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ...

Sections and locally free sheaves

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 πs = id_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. Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ...

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. In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

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 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. In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

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 k-tuple of continuous functions U → R.) A sheaf of -modules on a ringed space is called locally free if for each point , there is an open neighborhood (mathematics) of such that is free as an -module, or equivalently, , the stalk of at , is free as a -module. ...

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. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

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. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... For functors in computer science, see the function object article. ...

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 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 Hom-bundle 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 Hom-bundle is so-called (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.

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. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ...

Variants and generalizations

Vector bundles are special fiber bundles, loosely speaking those where the fibers are vector spaces. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π : E → X be a smooth map, and the local trivialization maps φ be diffeomorphisms. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

Replacing real vector spaces with complex ones, we obtain complex vector bundles. This is a special case of reduction of the structure group of a bundle. Vector spaces over other topological fields may also be used, but that is comparatively rare. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ...

If we allow arbitrary Banach spaces in the local trivialization (rather than only Rⁿ), we obtain Banach bundles. In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

K-theory

The K-theory 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 In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

0 → A → B → C → 0

then

[B]=[A]+[C]

in topological K-theory. KO-theory is a version of this construction which considers real vector bundles. K-theory with compact supports can also be defined, as well as higher K-theory groups. In mathematics, topological K-theory is a branch of algebraic topology. ... In mathematics, topological K-theory is a branch of algebraic topology. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

The famous periodicity theorem of Raoul Bott asserts that the K-theory of any space X is isomorphic to that of the Cartesian product In mathematics, the Bott periodicity theorem is a result from homotopy theory which was discovered by Raoul Bott during the latter part of the 1950s, and proved to be of foundational significance for much further research, in particular in K-theory. ... Raoul Bott (Harvard University News Office) Raoul Bott, FRS (born September 24, 1923, died December 20, 2005) was a mathematician known for numerous basic contributions to geometry in its broad sense. ... In mathematics, the Cartesian product is a direct product of sets. ...

X × S²,

where S² denotes the 2-sphere. For other uses, see sphere (disambiguation). ...

In algebraic geometry, one considers the K-theory groups consisting of coherent sheaves on a scheme X, as well as the K-theory 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. Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

See also

• Abelian category
• Grothendieck group

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way. ...

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 See section 1.5.
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 1.5.