# Vector bundle

A fibre space each fibre of which is endowed with the structure of a (finite-dimensional) vector space over a skew-field such that the following local triviality condition is satisfied. Each point has an open neighbourhood and an -isomorphism of fibre bundles such that is an isomorphism of vector spaces for each ; is said to be the dimension of the vector bundle. The sections of a vector bundle form a locally free module over the ring of continuous functions on with values in . A morphism of vector bundles is a morphism of fibre bundles for which the restriction to each fibre is linear. The set of vector bundles and their morphisms forms the category . The concept of a vector bundle arose as an extension of the tangent bundle and the normal bundle in differential geometry; by now it has become a basic tool for studies in various branches of mathematics: differential and algebraic topology, the theory of linear connections, algebraic geometry, the theory of (pseudo-) differential operators, etc.

A subset such that is a vector bundle and is a vector subspace in is said to be a subbundle of the vector bundle . For instance, let be a vector space and let be the Grassmann manifold of subspaces of of dimension ; the subspace of the product , consisting of pairs such that , will then be a subbundle of the trivial vector bundle . The union of all vector spaces , where is a subbundle of endowed with the quotient topology, is said to be a quotient bundle of . Let, furthermore, be a vector space and let be the Grassmann manifold of subspaces of of codimension ; the quotient bundle of the trivial vector bundle is defined as the quotient space of the product by the subbundle consisting of all pairs , . The concepts of a subbundle and a quotient bundle are used in contraction and glueing operations used to construct vector bundles over quotient spaces.

A -morphism of vector bundles is said to be of constant rank (pure) if is locally constant on . Injective and surjective morphisms are exact and are said to be monomorphisms and epimorphisms of the vector bundle, respectively. The following vector bundles are uniquely defined for a morphism of locally constant rank: (the kernel of ), which is a subbundle of ; (the image of ), which is a subbundle of ; (the cokernel of ), which is a quotient bundle of ; and (the co-image of ), which is a quotient bundle of . Any subbundle is the image of some monomorphism , while any quotient bundle is the cokernel of some epimorphism . A sequence of -morphisms of vector bundles

is said to be exact if the sequence

is exact for all . In particular, the sequence

where is the zero vector bundle, is exact if is a monomorphism, is an epimorphism and . The set of vector bundles over and -morphisms of locally constant rank forms an exact subcategory of the category .

For any vector bundle and mapping , the induced fibre bundle is endowed with a vector bundle structure such that the morphism is a vector bundle morphism. This structure is unique and has the following property: Every fibre mapping is an isomorphism of vector spaces. For instance, a vector bundle of dimension over a paracompact space is isomorphic to one of the vector bundles and induced by certain mappings and , respectively; moreover, homotopic mappings induce isomorphic vector bundles and, if , the converse is true: To isomorphic vector bundles there correspond homotopic mappings and . This is one of the fundamental theorems in the homotopic classification of vector bundles, expressing the universal character of the vector bundles and with respect to the classifying mappings and .

Any continuous operation (functor) on the category of vector spaces uniquely determines a continuous functor on the category of vector bundles over ; in this way it is possible to construct bundles associated with a given vector bundle: tensor bundles, vector bundles of morphisms and, in particular, the dual vector bundle , exterior powers of vector bundles, etc., whose sections are vector bundles with supplementary structures. These are extensively employed in practical applications.

A direct sum (Whitney sum) and tensor product have been defined for two vector bundles and . With respect to these operations the set of classes of isomorphic vector bundles over forms a semi-ring which plays an important part in the construction of a -functor; thus, if for vector bundles and there exist trivial vector bundles and such that the vector bundles and are isomorphic (i.e. and are stably equivalent), then their images in the "completion" of the semi-ring are identical; moreover, the fact that the ring and the set of classes of stably-equivalent vector bundles coincide follows from the existence of an inverse vector bundle for any vector bundle over a paracompact space.

For any vector bundle over a paracompact space there exists a section of the vector bundle

where is a trivial one-dimensional vector bundle, which on each fibre is a positive-definite form, i.e. is metrizable; this makes it possible to establish, in particular, the splittability of an arbitrary exact sequence of vector bundles

in which is metrizable, that is, the existence of a morphism such that , , where is the imbedding into the first term and is the projection onto the second term.

If, in each fibre of the vector bundle , one identifies the points lying on the same line passing through zero, one obtains a bundle , which is associated with and is said to be its projectivization; a fibre of is the projective space which is associated with . This bundle is used to study Thom spaces (cf. Thom space), used in the homotopic interpretation of classes of bordant manifolds, characteristic classes of vector bundles describing the homological properties of manifolds, etc.

The concept of a vector bundle can be generalized to the case when the fibre is an infinite-dimensional vector space; in doing so, one must distinguish between the different topologies of the space of morphisms , suitably modify the definitions of a pure morphism and an exact sequence of morphisms, and also the construction of vector bundles associated with continuous functors on the category of infinite-dimensional vector spaces.

#### References

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