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
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is said to be exact if the sequence
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is exact for all . In particular, the sequence
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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
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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
[1] | C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) MR0242081 Zbl 0653.53001 Zbl 0284.53018 |
[2] | M.F. Atiyah, "![]() |
[3] | S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III MR1931083 MR1532744 MR0155257 Zbl 1008.57001 Zbl 0103.15101 |
[4] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804 |
[5] | S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) MR0533884 Zbl 0444.32004 |
[6] | F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) MR1335917 MR0202713 Zbl 0376.14001 |
Comments
For more on the universality and classifying properties of the bundles and
cf. Classifying space or [a1].
References
[a1] | J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) MR0440554 Zbl 0298.57008 |
Vector bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_bundle&oldid=24005