Vector bundle

A fibre space $ \pi : X \rightarrow B $ each fibre $ \pi ^ {-1} ( b) $ of which is endowed with the structure of a (finite-dimensional) vector space $ V $ over a skew-field $ {\mathcal P} $ such that the following local triviality condition is satisfied. Each point $ b \in B $ has an open neighbourhood $ U $ and an $ U $-isomorphism of fibre bundles $ \phi : \pi ^ {-1} ( U) \rightarrow U \times V $ such that $ \phi \mid _ {\pi ^ {- 1} ( b) } : \pi ^ {-1} ( b) \rightarrow b \times V $ is an isomorphism of vector spaces for each $ b \in B $; $ \mathop{\rm dim} V $ is said to be the dimension of the vector bundle. The sections of a vector bundle $ \pi $ form a locally free module $ \Gamma ( \pi ) $ over the ring of continuous functions on $ B $ with values in $ {\mathcal P} $. A morphism of vector bundles is a morphism of fibre bundles $ f: \pi \rightarrow \pi ^ \prime $ for which the restriction to each fibre is linear. The set of vector bundles and their morphisms forms the category $ \mathbf{Bund} $. 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 $ X ^ \prime \subset X $ such that $ \pi \mid _ {X} ^ \prime : X ^ \prime \rightarrow B $ is a vector bundle and $ X ^ \prime \cap \pi ^ {-1} ( b) $ is a vector subspace in $ \pi ^ {-1} ( b) $ is said to be a subbundle of the vector bundle $ \pi $. For instance, let $ V $ be a vector space and let $ G _ {k} ( V) $ be the Grassmann manifold of subspaces of $ V $ of dimension $ k $; the subspace of the product $ G _ {k} ( V) \times V $, consisting of pairs $ ( p, v) $ such that $ v \in p $, will then be a subbundle $ \gamma _ {k} $ of the trivial vector bundle $ G _ {k} ( V) \times V $. The union of all vector spaces $ \pi ^ {-1} ( b) / \pi _ {2} ^ {-1} ( b) $, where $ \pi _ {2} $ is a subbundle of $ \pi $ endowed with the quotient topology, is said to be a quotient bundle of $ \pi $. Let, furthermore, $ V $ be a vector space and let $ G ^ {k} ( V) $ be the Grassmann manifold of subspaces of $ V $ of codimension $ k $; the quotient bundle $ \gamma ^ {k} $ of the trivial vector bundle $ G ^ {k} ( V) \times V $ is defined as the quotient space of the product $ G ^ {k} ( V) \times V $ by the subbundle consisting of all pairs $ ( p, v) $, $ v \in p $. 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 $ B $- morphism of vector bundles $ f : \pi \rightarrow \pi ^ \prime $ is said to be of constant rank (pure) if $ \mathop{\rm dim} \mathop{\rm ker} f \ \mid _ {\pi ^ {- 1} ( b) } $ is locally constant on $ B $. 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 $ f $ of locally constant rank: $ \mathop{\rm Ker} f $ (the kernel of $ f $), which is a subbundle of $ \pi $; $ \mathop{\rm Im} f $ (the image of $ f $), which is a subbundle of $ \pi ^ \prime $; $ \mathop{\rm Coker} f $ (the cokernel of $ f $), which is a quotient bundle of $ \pi $; and $ \mathop{\rm Coim} f $ (the co-image of $ f $), which is a quotient bundle of $ \pi ^ \prime $. Any subbundle $ \pi _ {1} $ is the image of some monomorphism $ i: \pi _ {1} \rightarrow \pi $, while any quotient bundle $ \pi _ {2} $ is the cokernel of some epimorphism $ j : \pi \rightarrow \pi _ {2} $. A sequence of $ B $- morphisms of vector bundles

$$ {} \cdots \rightarrow \pi ^ \prime \rightarrow \pi \rightarrow \pi ^ {\prime\prime} \rightarrow \cdots $$

is said to be exact if the sequence

$$ {} \cdots \rightarrow ( \pi ^ \prime ) ^ {-1} ( b) \rightarrow \pi ^ {-1} ( b) \rightarrow ( \pi ^ {\prime\prime} ) ^ {-1} ( b) \rightarrow \cdots $$

is exact for all $ b \in B $. In particular, the sequence

$$ 0 \rightarrow \pi _ {1} \rightarrow ^ { i } \pi \rightarrow ^ { j } \pi _ {2} \rightarrow 0 , $$

where $ 0 $ is the zero vector bundle, is exact if $ i $ is a monomorphism, $ j $ is an epimorphism and $ \mathop{\rm Im} i = { \mathop{\rm Ker} } j $. The set of vector bundles over $ B $ and $ B $- morphisms of locally constant rank forms an exact subcategory $ \mathbf{Bund} _ {B} $ of the category $ \mathbf{Bund} $.

For any vector bundle $ \pi : X \rightarrow B $ and mapping $ u: B _ {1} \rightarrow B $, the induced fibre bundle $ u ^ {*} ( \pi ) $ is endowed with a vector bundle structure such that the morphism $ U: u ^ {*} ( \pi ) \rightarrow \pi $ is a vector bundle morphism. This structure is unique and has the following property: Every fibre mapping $ {( u ^ {*} ( \pi )) } ^ {-1} ( b) \rightarrow \pi ^ {-1} ( u( b)) $ is an isomorphism of vector spaces. For instance, a vector bundle of dimension $ k $ over a paracompact space $ B $ is isomorphic to one of the vector bundles $ u ^ {*} ( \gamma _ {k} ) $ and $ \widetilde{u} {} ^ {*} ( \gamma ^ {k} ) $ induced by certain mappings $ u: B \rightarrow G _ {k} ( V) $ and $ \widetilde{u} : B \rightarrow G ^ {k} ( V) $, respectively; moreover, homotopic mappings induce isomorphic vector bundles and, if $ \mathop{\rm dim} V \neq \infty $, the converse is true: To isomorphic vector bundles there correspond homotopic mappings $ u $ and $ \widetilde{u} $. This is one of the fundamental theorems in the homotopic classification of vector bundles, expressing the universal character of the vector bundles $ \gamma _ {k} $ and $ \gamma ^ {k} $ with respect to the classifying mappings $ u $ and $ \widetilde{u} $.

Any continuous operation (functor) $ T $ on the category of vector spaces uniquely determines a continuous functor on the category of vector bundles over $ B $; in this way it is possible to construct bundles associated with a given vector bundle: tensor bundles, vector bundles of morphisms $ { \mathop{\rm Hom} } _ {B} ( \pi , \pi ^ \prime ) $ and, in particular, the dual vector bundle $ \pi ^ {*} $, 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) $ \pi \oplus \pi ^ \prime $ and tensor product $ \pi \otimes \pi ^ \prime $ have been defined for two vector bundles $ \pi $ and $ \pi ^ \prime $. With respect to these operations the set of classes $ { \mathop{\rm Vekt} } _ {B} $ of isomorphic vector bundles over $ B $ forms a semi-ring which plays an important part in the construction of a $ K $-functor; thus, if for vector bundles $ \pi $ and $ \pi ^ \prime $ there exist trivial vector bundles $ \theta $ and $ \theta ^ \prime $ such that the vector bundles $ \pi \oplus \theta $ and $ \pi ^ \prime \oplus \theta ^ \prime $ are isomorphic (i.e. $ \pi $ and $ \pi ^ \prime $ are stably equivalent), then their images in the "completion" $ K ( B) $ of the semi-ring $ { \mathop{\rm Vekt} } _ {B} $ are identical; moreover, the fact that the ring $ K( B) $ 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 $ \pi : X \rightarrow B $ over a paracompact space $ B $ there exists a section $ \beta $ of the vector bundle

$$ \pi ^ {*} \oplus \pi ^ {*} = \mathop{\rm Hom} ( \pi \oplus \pi , P ), $$

where $ P $ is a trivial one-dimensional vector bundle, which on each fibre $ \pi ^ {-1} ( b) $ is a positive-definite form, i.e. $ \pi $ is metrizable; this makes it possible to establish, in particular, the splittability of an arbitrary exact sequence of vector bundles

$$ 0 \rightarrow \xi \rightarrow ^ { u } \pi \rightarrow ^ { v } \zeta \rightarrow 0 $$

in which $ \pi $ is metrizable, that is, the existence of a morphism $ w : \xi \oplus \zeta \rightarrow \pi $ such that $ wi = u $, $ vw = j $, where $ i $ is the imbedding into the first term and $ j $ is the projection onto the second term.

If, in each fibre $ \pi ^ {-1} ( b) $ of the vector bundle $ \pi : X \rightarrow B $, one identifies the points lying on the same line passing through zero, one obtains a bundle $ \pi _ {0} : \Pi _ {( \pi ) } \rightarrow B $, which is associated with $ \pi $ and is said to be its projectivization; a fibre of $ \pi _ {0} $ is the projective space $ \Pi ( V) $ which is associated with $ V $. This bundle is used to study Thom spaces $ T ( \pi ) = \Pi ( \pi \oplus P)/ \Pi ( \pi ) $ (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 $ { \mathop{\rm Hom} } ( \pi , \pi ^ \prime ) $, 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, "$K$-theory: lectures" , Benjamin (1967) MR224083
[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 $ \gamma ^ {k} $ and $ \gamma _ {k} $ cf. Classifying space or [a1].

References

[a1]	J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) MR0440554 Zbl 0298.57008