# 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

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

is said to be exact if the sequence

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

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

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For more on the universality and classifying properties of the bundles $\gamma ^ {k}$ and $\gamma _ {k}$ cf. Classifying space or [a1].