Namespaces
Variants
Actions

Difference between revisions of "Vector bundle"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
Line 1: Line 1:
A [[Fibre space|fibre space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963801.png" /> each fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963802.png" /> of which is endowed with the structure of a (finite-dimensional) [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963803.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963804.png" /> such that the following local triviality condition is satisfied. Each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963805.png" /> has an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963806.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963807.png" />-isomorphism of fibre bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963808.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v0963809.png" /> is an isomorphism of vector spaces for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638010.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638011.png" /> is said to be the dimension of the vector bundle. The sections of a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638012.png" /> form a locally free module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638013.png" /> over the ring of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638014.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638015.png" />. A morphism of vector bundles is a morphism of fibre bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638016.png" /> for which the restriction to each fibre is linear. The set of vector bundles and their morphisms forms the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638017.png" />. The concept of a vector bundle arose as an extension of the [[Tangent bundle|tangent bundle]] and the [[Normal bundle|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.
+
<!--
 +
v0963801.png
 +
$#A+1 = 143 n = 1
 +
$#C+1 = 143 : ~/encyclopedia/old_files/data/V096/V.0906380 Vector bundle
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638019.png" /> is a vector bundle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638020.png" /> is a vector subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638021.png" /> is said to be a subbundle of the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638022.png" />. For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638023.png" /> be a vector space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638024.png" /> be the [[Grassmann manifold|Grassmann manifold]] of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638025.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638026.png" />; the subspace of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638027.png" />, consisting of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638029.png" />, will then be a subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638030.png" /> of the trivial vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638031.png" />. The union of all vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638033.png" /> is a subbundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638034.png" /> endowed with the quotient topology, is said to be a quotient bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638035.png" />. Let, furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638036.png" /> be a vector space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638037.png" /> be the Grassmann manifold of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638038.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638039.png" />; the quotient bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638040.png" /> of the trivial vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638041.png" /> is defined as the quotient space of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638042.png" /> by the subbundle consisting of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638044.png" />. The concepts of a subbundle and a quotient bundle are used in contraction and glueing operations used to construct vector bundles over quotient spaces.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638045.png" />-morphism of vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638046.png" /> is said to be of constant rank (pure) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638047.png" /> is locally constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638048.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638049.png" /> of locally constant rank: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638050.png" /> (the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638051.png" />), which is a subbundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638052.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638053.png" /> (the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638054.png" />), which is a subbundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638055.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638056.png" /> (the cokernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638057.png" />), which is a quotient bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638058.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638059.png" /> (the co-image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638060.png" />), which is a quotient bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638061.png" />. Any subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638062.png" /> is the image of some monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638063.png" />, while any quotient bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638064.png" /> is the cokernel of some epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638065.png" />. A sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638066.png" />-morphisms of vector bundles
+
A [[Fibre space|fibre space]]  $  \pi : X \rightarrow B $
 +
each fibre  $  \pi  ^ {-} 1 ( b) $
 +
of which is endowed with the structure of a (finite-dimensional) [[Vector space|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|tangent bundle]] and the [[Normal bundle|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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638067.png" /></td> </tr></table>
+
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|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
 
is said to be exact if the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638068.png" /></td> </tr></table>
+
$$
 +
{} \dots \rightarrow  ( \pi  ^  \prime  )  ^ {-} 1 ( b)  \rightarrow  \pi  ^ {-} 1 ( b)
 +
\rightarrow  ( \pi  ^ {\prime\prime} )  ^ {-} 1 ( b)  \rightarrow \dots
 +
$$
  
is exact for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638069.png" />. In particular, the sequence
+
is exact for all $  b \in B $.  
 +
In particular, the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638070.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  \pi _ {1}  \rightarrow ^ { i }  \pi  \rightarrow ^ { j }  \pi _ {2}  \rightarrow  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638071.png" /> is the zero vector bundle, is exact if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638072.png" /> is a monomorphism, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638073.png" /> is an epimorphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638074.png" />. The set of vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638076.png" />-morphisms of locally constant rank forms an exact subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638077.png" /> of the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638078.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638079.png" /> and mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638080.png" />, the [[Induced fibre bundle|induced fibre bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638081.png" /> is endowed with a vector bundle structure such that the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638082.png" /> is a vector bundle morphism. This structure is unique and has the following property: Every fibre mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638083.png" /> is an isomorphism of vector spaces. For instance, a vector bundle of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638084.png" /> over a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638085.png" /> is isomorphic to one of the vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638087.png" /> induced by certain mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638089.png" />, respectively; moreover, homotopic mappings induce isomorphic vector bundles and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638090.png" />, the converse is true: To isomorphic vector bundles there correspond homotopic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638092.png" />. This is one of the fundamental theorems in the homotopic classification of vector bundles, expressing the universal character of the vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638094.png" /> with respect to the classifying mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638096.png" />.
+
For any vector bundle $  \pi : X \rightarrow B $
 +
and mapping $  u: B _ {1} \rightarrow B $,  
 +
the [[Induced fibre bundle|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|functor]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638097.png" /> on the category of vector spaces uniquely determines a continuous functor on the category of vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638098.png" />; in this way it is possible to construct bundles associated with a given vector bundle: tensor bundles, vector bundles of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v09638099.png" /> and, in particular, the dual vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380100.png" />, exterior powers of vector bundles, etc., whose sections are vector bundles with supplementary structures. These are extensively employed in practical applications.
+
Any continuous operation ([[Functor|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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380101.png" /> and tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380102.png" /> have been defined for two vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380104.png" />. With respect to these operations the set of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380105.png" /> of isomorphic vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380106.png" /> forms a semi-ring which plays an important part in the construction of a [[K-functor|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380107.png" />-functor]]; thus, if for vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380109.png" /> there exist trivial vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380111.png" /> such that the vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380113.png" /> are isomorphic (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380115.png" /> are stably equivalent), then their images in the "completion" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380116.png" /> of the semi-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380117.png" /> are identical; moreover, the fact that the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380118.png" /> 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.
+
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| $  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380119.png" /> over a paracompact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380120.png" /> there exists a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380121.png" /> of the vector bundle
+
For any vector bundle $  \pi : X \rightarrow B $
 +
over a paracompact space $  B $
 +
there exists a section $  \beta $
 +
of the vector bundle
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380122.png" /></td> </tr></table>
+
$$
 +
\pi  ^ {*} \oplus \pi  ^ {*}  =   \mathop{\rm Hom} ( \pi \oplus \pi , P ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380123.png" /> is a trivial one-dimensional vector bundle, which on each fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380124.png" /> is a positive-definite form, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380125.png" /> is metrizable; this makes it possible to establish, in particular, the splittability of an arbitrary exact sequence of vector bundles
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380126.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  \xi  \rightarrow ^ { u }  \pi  \rightarrow ^ { v }  \zeta  \rightarrow  0
 +
$$
  
in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380127.png" /> is metrizable, that is, the existence of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380128.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380130.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380131.png" /> is the imbedding into the first term and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380132.png" /> is the projection onto the second term.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380133.png" /> of the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380134.png" />, one identifies the points lying on the same line passing through zero, one obtains a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380135.png" />, which is associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380136.png" /> and is said to be its projectivization; a fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380137.png" /> is the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380138.png" /> which is associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380139.png" />. This bundle is used to study Thom spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380140.png" /> (cf. [[Thom space|Thom space]]), used in the homotopic interpretation of classes of bordant manifolds, characteristic classes of vector bundles describing the homological properties of manifolds, etc.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380141.png" />, 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.
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) {{MR|0242081}} {{ZBL|0653.53001}} {{ZBL|0284.53018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380142.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III {{MR|1931083}} {{MR|1532744}} {{MR|0155257}} {{ZBL|1008.57001}} {{ZBL|0103.15101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) {{MR|1335917}} {{MR|0202713}} {{ZBL|0376.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) {{MR|0242081}} {{ZBL|0653.53001}} {{ZBL|0284.53018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380142.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III {{MR|1931083}} {{MR|1532744}} {{MR|0155257}} {{ZBL|1008.57001}} {{ZBL|0103.15101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) {{MR|1335917}} {{MR|0202713}} {{ZBL|0376.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For more on the universality and classifying properties of the bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380143.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096380/v096380144.png" /> cf. [[Classifying space|Classifying space]] or [[#References|[a1]]].
+
For more on the universality and classifying properties of the bundles $  \gamma  ^ {k} $
 +
and $  \gamma _ {k} $
 +
cf. [[Classifying space|Classifying space]] or [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) {{MR|0440554}} {{ZBL|0298.57008}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) {{MR|0440554}} {{ZBL|0298.57008}} </TD></TR></table>

Revision as of 08:28, 6 June 2020


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

[1] C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) MR0242081 Zbl 0653.53001 Zbl 0284.53018
[2] M.F. Atiyah, "-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
How to Cite This Entry:
Vector bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_bundle&oldid=49135
This article was adapted from an original article by A.F. Shchekut'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article