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A [[Characteristic number|characteristic number]] of a closed manifold taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877801.png" />, the integers modulo 2. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877802.png" /> be an arbitrary stable [[Characteristic class|characteristic class]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877803.png" /> be a closed manifold. The residue modulo 2 defined by
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<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/s/s087/s087780/s0877804.png" /></td> </tr></table>
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is called the Stiefel number (or Stiefel–Whitney number) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877805.png" /> corresponding to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877806.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877807.png" /> is the tangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877808.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s0877809.png" /> is the [[Fundamental class|fundamental class]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778010.png" />-dimensional manifolds, the Stiefel number depends only on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778011.png" />-th homogeneous component of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778012.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778013.png" /> is isomorphic to a vector space over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778014.png" /> whose basis is in one-to-one correspondence with the set of all partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778015.png" /> of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778016.png" />, i.e. tuples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778017.png" /> of non-negative integers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778018.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778019.png" /> would be a natural choice for a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778020.png" />. Thus, to characterize a manifold by its Stiefel numbers it is sufficient to consider the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778022.png" /> is a partition of the dimension of the manifold.
+
A [[Characteristic number|characteristic number]] of a closed manifold taking values in  $  \mathbf Z _ {2} $,
 +
the integers modulo 2. Let  $  x \in H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
be an arbitrary stable [[Characteristic class|characteristic class]], and let  $  M $
 +
be a closed manifold. The residue modulo 2 defined by
  
Bordant manifolds have the same Stiefel numbers, since each characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778023.png" /> determines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778025.png" /> is the group of classes of bordant non-oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778026.png" />-dimensional manifolds. If for two closed manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778028.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778029.png" /> holds for all partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778031.png" />, then the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778033.png" /> are bordant (Thom's theorem).
+
$$
 +
x[ M]  = \langle  x( \tau M), [ M]\rangle
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778034.png" /> be the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778035.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778036.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778037.png" /> be the basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778038.png" /> dual to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778041.png" />, here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778042.png" /> are partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778043.png" />; and let a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778044.png" /> be defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778045.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778046.png" /> is monomorphic, and for a complete description of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778047.png" /> by the Stiefel numbers it is necessary to find its image. This problem is analogous to the Milnor–Hirzebruch problem for Chern classes (cf. [[Chern class|Chern class]]). For a closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778048.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778049.png" /> be the so-called Wu class, uniquely defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778050.png" />, which should hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778051.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778053.png" /> is the tangent bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778054.png" /> (Wu's theorem).
+
is called the Stiefel number (or Stiefel–Whitney number) of  $  M $
 +
corresponding to the class  $  x $.  
 +
Here  $  \tau M $
 +
is the tangent bundle of  $  M $,
 +
and  $  [ M] \in H _  \star  ( M;  \mathbf Z _ {2} ) $
 +
is the [[Fundamental class|fundamental class]]. For  $  n $-
 +
dimensional manifolds, the Stiefel number depends only on the  $  n $-
 +
th homogeneous component of the class  $  x $.  
 +
The group  $  H  ^ {n} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
is isomorphic to a vector space over the field  $  \mathbf Z _ {2} $
 +
whose basis is in one-to-one correspondence with the set of all partitions  $  w = \{ i _ {1} \dots i _ {k} \} $
 +
of the number  $  n $,
 +
i.e. tuples  $  \{ i _ {1} \dots i _ {k} \} $
 +
of non-negative integers such that  $  i _ {1} + \dots + i _ {k} = n $.
 +
The classes  $  w _  \omega  = w _ {i _ {1}  } \dots w _ {i _ {k}  } $
 +
would be a natural choice for a basis of  $  H  ^ {n} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $.  
 +
Thus, to characterize a manifold by its Stiefel numbers it is sufficient to consider the classes  $  w _  \omega  $,
 +
where  $  \omega $
 +
is a partition of the dimension of the manifold.
 +
 
 +
Bordant manifolds have the same Stiefel numbers, since each characteristic class  $  x $
 +
determines a homomorphism  $  x[ {} ] :  \mathfrak N  ^ {n} \rightarrow \mathbf Z _ {2} $,
 +
where  $  \mathfrak N  ^ {n} $
 +
is the group of classes of bordant non-oriented  $  n $-
 +
dimensional manifolds. If for two closed manifolds  $  M $,
 +
$  N $
 +
the equality  $  w _  \omega  [ M] = w _  \omega  [ N] $
 +
holds for all partitions  $  \omega $
 +
of  $  n = \mathop{\rm dim}  M = \mathop{\rm dim}  N $,
 +
then the manifolds  $  M $
 +
and  $  N $
 +
are bordant (Thom's theorem).
 +
 
 +
Let  $  A $
 +
be the vector space  $  \mathop{\rm Hom} ( H  ^ {n} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ), \mathbf Z _ {2} ) $
 +
over the field  $  \mathbf Z _ {2} $.  
 +
Let $  \{ e _  \omega  \} $
 +
be the basis in $  A $
 +
dual to the basis $  \{ w _  \omega  \} $
 +
in $  H  ^ {n} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $,  
 +
$  e _  \omega  ( w _ {\omega  ^  \prime  } ) = \delta _  \omega  ^ {\omega  ^  \prime  } $,  
 +
here $  \omega , \omega  ^  \prime  $
 +
are partitions of $  n $;  
 +
and let a mapping $  \phi : \mathfrak N \rightarrow A $
 +
be defined by $  \phi ([ M]) = \sum _  \omega  w _  \omega  [ M] e _  \omega  $.  
 +
The mapping $  \phi $
 +
is monomorphic, and for a complete description of the group $  \mathfrak N  ^ {n} $
 +
by the Stiefel numbers it is necessary to find its image. This problem is analogous to the Milnor–Hirzebruch problem for Chern classes (cf. [[Chern class|Chern class]]). For a closed manifold $  M $,  
 +
let $  v \in H  ^  \star  ( M;  \mathbf Z _ {2} ) $
 +
be the so-called Wu class, uniquely defined by $  \langle  \alpha \cup v, [ M]\rangle = \langle  Sq  \alpha [ M]\rangle $,  
 +
which should hold for all $  \alpha \in H  ^  \star  ( M;  \mathbf Z _ {2} ) $.  
 +
Then $  w( \tau M) = Sqv $,  
 +
where $  \tau M $
 +
is the tangent bundle to $  M $(
 +
Wu's theorem).
  
 
This theorem implies that the Wu class can be defined as a characteristic class: Let
 
This theorem implies that the Wu class can be defined as a characteristic class: Let
  
<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/s/s087/s087780/s08778055.png" /></td> </tr></table>
+
$$
 +
= Sq  ^ {-} 1 w  \in  H  ^  \star  (  \mathop{\rm BO} ; \mathbf Z _ {2} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778056.png" /> is the complete Stiefel–Whitney class and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778057.png" /> is the cohomology operation inverse to the complete [[Steenrod square|Steenrod square]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778058.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778059.png" /> be an arbitrary characteristic class. Then for any closed manifold the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778061.png" /> coincide. Thus, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778063.png" /> can be in the image of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778064.png" /> only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778065.png" /> holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778066.png" />. For a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778067.png" /> there exists a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778069.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778070.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778071.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778072.png" /> (Dold's theorem).
+
where $  w \in H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
is the complete Stiefel–Whitney class and $  Sq  ^ {-} 1 = 1 + Sq  ^ {1} + Sq  ^ {2} + Sq  ^ {2} Sq  ^ {1} + \dots $
 +
is the cohomology operation inverse to the complete [[Steenrod square|Steenrod square]] $  Sq $.  
 +
Let $  \alpha \in H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
be an arbitrary characteristic class. Then for any closed manifold the numbers $  ( \alpha \cup v)[ M] $
 +
and $  ( Sq \alpha )[ M] $
 +
coincide. Thus, an element $  a \in A $,  
 +
$  a = \sum a _  \omega  e _  \omega  $
 +
can be in the image of the mapping $  \phi $
 +
only if $  a( \alpha \cup v) = a( Sq \alpha ) $
 +
holds for all $  \alpha \in H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $.  
 +
For a homomorphism $  a: H  ^ {n} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) \rightarrow \mathbf Z _ {2} $
 +
there exists a manifold $  M  ^ {n} $
 +
such that $  x[ M  ^ {n} ] = a( x) $
 +
for all $  x \in H  ^ {n} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
if and only if $  a( \alpha \cup v) = a( Sq \alpha ) $
 +
for all $  \alpha \in H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $(
 +
Dold's theorem).
  
 
For references, see [[Stiefel–Whitney class|Stiefel–Whitney class]].
 
For references, see [[Stiefel–Whitney class|Stiefel–Whitney class]].
 
 
  
 
====Comments====
 
====Comments====
As is customary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778073.png" /> denotes the direct product of the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778074.png" /> of the [[Classifying space|classifying space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778075.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087780/s08778076.png" /> is the direct sum.
+
As is customary $  H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
denotes the direct product of the cohomology groups $  H  ^ {n} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
of the [[Classifying space|classifying space]] $  \mathop{\rm BO} $,  
 +
while $  H  ^  \star  (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
 +
is the direct sum.

Latest revision as of 08:23, 6 June 2020


A characteristic number of a closed manifold taking values in $ \mathbf Z _ {2} $, the integers modulo 2. Let $ x \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ be an arbitrary stable characteristic class, and let $ M $ be a closed manifold. The residue modulo 2 defined by

$$ x[ M] = \langle x( \tau M), [ M]\rangle $$

is called the Stiefel number (or Stiefel–Whitney number) of $ M $ corresponding to the class $ x $. Here $ \tau M $ is the tangent bundle of $ M $, and $ [ M] \in H _ \star ( M; \mathbf Z _ {2} ) $ is the fundamental class. For $ n $- dimensional manifolds, the Stiefel number depends only on the $ n $- th homogeneous component of the class $ x $. The group $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ is isomorphic to a vector space over the field $ \mathbf Z _ {2} $ whose basis is in one-to-one correspondence with the set of all partitions $ w = \{ i _ {1} \dots i _ {k} \} $ of the number $ n $, i.e. tuples $ \{ i _ {1} \dots i _ {k} \} $ of non-negative integers such that $ i _ {1} + \dots + i _ {k} = n $. The classes $ w _ \omega = w _ {i _ {1} } \dots w _ {i _ {k} } $ would be a natural choice for a basis of $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $. Thus, to characterize a manifold by its Stiefel numbers it is sufficient to consider the classes $ w _ \omega $, where $ \omega $ is a partition of the dimension of the manifold.

Bordant manifolds have the same Stiefel numbers, since each characteristic class $ x $ determines a homomorphism $ x[ {} ] : \mathfrak N ^ {n} \rightarrow \mathbf Z _ {2} $, where $ \mathfrak N ^ {n} $ is the group of classes of bordant non-oriented $ n $- dimensional manifolds. If for two closed manifolds $ M $, $ N $ the equality $ w _ \omega [ M] = w _ \omega [ N] $ holds for all partitions $ \omega $ of $ n = \mathop{\rm dim} M = \mathop{\rm dim} N $, then the manifolds $ M $ and $ N $ are bordant (Thom's theorem).

Let $ A $ be the vector space $ \mathop{\rm Hom} ( H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ), \mathbf Z _ {2} ) $ over the field $ \mathbf Z _ {2} $. Let $ \{ e _ \omega \} $ be the basis in $ A $ dual to the basis $ \{ w _ \omega \} $ in $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $, $ e _ \omega ( w _ {\omega ^ \prime } ) = \delta _ \omega ^ {\omega ^ \prime } $, here $ \omega , \omega ^ \prime $ are partitions of $ n $; and let a mapping $ \phi : \mathfrak N \rightarrow A $ be defined by $ \phi ([ M]) = \sum _ \omega w _ \omega [ M] e _ \omega $. The mapping $ \phi $ is monomorphic, and for a complete description of the group $ \mathfrak N ^ {n} $ by the Stiefel numbers it is necessary to find its image. This problem is analogous to the Milnor–Hirzebruch problem for Chern classes (cf. Chern class). For a closed manifold $ M $, let $ v \in H ^ \star ( M; \mathbf Z _ {2} ) $ be the so-called Wu class, uniquely defined by $ \langle \alpha \cup v, [ M]\rangle = \langle Sq \alpha [ M]\rangle $, which should hold for all $ \alpha \in H ^ \star ( M; \mathbf Z _ {2} ) $. Then $ w( \tau M) = Sqv $, where $ \tau M $ is the tangent bundle to $ M $( Wu's theorem).

This theorem implies that the Wu class can be defined as a characteristic class: Let

$$ v = Sq ^ {-} 1 w \in H ^ \star ( \mathop{\rm BO} ; \mathbf Z _ {2} ), $$

where $ w \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ is the complete Stiefel–Whitney class and $ Sq ^ {-} 1 = 1 + Sq ^ {1} + Sq ^ {2} + Sq ^ {2} Sq ^ {1} + \dots $ is the cohomology operation inverse to the complete Steenrod square $ Sq $. Let $ \alpha \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ be an arbitrary characteristic class. Then for any closed manifold the numbers $ ( \alpha \cup v)[ M] $ and $ ( Sq \alpha )[ M] $ coincide. Thus, an element $ a \in A $, $ a = \sum a _ \omega e _ \omega $ can be in the image of the mapping $ \phi $ only if $ a( \alpha \cup v) = a( Sq \alpha ) $ holds for all $ \alpha \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $. For a homomorphism $ a: H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) \rightarrow \mathbf Z _ {2} $ there exists a manifold $ M ^ {n} $ such that $ x[ M ^ {n} ] = a( x) $ for all $ x \in H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ if and only if $ a( \alpha \cup v) = a( Sq \alpha ) $ for all $ \alpha \in H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $( Dold's theorem).

For references, see Stiefel–Whitney class.

Comments

As is customary $ H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ denotes the direct product of the cohomology groups $ H ^ {n} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ of the classifying space $ \mathop{\rm BO} $, while $ H ^ \star ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ is the direct sum.

How to Cite This Entry:
Stiefel number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stiefel_number&oldid=48839
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article