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A [[Characteristic class|characteristic class]] with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877901.png" />, defined for real vector bundles. The Stiefel–Whitney classes are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877903.png" />, and for a real vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877904.png" /> over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877905.png" />, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877906.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877907.png" />. These classes were introduced by E. Stiefel [[#References|[1]]] and H. Whitney [[#References|[2]]] and have the following properties. 1) For two real vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s0877908.png" /> over a common base,
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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/s087790/s0877909.png" /></td> </tr></table>
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in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779011.png" /> is the complete Stiefel–Whitney class. 2) For the one-dimensional universal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779013.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779014.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779015.png" /> is the non-zero element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779016.png" />. These two properties together with naturality for induced bundles define the Stiefel–Whitney classes uniquely. The Stiefel–Whitney classes are stable, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779018.png" /> is the trivial bundle, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779020.png" />. For an oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779021.png" />-dimensional vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779022.png" /> over a base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779024.png" /> coincides with the reduction modulo 2 of the [[Euler class|Euler class]].
+
A [[Characteristic class|characteristic class]] with values in $  H  ^  \star  (  ;  \mathbf Z _ {2} ) $,  
 +
defined for real vector bundles. The Stiefel–Whitney classes are denoted by  $  w _ {i} $,  
 +
i > 0 $,  
 +
and for a real vector bundle $  \xi $
 +
over a topological space  $  B $,  
 +
the class  $  w _ {i} ( \xi ) $
 +
lies in  $  H  ^ {i} ( B;  \mathbf Z _ {2} ) $.  
 +
These classes were introduced by E. Stiefel [[#References|[1]]] and H. Whitney [[#References|[2]]] and have the following properties. 1) For two real vector bundles  $  \xi , \eta $
 +
over a common base,
  
For a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779025.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779027.png" /> be the [[Thom space|Thom space]] of this bundle. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779028.png" /> be the [[Thom isomorphism|Thom isomorphism]]. Then the complete Stiefel–Whitney class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779029.png" /> coincides with
+
$$
 +
w _ {k} ( \xi \oplus \eta )  = \sum _ { i } w _ {i} ( \xi ) w _ {k- i} ( \eta ),\ \
 +
w _ {0= 1;
 +
$$
  
<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/s087790/s08779030.png" /></td> </tr></table>
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in other words,  $  w( \xi \oplus \eta ) = w( \xi ) w( \eta ) $,
 +
where  $  w = 1+ w _ {1} + w _ {2} + \dots $
 +
is the complete Stiefel–Whitney class. 2) For the one-dimensional universal bundle  $  \zeta _ {1} $
 +
over  $  \mathbf R P  ^  \infty  $
 +
the equality  $  w( \zeta _ {1} ) = 1 + y $
 +
holds, where  $  y $
 +
is the non-zero element of the group  $  H  ^ {1} ( \mathbf R P  ^  \infty  ; \mathbf Z _ {2} ) = \mathbf Z _ {2} $.
 +
These two properties together with naturality for induced bundles define the Stiefel–Whitney classes uniquely. The Stiefel–Whitney classes are stable, i.e.  $  w( \xi \oplus \theta ) = w ( \xi ) $,
 +
where  $  \theta $
 +
is the trivial bundle, and  $  w _ {i} ( \xi ) = 0 $
 +
for  $  i >  \mathop{\rm dim}  \xi $.  
 +
For an oriented  $  n $-
 +
dimensional vector bundle  $  \xi $
 +
over a base  $  B $,
 +
$  w _ {n} ( \xi ) \in H  ^ {n} ( B;  \mathbf Z _ {2} ) $
 +
coincides with the reduction modulo 2 of the [[Euler class|Euler class]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779031.png" /> is the complete [[Steenrod square|Steenrod square]]. This property of Stiefel–Whitney classes can be used as their definition. Stiefel–Whitney classes are homotopy invariants in the sense that they coincide for fibre-wise homotopically-equivalent bundles over a common base.
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For a vector bundle  $  \xi $
 +
over  $  B $,
 +
let  $  B  ^  \xi  $
 +
be the [[Thom space|Thom space]] of this bundle. Further, let  $  \Phi :  H  ^  \star  ( B;  \mathbf Z _ {2} ) \rightarrow \widetilde{H}  {} ^ {\star+ n } ( B  ^  \xi  ;  \mathbf Z _ {2} ) $
 +
be the [[Thom isomorphism|Thom isomorphism]]. Then the complete Stiefel–Whitney class  $  w( \xi ) $
 +
coincides with
  
Any characteristic class with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779032.png" />, defined for real vector bundles, can be expressed by Stiefel–Whitney classes: The rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779034.png" /> are rings of formal power series in the Stiefel–Whitney classes:
+
$$
 +
\Phi  ^ {-} 1 Sq \Phi ( 1)  \in H  ^  \star  ( B;  \mathbf Z _ {2} ),
 +
$$
  
<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/s087790/s08779035.png" /></td> </tr></table>
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where  $  Sq = 1 + Sq  ^ {1} + Sq  ^ {2} + \dots $
 +
is the complete [[Steenrod square|Steenrod square]]. This property of Stiefel–Whitney classes can be used as their definition. Stiefel–Whitney classes are homotopy invariants in the sense that they coincide for fibre-wise homotopically-equivalent bundles over a common base.
  
<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/s087790/s08779036.png" /></td> </tr></table>
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Any characteristic class with values in  $  H  ^  \star  (  ;  \mathbf Z _ {2} ) $,
 +
defined for real vector bundles, can be expressed by Stiefel–Whitney classes: The rings  $  H  ^ {\star\star} (  \mathop{\rm BO} _ {n} ; \mathbf Z _ {2} ) $
 +
and  $  H  ^ {\star\star} (  \mathop{\rm BO} ; \mathbf Z _ {2} ) $
 +
are rings of formal power series in the Stiefel–Whitney classes:
 +
 
 +
$$
 +
H  ^ {\star\star} (  \mathop{\rm BO} _ {n} ; \mathbf Z _ {2} )  = \mathbf Z _ {2} [[ w _ {1} \dots w _ {n} ]],
 +
$$
 +
 
 +
$$
 +
H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} )  = \mathbf Z _ {2} [[ w _ {1} ,\dots ]].
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Stiefel,  "Richtungsfelden und Fernparallelismus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779037.png" />-dimensionalen Mannigfaltigkeiten"  ''Comm. Math. Helv.'' , '''8''' :  4  (1935–1936)  pp. 305–353</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Whitney,  "Topological properties of differentiable manifolds"  ''Bull. Amer. Math. Soc.'' , '''43'''  (1937)  pp. 785–805</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W. Milnor,  J.D. Stasheff,  "Characteristic classes" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Steenrod,  "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Stiefel,  "Richtungsfelden und Fernparallelismus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779037.png" />-dimensionalen Mannigfaltigkeiten"  ''Comm. Math. Helv.'' , '''8''' :  4  (1935–1936)  pp. 305–353</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Whitney,  "Topological properties of differentiable manifolds"  ''Bull. Amer. Math. Soc.'' , '''43'''  (1937)  pp. 785–805</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W. Milnor,  J.D. Stasheff,  "Characteristic classes" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Steenrod,  "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779038.png" /> denotes the product of the Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779039.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779040.png" /> is the direct sum; the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779041.png" /> means that there is a graded homomorphism of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779042.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779043.png" />. For the classifying spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779045.png" /> see [[Classifying space|Classifying space]].
+
The notation $  H  ^ {\star\star} ( X;  G) $
 +
denotes the product of the Abelian groups $  H  ^ {n} ( X;  G) $,  
 +
while $  H  ^  \star  ( X;  G) $
 +
is the direct sum; the notation $  H  ^  \star  ( B;  \mathbf Z _ {2} ) \rightarrow \widetilde{H}  {}  ^ {\star+} n ( B  ^  \xi  ;  \mathbf Z _ {2} ) $
 +
means that there is a graded homomorphism of degree $  n $:  
 +
$  H  ^ {m} ( B;  \mathbf Z _ {2} ) \rightarrow \widetilde{H}  {}  ^ {m+} n ( B  ^  \xi  ;  \mathbf Z _ {2} ) $.  
 +
For the classifying spaces $  \mathop{\rm BO} _ {n} $
 +
and $  \mathop{\rm BO} $
 +
see [[Classifying space|Classifying space]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>

Latest revision as of 17:18, 6 January 2024


A characteristic class with values in $ H ^ \star ( ; \mathbf Z _ {2} ) $, defined for real vector bundles. The Stiefel–Whitney classes are denoted by $ w _ {i} $, $ i > 0 $, and for a real vector bundle $ \xi $ over a topological space $ B $, the class $ w _ {i} ( \xi ) $ lies in $ H ^ {i} ( B; \mathbf Z _ {2} ) $. These classes were introduced by E. Stiefel [1] and H. Whitney [2] and have the following properties. 1) For two real vector bundles $ \xi , \eta $ over a common base,

$$ w _ {k} ( \xi \oplus \eta ) = \sum _ { i } w _ {i} ( \xi ) w _ {k- i} ( \eta ),\ \ w _ {0} = 1; $$

in other words, $ w( \xi \oplus \eta ) = w( \xi ) w( \eta ) $, where $ w = 1+ w _ {1} + w _ {2} + \dots $ is the complete Stiefel–Whitney class. 2) For the one-dimensional universal bundle $ \zeta _ {1} $ over $ \mathbf R P ^ \infty $ the equality $ w( \zeta _ {1} ) = 1 + y $ holds, where $ y $ is the non-zero element of the group $ H ^ {1} ( \mathbf R P ^ \infty ; \mathbf Z _ {2} ) = \mathbf Z _ {2} $. These two properties together with naturality for induced bundles define the Stiefel–Whitney classes uniquely. The Stiefel–Whitney classes are stable, i.e. $ w( \xi \oplus \theta ) = w ( \xi ) $, where $ \theta $ is the trivial bundle, and $ w _ {i} ( \xi ) = 0 $ for $ i > \mathop{\rm dim} \xi $. For an oriented $ n $- dimensional vector bundle $ \xi $ over a base $ B $, $ w _ {n} ( \xi ) \in H ^ {n} ( B; \mathbf Z _ {2} ) $ coincides with the reduction modulo 2 of the Euler class.

For a vector bundle $ \xi $ over $ B $, let $ B ^ \xi $ be the Thom space of this bundle. Further, let $ \Phi : H ^ \star ( B; \mathbf Z _ {2} ) \rightarrow \widetilde{H} {} ^ {\star+ n } ( B ^ \xi ; \mathbf Z _ {2} ) $ be the Thom isomorphism. Then the complete Stiefel–Whitney class $ w( \xi ) $ coincides with

$$ \Phi ^ {-} 1 Sq \Phi ( 1) \in H ^ \star ( B; \mathbf Z _ {2} ), $$

where $ Sq = 1 + Sq ^ {1} + Sq ^ {2} + \dots $ is the complete Steenrod square. This property of Stiefel–Whitney classes can be used as their definition. Stiefel–Whitney classes are homotopy invariants in the sense that they coincide for fibre-wise homotopically-equivalent bundles over a common base.

Any characteristic class with values in $ H ^ \star ( ; \mathbf Z _ {2} ) $, defined for real vector bundles, can be expressed by Stiefel–Whitney classes: The rings $ H ^ {\star\star} ( \mathop{\rm BO} _ {n} ; \mathbf Z _ {2} ) $ and $ H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) $ are rings of formal power series in the Stiefel–Whitney classes:

$$ H ^ {\star\star} ( \mathop{\rm BO} _ {n} ; \mathbf Z _ {2} ) = \mathbf Z _ {2} [[ w _ {1} \dots w _ {n} ]], $$

$$ H ^ {\star\star} ( \mathop{\rm BO} ; \mathbf Z _ {2} ) = \mathbf Z _ {2} [[ w _ {1} ,\dots ]]. $$

References

[1] E. Stiefel, "Richtungsfelden und Fernparallelismus in -dimensionalen Mannigfaltigkeiten" Comm. Math. Helv. , 8 : 4 (1935–1936) pp. 305–353
[2] H. Whitney, "Topological properties of differentiable manifolds" Bull. Amer. Math. Soc. , 43 (1937) pp. 785–805
[3] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974)
[4] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)
[5] N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)

Comments

The notation $ H ^ {\star\star} ( X; G) $ denotes the product of the Abelian groups $ H ^ {n} ( X; G) $, while $ H ^ \star ( X; G) $ is the direct sum; the notation $ H ^ \star ( B; \mathbf Z _ {2} ) \rightarrow \widetilde{H} {} ^ {\star+} n ( B ^ \xi ; \mathbf Z _ {2} ) $ means that there is a graded homomorphism of degree $ n $: $ H ^ {m} ( B; \mathbf Z _ {2} ) \rightarrow \widetilde{H} {} ^ {m+} n ( B ^ \xi ; \mathbf Z _ {2} ) $. For the classifying spaces $ \mathop{\rm BO} _ {n} $ and $ \mathop{\rm BO} $ see Classifying space.

References

[a1] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
How to Cite This Entry:
Stiefel-Whitney class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stiefel-Whitney_class&oldid=23045
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article