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A [[Characteristic class|characteristic class]] defined for real vector bundles (cf. [[Vector bundle|Vector bundle]]). Pontryagin classes were introduced by L.S. Pontryagin [[#References|[1]]] in 1947. For a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737501.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737502.png" /> the Pontryagin classes are denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737503.png" /> and are defined to be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737504.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737505.png" /> is the complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737507.png" /> are the Chern classes (cf. [[Chern class|Chern class]]). The total Pontryagin class is the non-homogeneous characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737508.png" />. In other words, the Pontryagin classes are defined as homology classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p0737509.png" /> determined by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375011.png" /> is the mapping corresponding to the complexification of the universal vector bundle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375012.png" /> are the Chern classes.
+
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$#C+1 = 124 : ~/encyclopedia/old_files/data/P073/P.0703750 Pontryagin class
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375013.png" /> be the real bundle of the universal vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375015.png" />. The total Pontryagin class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375016.png" /> of the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375017.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375019.png" /> are the Wu generators (see [[Characteristic class|Characteristic class]]).
+
{{TEX|auto}}
 +
{{TEX|done}}
  
A partial description of the cohomology ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375020.png" /> can be obtained in terms of Wu generators in the following way. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375021.png" /> corresponding to the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375023.png" /> is the one-dimensional trivial vector bundle, induces a ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375024.png" />, under which the subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375025.png" /> generated by the Pontryagin classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375026.png" /> is mapped monomorphically onto the subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375027.png" /> consisting of all even symmetric polynomials in the Wu generators. Evenness is understood in the sense that the degree of every variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375028.png" /> in the polynomial should be even. Thus, an expression in Wu generators is obtained for any element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375029.png" />. This is important for practical calculations with Pontryagin classes. The characteristic class determined by an even symmetric polynomial in the Wu generators can be expressed in Pontryagin classes as follows. First, the polynomial is written in elementary symmetric functions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375030.png" /> and then the elementary symmetric functions are replaced by Pontryagin classes.
+
A [[Characteristic class|characteristic class]] defined for real vector bundles (cf. [[Vector bundle|Vector bundle]]). Pontryagin classes were introduced by L.S. Pontryagin [[#References|[1]]] in 1947. For a vector bundle  $  \xi $
 +
with base  $  B $
 +
the Pontryagin classes are denoted by the symbol  $  p _ {i} ( \xi ) \in H  ^ {4i} ( B) $
 +
and are defined to be equal to $  p _ {i} ( \xi ) = ( - 1 )  ^ {i} c _ {2i} ( \xi \otimes \mathbf C ) $,  
 +
where $  \xi \otimes \mathbf C $
 +
is the complexification of $  \xi $
 +
and  $  c _ {k} $
 +
are the Chern classes (cf. [[Chern class|Chern class]]). The total Pontryagin class is the non-homogeneous characteristic class  $  p = 1 + p _ {1} + p _ {2} + \dots $.  
 +
In other words, the Pontryagin classes are defined as homology classes  $  p _ {i} \in H  ^ {4i} (  \mathop{\rm BO} _ {n} ) $
 +
determined by the equality  $  p _ {i} = f ^ { * } ( ( - 1 )  ^ {i} c _ {2i} ) $,  
 +
where  $  f :  \mathop{\rm BO} _ {n} \rightarrow  \mathop{\rm BU} _ {n} $
 +
is the mapping corresponding to the complexification of the universal vector bundle and $  c _ {n} \in H  ^ {2k} (  \mathop{\rm BU} _ {n} ) $
 +
are the Chern classes.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375031.png" /> are two real vector bundles over a common base, then the cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375032.png" /> is of order at most two; this is due to the fact that for the first Chern class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375033.png" />.
+
Let  $  ( \kappa _ {n} ) _ {\mathbf R }  $
 +
be the real bundle of the universal vector bundle  $  \kappa _ {n} $
 +
over $  \mathop{\rm BU} _ {n} $.
 +
The total Pontryagin class p ( ( \kappa _ {n} ) _ {\mathbf R }  ) $
 +
of the vector bundle  $  ( \kappa _ {n} ) _ {\mathbf R }  $
 +
coincides with  $  \prod _{i=1}  ^ {n} ( 1 + x _ {i}  ^ {2} ) \in H  ^ {*} (  \mathop{\rm BU} _ {n} ) $,
 +
where  $  x _ {1} \dots x _ {n} $
 +
are the Wu generators (see [[Characteristic class|Characteristic class]]).
  
Let a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375034.png" />, containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375035.png" />, be considered as the ring of coefficients, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375036.png" /> be a Pontryagin class with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375037.png" />. In this case the following equality is valid:
+
A partial description of the cohomology ring $  H  ^ {*} (  \mathop{\rm BO} _ {n} ) $
 +
can be obtained in terms of Wu generators in the following way. The mapping  $  g :   \mathop{\rm BU} _ {[} n/2] \rightarrow  \mathop{\rm BO} _ {n} $
 +
corresponding to the vector bundle  $  ( \kappa _ {[} n/2] ) _ {\mathbf R} \otimes \theta _ {1} $,
 +
where  $  \theta _ {1} $
 +
is the one-dimensional trivial vector bundle, induces a ring homomorphism  $  g  ^ {*} : H  ^ {*} (  \mathop{\rm BO} _ {n} ) \rightarrow H  ^ {*} (  \mathop{\rm BU} _ {[} n/2] ) $,
 +
under which the subring of  $  H  ^ {*} (  \mathop{\rm BO} _ {n} ) $
 +
generated by the Pontryagin classes  $  p _ {1} \dots p  ^ {[} n/2] $
 +
is mapped monomorphically onto the subring of  $  H  ^ {*} (  \mathop{\rm BU} _ {n} ) $
 +
consisting of all even symmetric polynomials in the Wu generators. Evenness is understood in the sense that the degree of every variable  $  x _ {i} $
 +
in the polynomial should be even. Thus, an expression in Wu generators is obtained for any element of the ring $  \mathbf Z ( p _ {1} \dots p _ {[} n/2] ) \subset  H  ^ {*} (  \mathop{\rm BO} _ {n} ) $.  
 +
This is important for practical calculations with Pontryagin classes. The characteristic class determined by an even symmetric polynomial in the Wu generators can be expressed in Pontryagin classes as follows. First, the polynomial is written in elementary symmetric functions of the variables  $  x _ {1}  ^ {2} \dots x _ {n}  ^ {2} $
 +
and then the elementary symmetric functions are replaced by Pontryagin classes.
  
<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/p/p073/p073750/p07375038.png" /></td> </tr></table>
+
If  $  \xi , \eta $
 +
are two real vector bundles over a common base, then the cohomology class $  p ( \xi \oplus \eta ) - p ( \xi ) p ( \eta ) $
 +
is of order at most two; this is due to the fact that for the first Chern class  $  c _ {1} ( \lambda ) = - c _ {1} ( \overline \lambda \; ) $.
 +
 
 +
Let a ring  $  \Lambda $,
 +
containing  $  1/2 $,
 +
be considered as the ring of coefficients, and let  $  p _ {i} $
 +
be a Pontryagin class with values in  $  H  ^ {*} ( \cdot ;  \Lambda ) $.  
 +
In this case the following equality is valid:
 +
 
 +
$$
 +
p ( \xi \oplus \eta )  =  p ( \xi ) p ( \eta ),
 +
$$
  
 
or
 
or
  
<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/p/p073/p073750/p07375039.png" /></td> </tr></table>
+
$$
 +
p _ {k} ( \xi \oplus \eta )  = \sum_i p_{k-i} ( \xi ) p _ {i} ( \eta ) ,\  p _ {0= 1 .
 +
$$
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375040.png" /> is monomorphically mapped into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375041.png" /> and the image of this mapping coincides with the subring of all even symmetric series in Wu generators as variables. Then the total Pontryagin class is mapped to the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375042.png" />, and the Pontryagin classes — to elementary symmetric functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375043.png" />. Theorem:
+
The ring $  H  ^ {**} (  \mathop{\rm BO} _ {n} ;  \Lambda ) $
 +
is monomorphically mapped into $  H  ^ {**} (  \mathop{\rm BU} _ {[} n/2] ;  \Lambda ) $
 +
and the image of this mapping coincides with the subring of all even symmetric series in Wu generators as variables. Then the total Pontryagin class is mapped to the polynomial $  \prod _ {i=1}  ^ {[n/2]} ( 1 + x _ {i}  ^ {2} ) $,  
 +
and the Pontryagin classes — to elementary symmetric functions of $  x _ {1}  ^ {2} \dots x _ {n}  ^ {2} $.  
 +
Theorem:
  
<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/p/p073/p073750/p07375044.png" /></td> </tr></table>
+
$$
 +
H  ^ {**} (  \mathop{\rm BO} _ {n} ; \Lambda )  = \Lambda [[ p _ {1} \dots
 +
p _ {[} n/2] ]].
 +
$$
  
The cohomology ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375045.png" /> contains, beside a Pontryagin class, also the [[Euler class|Euler class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375046.png" />. Theorem:
+
The cohomology ring $  H  ^ {*} (  \mathop{\rm BSO} _ {n} ) $
 +
contains, beside a Pontryagin class, also the [[Euler class|Euler class]] $  e \in H  ^ {n} (  \mathop{\rm BSO} _ {n} ) $.  
 +
Theorem:
  
<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/p/p073/p073750/p07375047.png" /></td> </tr></table>
+
$$
 +
H  ^ {**} (  \mathop{\rm BSO} _ {2k+} 1 ; \Lambda )  = \
 +
\Lambda [[ p _ {1} \dots p _ {k} , e ]] ,
 +
$$
  
<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/p/p073/p073750/p07375048.png" /></td> </tr></table>
+
$$
 +
H  ^ {**} (  \mathop{\rm BSO} _ {2k} ; \Lambda )  = \Lambda [[ p_1 \dots p_{k-1}, e ]] ,
 +
$$
  
for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375049.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375050.png" /> holds.
+
for the space $  \mathop{\rm BSO} _ {2k} $
 +
the equality $  p _ {k} = e  ^ {2} $
 +
holds.
  
The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375051.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375052.png" />. The induced mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375053.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375054.png" /> to zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375055.png" /> odd, and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375056.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375057.png" /> even.
+
The mapping $  g :   \mathop{\rm BU} _ {[} n/2] \rightarrow  \mathop{\rm BO} _ {n} $
 +
can be extended to $  \mathop{\rm BU} _ {[} n/2] \rightarrow  \mathop{\rm BSO} _ {n} $.  
 +
The induced mapping $  H  ^ {**} (  \mathop{\rm BSO} _ {n} ) \rightarrow H  ^ {**} (  \mathop{\rm BU} _ {[ n/2]} ) $
 +
maps $  e $
 +
to zero for $  n $
 +
odd, and to $  \prod _{i=1}  ^ {n/2} x _ {i} $
 +
for $  n $
 +
even.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375058.png" /> be a [[Formal power series|formal power series]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375059.png" />. Then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375060.png" /> determines some non-homogeneous element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375061.png" />, i.e. a characteristic class. Admitting a certain freedom, one can write
+
Let $  f ( t) \in \mathbf Q [[ t]] $
 +
be a [[Formal power series|formal power series]] over the field $  \mathbf Q $.  
 +
Then the series $  \prod _{i=1}  ^ {[ n/2] }f ( x _ {i} ) $
 +
determines some non-homogeneous element of the ring $  H  ^ {**} (  \mathop{\rm BO} _ {n} ;  \mathbf Q ) $,  
 +
i.e. a characteristic class. Admitting a certain freedom, one can write
  
<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/p/p073/p073750/p07375062.png" /></td> </tr></table>
+
$$
 +
= \prod _{i=1} ^ { [  n/2]} f ( x _ {i} )  \in  H  ^ {**} (
 +
\mathop{\rm BO} _ {n} ; \mathbf Q ).
 +
$$
  
The characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375063.png" /> is stable (that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375065.png" /> is the trivial vector bundle) if and only if the constant term in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375066.png" /> is equal to one. If one assumes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375067.png" />, then the characteristic class constructed by the above-described method is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375068.png" /> and is called the Hirzebruch <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375070.png" />-class,
+
The characteristic class $  x $
 +
is stable (that is $  x ( \xi \oplus \theta ) = x ( \xi ) $,  
 +
where $  \theta $
 +
is the trivial vector bundle) if and only if the constant term in $  f ( t) $
 +
is equal to one. If one assumes $  f ( t) = t / \mathop{\rm tanh}  t $,  
 +
then the characteristic class constructed by the above-described method is denoted by $  L $
 +
and is called the Hirzebruch $  L $-
 +
class,
  
<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/p/p073/p073750/p07375071.png" /></td> </tr></table>
+
$$
 +
= \prod _{i=1} ^ { [n/2]}
  
The standard procedure of expressing the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375072.png" /> in elementary symmetric functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375073.png" /> leads to the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375074.png" /> in the form of a series in Pontryagin classes. Another characteristic class that is important for applications is obtained if it is assumed that
+
\frac{x _ {i} }{ \mathop{\rm tanh}  x _ {i} }
 +
  \in \
 +
H  ^ {**} (  \mathop{\rm BO} _ {n} ;  \mathbf Q ) .
 +
$$
  
<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/p/p073/p073750/p07375075.png" /></td> </tr></table>
+
The standard procedure of expressing the series  $  \prod f ( x _ {i} ) $
 +
in elementary symmetric functions of  $  x _ {1}  ^ {2} \dots x _ {n}  ^ {2} $
 +
leads to the representation of  $  L $
 +
in the form of a series in Pontryagin classes. Another characteristic class that is important for applications is obtained if it is assumed that
 +
 
 +
$$
 +
f ( t)  = t/
 +
\frac{2}{\sinh ( t / 2 ) }
 +
  = \frac{1}{e  ^ {t/2} - e  ^ {-t/2} }
 +
.
 +
$$
  
 
The class determined by the even symmetric series
 
The class determined by the even symmetric series
  
<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/p/p073/p073750/p07375076.png" /></td> </tr></table>
+
$$
 +
\prod f ( x _ {i} )  = \prod
 +
\frac{x _ {i} / 2 }{\sinh ( x _ {i} / 2 ) }
  
is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375078.png" />-class. Similarly, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375080.png" />-class is the characteristic class determined by the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375081.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375082.png" />. Both these classes, as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375083.png" />, can be expressed in Pontryagin classes.
+
$$
 +
 
 +
is called the $  \widehat{A}  $-
 +
class. Similarly, the $  A $-
 +
class is the characteristic class determined by the series $  \prod f ( x _ {i} ) $
 +
where $  f ( t ) = 2t / \sinh ( 2t ) $.  
 +
Both these classes, as well as $  L $,  
 +
can be expressed in Pontryagin classes.
  
 
==Topological invariance.==
 
==Topological invariance.==
 
In 1965 S.P. Novikov [[#References|[2]]] proved that the Pontryagin classes with rational coefficients coincide for two homeomorphic manifolds. It was already known that rational Pontryagin classes are piecewise-linearly invariant, i.e. coincide for two piecewise-linear homeomorphic manifolds. Moreover, rational Pontryagin classes were defined (see [[#References|[4]]]) for piecewise-linear manifolds (possibly with a boundary). An example was given (see ) of integer Pontryagin classes which are not topological invariants.
 
In 1965 S.P. Novikov [[#References|[2]]] proved that the Pontryagin classes with rational coefficients coincide for two homeomorphic manifolds. It was already known that rational Pontryagin classes are piecewise-linearly invariant, i.e. coincide for two piecewise-linear homeomorphic manifolds. Moreover, rational Pontryagin classes were defined (see [[#References|[4]]]) for piecewise-linear manifolds (possibly with a boundary). An example was given (see ) of integer Pontryagin classes which are not topological invariants.
  
In 1969 it was shown (see [[#References|[7]]]) that the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375084.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375085.png" /> has the homotopy type of the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375086.png" />. From this the topological invariance of rational Pontryagin classes follows, as well as a disproof of the fundamental hypothesis of combinatorial topology (the Hauptvermutung).
+
In 1969 it was shown (see [[#References|[7]]]) that the fibre $  \mathop{\rm Top} / \mathop{\rm PL} $
 +
of the bundle $  \mathop{\rm BPL} \rightarrow  \mathop{\rm B}  \mathop{\rm Top} $
 +
has the homotopy type of the [[Eilenberg–MacLane space]] $  K ( \mathbf Z _ {2} , 3 ) $.  
 +
From this the topological invariance of rational Pontryagin classes follows, as well as a disproof of the fundamental hypothesis of combinatorial topology (the Hauptvermutung).
  
 
==Generalized Pontryagin classes.==
 
==Generalized Pontryagin classes.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375087.png" /> be a generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) in which Chern classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375088.png" /> are defined. If for a one-dimensional complex vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375089.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375090.png" /> holds, the Pontryagin classes with values in the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375091.png" /> can be defined via the above-mentioned formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375092.png" />. The classes thus defined will have the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375094.png" /> is the total Pontryagin class considered in the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375095.png" />.
+
Let $  h  ^ {*} $
 
+
be a generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) in which Chern classes $  \sigma _ {i} $
However, in many generalized theories used in practice (for example, in [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375096.png" />-theory]]) the equality proposed for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375097.png" /> does not hold. In such theories it does not make sense to define Pontryagin classes in the above-described manner, since under such a definition the usual formula for the total class of the sum of two vector bundles, even after including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375098.png" /> in the coefficients, is not valid. One can define generalized Pontryagin classes in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p07375099.png" /> be a multiplicative cohomology theory in which an orientation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750100.png" /> of a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750102.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750103.png" />-dimensional real vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750104.png" />, is universally given. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750105.png" /> be the Euler class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750108.png" /> is the inclusion of a zero section. Pontryagin classes in the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750109.png" /> are the characteristic classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750110.png" />, defined for real vector bundles and satisfying the following conditions:
+
are defined. If for a one-dimensional complex vector bundle $  \lambda $
 
+
the equality $  \sigma _ {1} ( \lambda ) = - \sigma _ {1} ( \overline \lambda \; ) $
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750111.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750112.png" />;
+
holds, the Pontryagin classes with values in the theory $  h  ^ {*} $
 +
can be defined via the above-mentioned formula $  P _ {i} ( \xi ) = ( - 1 )  ^ {i} \sigma _ {2i} ( \xi \otimes \mathbf C ) $.  
 +
The classes thus defined will have the property $  P ( \xi \otimes \eta ) = P ( \xi ) P ( \eta ) $,  
 +
where $  P = 1 + P _ {1} + P _ {2} + \dots $
 +
is the total Pontryagin class considered in the theory $  h  ^ {*} \otimes \mathbf Z [ 1 /2 ] $.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750113.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750114.png" /> is the trivial bundle;
+
However, in many generalized theories used in practice (for example, in [[K-theory| $  K $-
 +
theory]]) the equality proposed for  $  \sigma _ {1} $
 +
does not hold. In such theories it does not make sense to define Pontryagin classes in the above-described manner, since under such a definition the usual formula for the total class of the sum of two vector bundles, even after including  $  1/2 $
 +
in the coefficients, is not valid. One can define generalized Pontryagin classes in the following way. Let  $  h  ^ {*} $
 +
be a multiplicative cohomology theory in which an orientation  $  u ( \xi \otimes \mathbf C ) \in \widehat{h}  {}  ^ {2n} ( B ^ {\xi \otimes \mathbf C } ) $
 +
of a vector bundle  $  \xi \otimes \mathbf C $,
 +
where  $  \xi $
 +
is an arbitrary  $  n $-
 +
dimensional real vector bundle over  $  B $,
 +
is universally given. Let  $  e ( \xi \otimes \mathbf C ) $
 +
be the Euler class of  $  \xi \otimes \mathbf C $,
 +
$  e ( \xi \otimes \mathbf C ) = i  ^ {*} u ( \xi \otimes \mathbf C ) $,  
 +
where $  i : B \rightarrow B ^ {\xi \otimes \mathbf C } $
 +
is the inclusion of a zero section. Pontryagin classes in the theory  $  h  ^ {*} $
 +
are the characteristic classes  $  P _ {i} $,
 +
defined for real vector bundles and satisfying the following conditions:
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750115.png" /> is an element of order a power of two;
+
1)  $  P _ {i} ( \xi ) = 0 $
 +
if  $  i > 2  \mathop{\rm dim}  \xi $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750116.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750117.png" />.
+
2)  $  P _ {i} ( \xi \oplus \theta ) = P _ {i} ( \xi ) $,  
 +
where $  \theta $
 +
is the trivial bundle;
  
The uniqueness and existence of characteristic classes with the above properties has been proved. From this point of view, Pontryagin classes lead to the notion of a two-valued [[Formal group|formal group]] over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750118.png" /> corresponding to the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750119.png" />.
+
3)  $  P _ {k} ( \xi \oplus \eta ) - \sum_i P_i ( \xi ) P_{k-i} ( \eta ) $
 +
is an element of order a power of two;
  
The characteristic classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750120.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750121.png" />-theory are defined by the following formula:
+
4)  $  P _ {n} ( \xi ) = ( - 1 )  ^ {n} e ( \xi \otimes \mathbf C ) $,
 +
where  $  n = \mathop{\rm dim}  \xi $.
  
<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/p/p073/p073750/p073750122.png" /></td> </tr></table>
+
The uniqueness and existence of characteristic classes with the above properties has been proved. From this point of view, Pontryagin classes lead to the notion of a two-valued [[formal group]] over the ring  $  h  ^ {*} ( \mathop{\rm pt} ) $
 
+
corresponding to the theory $  h  ^ {*} $.
<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/p/p073/p073750/p073750123.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750124.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750125.png" /> are the Chern classes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750126.png" />-graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073750/p073750127.png" />-theory.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Characteristic classes of differentiable manifolds" ''Transl. Amer. Math. Soc. (1)'' , '''7''' (1962) pp. 279–331 ''Mat. Sb.'' , '''21''' (1947) pp. 233–284</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.P. Novikov, "Topological invariance of rational Pontrjagin classes" ''Soviet Math. Dokl.'' , '''6''' : 4 (1965) pp. 921–923 ''Dokl. Akad. Nauk SSSR'' , '''163''' (1965) pp. 298–300 {{MR|}} {{ZBL|0146.19503}} {{ZBL|0146.19502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.M. Bukhshtaber, "The Chern–Dold character in cobordisms. I" ''Math. USSR Sb.'' , '''12''' : 4 (1970) pp. 573–594 ''Mat. Sb.'' , '''83''' (1970) pp. 575–595 {{MR|}} {{ZBL|0219.57027}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.A. Rokhlin, A.S. Shvarč, "The combinatorial invariance of Pontrjagin classes" ''Dokl. Akad. Nauk SSSR'' , '''114''' (1957) pp. 490–493 (In Russian) {{MR|}} {{ZBL|0078.36803}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> J. Milnor, ''Matematika'' , '''3''' : 4 (1959) pp. 3–53 {{MR|0339964}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> J. Milnor, ''Matematika'' , '''9''' : 4 (1965) pp. 3–40 {{MR|0339964}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977) {{MR|0645390}} {{ZBL|0361.57004}} </TD></TR></table>
 
  
 +
The characteristic classes  $  \pi _ {i} $
 +
in  $  K $-
 +
theory are defined by the following formula:
  
 +
$$
 +
\sum _ { i } \pi _ {i} ( \xi ) s  ^ {i}  =  \sum _ { i } (- 1)  ^ {i} t  ^ {i}
 +
\gamma _ {i} ( \xi \otimes \mathbf C )  =  \gamma _ {-t} ( \xi \otimes \mathbf C ) =
 +
$$
  
====Comments====
+
$$
 +
= \
 +
\lambda _ {t/(1- t)} ( ( \xi \oplus ( -  \mathop{\rm dim}  \xi ) ) \otimes \mathbf C ) ,
 +
$$
  
 +
where  $  s = t - t  ^ {2} $;
 +
here  $  \gamma _ {i} $
 +
are the Chern classes in  $  \mathbf Z _ {2} $-
 +
graded  $  K $-
 +
theory.
  
 
====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><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.W. Anderson, E.H. Brown, F.P. Peterson, "Spin cobordism" ''Ann. of Math.'' (To appear) {{MR|0219077}} {{MR|0190939}} {{ZBL|0156.21605}} {{ZBL|0136.44103}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) {{MR|0995842}} {{ZBL|0673.55002}} </TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Characteristic classes of differentiable manifolds" ''Transl. Amer. Math. Soc. (1)'' , '''7''' (1962) pp. 279–331 ''Mat. Sb.'' , '''21''' (1947) pp. 233–284</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.P. Novikov, "Topological invariance of rational Pontrjagin classes" ''Soviet Math. Dokl.'' , '''6''' : 4 (1965) pp. 921–923 ''Dokl. Akad. Nauk SSSR'' , '''163''' (1965) pp. 298–300 {{MR|}} {{ZBL|0146.19503}} {{ZBL|0146.19502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.M. Bukhshtaber, "The Chern–Dold character in cobordisms. I" ''Math. USSR Sb.'' , '''12''' : 4 (1970) pp. 573–594 ''Mat. Sb.'' , '''83''' (1970) pp. 575–595 {{MR|}} {{ZBL|0219.57027}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.A. Rokhlin, A.S. Shvarč, "The combinatorial invariance of Pontrjagin classes" ''Dokl. Akad. Nauk SSSR'' , '''114''' (1957) pp. 490–493 (In Russian) {{MR|}} {{ZBL|0078.36803}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> J. Milnor, ''Matematika'' , '''3''' : 4 (1959) pp. 3–53 {{MR|0339964}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> J. Milnor, ''Matematika'' , '''9''' : 4 (1965) pp. 3–40 {{MR|0339964}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR>
 +
<TR><TD valign="top">[7]</TD> <TD valign="top"> R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977) {{MR|0645390}} {{ZBL|0361.57004}} </TD></TR>
 +
<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>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> D.W. Anderson, E.H. Brown, F.P. Peterson, "Spin cobordism" ''Ann. of Math.'' (To appear) {{MR|0219077}} {{MR|0190939}} {{ZBL|0156.21605}} {{ZBL|0136.44103}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) {{MR|0995842}} {{ZBL|0673.55002}} </TD></TR></table>

Latest revision as of 19:15, 16 January 2024


A characteristic class defined for real vector bundles (cf. Vector bundle). Pontryagin classes were introduced by L.S. Pontryagin [1] in 1947. For a vector bundle $ \xi $ with base $ B $ the Pontryagin classes are denoted by the symbol $ p _ {i} ( \xi ) \in H ^ {4i} ( B) $ and are defined to be equal to $ p _ {i} ( \xi ) = ( - 1 ) ^ {i} c _ {2i} ( \xi \otimes \mathbf C ) $, where $ \xi \otimes \mathbf C $ is the complexification of $ \xi $ and $ c _ {k} $ are the Chern classes (cf. Chern class). The total Pontryagin class is the non-homogeneous characteristic class $ p = 1 + p _ {1} + p _ {2} + \dots $. In other words, the Pontryagin classes are defined as homology classes $ p _ {i} \in H ^ {4i} ( \mathop{\rm BO} _ {n} ) $ determined by the equality $ p _ {i} = f ^ { * } ( ( - 1 ) ^ {i} c _ {2i} ) $, where $ f : \mathop{\rm BO} _ {n} \rightarrow \mathop{\rm BU} _ {n} $ is the mapping corresponding to the complexification of the universal vector bundle and $ c _ {n} \in H ^ {2k} ( \mathop{\rm BU} _ {n} ) $ are the Chern classes.

Let $ ( \kappa _ {n} ) _ {\mathbf R } $ be the real bundle of the universal vector bundle $ \kappa _ {n} $ over $ \mathop{\rm BU} _ {n} $. The total Pontryagin class $ p ( ( \kappa _ {n} ) _ {\mathbf R } ) $ of the vector bundle $ ( \kappa _ {n} ) _ {\mathbf R } $ coincides with $ \prod _{i=1} ^ {n} ( 1 + x _ {i} ^ {2} ) \in H ^ {*} ( \mathop{\rm BU} _ {n} ) $, where $ x _ {1} \dots x _ {n} $ are the Wu generators (see Characteristic class).

A partial description of the cohomology ring $ H ^ {*} ( \mathop{\rm BO} _ {n} ) $ can be obtained in terms of Wu generators in the following way. The mapping $ g : \mathop{\rm BU} _ {[} n/2] \rightarrow \mathop{\rm BO} _ {n} $ corresponding to the vector bundle $ ( \kappa _ {[} n/2] ) _ {\mathbf R} \otimes \theta _ {1} $, where $ \theta _ {1} $ is the one-dimensional trivial vector bundle, induces a ring homomorphism $ g ^ {*} : H ^ {*} ( \mathop{\rm BO} _ {n} ) \rightarrow H ^ {*} ( \mathop{\rm BU} _ {[} n/2] ) $, under which the subring of $ H ^ {*} ( \mathop{\rm BO} _ {n} ) $ generated by the Pontryagin classes $ p _ {1} \dots p ^ {[} n/2] $ is mapped monomorphically onto the subring of $ H ^ {*} ( \mathop{\rm BU} _ {n} ) $ consisting of all even symmetric polynomials in the Wu generators. Evenness is understood in the sense that the degree of every variable $ x _ {i} $ in the polynomial should be even. Thus, an expression in Wu generators is obtained for any element of the ring $ \mathbf Z ( p _ {1} \dots p _ {[} n/2] ) \subset H ^ {*} ( \mathop{\rm BO} _ {n} ) $. This is important for practical calculations with Pontryagin classes. The characteristic class determined by an even symmetric polynomial in the Wu generators can be expressed in Pontryagin classes as follows. First, the polynomial is written in elementary symmetric functions of the variables $ x _ {1} ^ {2} \dots x _ {n} ^ {2} $ and then the elementary symmetric functions are replaced by Pontryagin classes.

If $ \xi , \eta $ are two real vector bundles over a common base, then the cohomology class $ p ( \xi \oplus \eta ) - p ( \xi ) p ( \eta ) $ is of order at most two; this is due to the fact that for the first Chern class $ c _ {1} ( \lambda ) = - c _ {1} ( \overline \lambda \; ) $.

Let a ring $ \Lambda $, containing $ 1/2 $, be considered as the ring of coefficients, and let $ p _ {i} $ be a Pontryagin class with values in $ H ^ {*} ( \cdot ; \Lambda ) $. In this case the following equality is valid:

$$ p ( \xi \oplus \eta ) = p ( \xi ) p ( \eta ), $$

or

$$ p _ {k} ( \xi \oplus \eta ) = \sum_i p_{k-i} ( \xi ) p _ {i} ( \eta ) ,\ p _ {0} = 1 . $$

The ring $ H ^ {**} ( \mathop{\rm BO} _ {n} ; \Lambda ) $ is monomorphically mapped into $ H ^ {**} ( \mathop{\rm BU} _ {[} n/2] ; \Lambda ) $ and the image of this mapping coincides with the subring of all even symmetric series in Wu generators as variables. Then the total Pontryagin class is mapped to the polynomial $ \prod _ {i=1} ^ {[n/2]} ( 1 + x _ {i} ^ {2} ) $, and the Pontryagin classes — to elementary symmetric functions of $ x _ {1} ^ {2} \dots x _ {n} ^ {2} $. Theorem:

$$ H ^ {**} ( \mathop{\rm BO} _ {n} ; \Lambda ) = \Lambda [[ p _ {1} \dots p _ {[} n/2] ]]. $$

The cohomology ring $ H ^ {*} ( \mathop{\rm BSO} _ {n} ) $ contains, beside a Pontryagin class, also the Euler class $ e \in H ^ {n} ( \mathop{\rm BSO} _ {n} ) $. Theorem:

$$ H ^ {**} ( \mathop{\rm BSO} _ {2k+} 1 ; \Lambda ) = \ \Lambda [[ p _ {1} \dots p _ {k} , e ]] , $$

$$ H ^ {**} ( \mathop{\rm BSO} _ {2k} ; \Lambda ) = \Lambda [[ p_1 \dots p_{k-1}, e ]] , $$

for the space $ \mathop{\rm BSO} _ {2k} $ the equality $ p _ {k} = e ^ {2} $ holds.

The mapping $ g : \mathop{\rm BU} _ {[} n/2] \rightarrow \mathop{\rm BO} _ {n} $ can be extended to $ \mathop{\rm BU} _ {[} n/2] \rightarrow \mathop{\rm BSO} _ {n} $. The induced mapping $ H ^ {**} ( \mathop{\rm BSO} _ {n} ) \rightarrow H ^ {**} ( \mathop{\rm BU} _ {[ n/2]} ) $ maps $ e $ to zero for $ n $ odd, and to $ \prod _{i=1} ^ {n/2} x _ {i} $ for $ n $ even.

Let $ f ( t) \in \mathbf Q [[ t]] $ be a formal power series over the field $ \mathbf Q $. Then the series $ \prod _{i=1} ^ {[ n/2] }f ( x _ {i} ) $ determines some non-homogeneous element of the ring $ H ^ {**} ( \mathop{\rm BO} _ {n} ; \mathbf Q ) $, i.e. a characteristic class. Admitting a certain freedom, one can write

$$ x = \prod _{i=1} ^ { [ n/2]} f ( x _ {i} ) \in H ^ {**} ( \mathop{\rm BO} _ {n} ; \mathbf Q ). $$

The characteristic class $ x $ is stable (that is $ x ( \xi \oplus \theta ) = x ( \xi ) $, where $ \theta $ is the trivial vector bundle) if and only if the constant term in $ f ( t) $ is equal to one. If one assumes $ f ( t) = t / \mathop{\rm tanh} t $, then the characteristic class constructed by the above-described method is denoted by $ L $ and is called the Hirzebruch $ L $- class,

$$ L = \prod _{i=1} ^ { [n/2]} \frac{x _ {i} }{ \mathop{\rm tanh} x _ {i} } \in \ H ^ {**} ( \mathop{\rm BO} _ {n} ; \mathbf Q ) . $$

The standard procedure of expressing the series $ \prod f ( x _ {i} ) $ in elementary symmetric functions of $ x _ {1} ^ {2} \dots x _ {n} ^ {2} $ leads to the representation of $ L $ in the form of a series in Pontryagin classes. Another characteristic class that is important for applications is obtained if it is assumed that

$$ f ( t) = t/ \frac{2}{\sinh ( t / 2 ) } = \frac{1}{e ^ {t/2} - e ^ {-t/2} } . $$

The class determined by the even symmetric series

$$ \prod f ( x _ {i} ) = \prod \frac{x _ {i} / 2 }{\sinh ( x _ {i} / 2 ) } $$

is called the $ \widehat{A} $- class. Similarly, the $ A $- class is the characteristic class determined by the series $ \prod f ( x _ {i} ) $ where $ f ( t ) = 2t / \sinh ( 2t ) $. Both these classes, as well as $ L $, can be expressed in Pontryagin classes.

Topological invariance.

In 1965 S.P. Novikov [2] proved that the Pontryagin classes with rational coefficients coincide for two homeomorphic manifolds. It was already known that rational Pontryagin classes are piecewise-linearly invariant, i.e. coincide for two piecewise-linear homeomorphic manifolds. Moreover, rational Pontryagin classes were defined (see [4]) for piecewise-linear manifolds (possibly with a boundary). An example was given (see ) of integer Pontryagin classes which are not topological invariants.

In 1969 it was shown (see [7]) that the fibre $ \mathop{\rm Top} / \mathop{\rm PL} $ of the bundle $ \mathop{\rm BPL} \rightarrow \mathop{\rm B} \mathop{\rm Top} $ has the homotopy type of the Eilenberg–MacLane space $ K ( \mathbf Z _ {2} , 3 ) $. From this the topological invariance of rational Pontryagin classes follows, as well as a disproof of the fundamental hypothesis of combinatorial topology (the Hauptvermutung).

Generalized Pontryagin classes.

Let $ h ^ {*} $ be a generalized cohomology theory (cf. Generalized cohomology theories) in which Chern classes $ \sigma _ {i} $ are defined. If for a one-dimensional complex vector bundle $ \lambda $ the equality $ \sigma _ {1} ( \lambda ) = - \sigma _ {1} ( \overline \lambda \; ) $ holds, the Pontryagin classes with values in the theory $ h ^ {*} $ can be defined via the above-mentioned formula $ P _ {i} ( \xi ) = ( - 1 ) ^ {i} \sigma _ {2i} ( \xi \otimes \mathbf C ) $. The classes thus defined will have the property $ P ( \xi \otimes \eta ) = P ( \xi ) P ( \eta ) $, where $ P = 1 + P _ {1} + P _ {2} + \dots $ is the total Pontryagin class considered in the theory $ h ^ {*} \otimes \mathbf Z [ 1 /2 ] $.

However, in many generalized theories used in practice (for example, in $ K $- theory) the equality proposed for $ \sigma _ {1} $ does not hold. In such theories it does not make sense to define Pontryagin classes in the above-described manner, since under such a definition the usual formula for the total class of the sum of two vector bundles, even after including $ 1/2 $ in the coefficients, is not valid. One can define generalized Pontryagin classes in the following way. Let $ h ^ {*} $ be a multiplicative cohomology theory in which an orientation $ u ( \xi \otimes \mathbf C ) \in \widehat{h} {} ^ {2n} ( B ^ {\xi \otimes \mathbf C } ) $ of a vector bundle $ \xi \otimes \mathbf C $, where $ \xi $ is an arbitrary $ n $- dimensional real vector bundle over $ B $, is universally given. Let $ e ( \xi \otimes \mathbf C ) $ be the Euler class of $ \xi \otimes \mathbf C $, $ e ( \xi \otimes \mathbf C ) = i ^ {*} u ( \xi \otimes \mathbf C ) $, where $ i : B \rightarrow B ^ {\xi \otimes \mathbf C } $ is the inclusion of a zero section. Pontryagin classes in the theory $ h ^ {*} $ are the characteristic classes $ P _ {i} $, defined for real vector bundles and satisfying the following conditions:

1) $ P _ {i} ( \xi ) = 0 $ if $ i > 2 \mathop{\rm dim} \xi $;

2) $ P _ {i} ( \xi \oplus \theta ) = P _ {i} ( \xi ) $, where $ \theta $ is the trivial bundle;

3) $ P _ {k} ( \xi \oplus \eta ) - \sum_i P_i ( \xi ) P_{k-i} ( \eta ) $ is an element of order a power of two;

4) $ P _ {n} ( \xi ) = ( - 1 ) ^ {n} e ( \xi \otimes \mathbf C ) $, where $ n = \mathop{\rm dim} \xi $.

The uniqueness and existence of characteristic classes with the above properties has been proved. From this point of view, Pontryagin classes lead to the notion of a two-valued formal group over the ring $ h ^ {*} ( \mathop{\rm pt} ) $ corresponding to the theory $ h ^ {*} $.

The characteristic classes $ \pi _ {i} $ in $ K $- theory are defined by the following formula:

$$ \sum _ { i } \pi _ {i} ( \xi ) s ^ {i} = \sum _ { i } (- 1) ^ {i} t ^ {i} \gamma _ {i} ( \xi \otimes \mathbf C ) = \gamma _ {-t} ( \xi \otimes \mathbf C ) = $$

$$ = \ \lambda _ {t/(1- t)} ( ( \xi \oplus ( - \mathop{\rm dim} \xi ) ) \otimes \mathbf C ) , $$

where $ s = t - t ^ {2} $; here $ \gamma _ {i} $ are the Chern classes in $ \mathbf Z _ {2} $- graded $ K $- theory.

References

[1] L.S. Pontryagin, "Characteristic classes of differentiable manifolds" Transl. Amer. Math. Soc. (1) , 7 (1962) pp. 279–331 Mat. Sb. , 21 (1947) pp. 233–284
[2] S.P. Novikov, "Topological invariance of rational Pontrjagin classes" Soviet Math. Dokl. , 6 : 4 (1965) pp. 921–923 Dokl. Akad. Nauk SSSR , 163 (1965) pp. 298–300 Zbl 0146.19503 Zbl 0146.19502
[3] V.M. Bukhshtaber, "The Chern–Dold character in cobordisms. I" Math. USSR Sb. , 12 : 4 (1970) pp. 573–594 Mat. Sb. , 83 (1970) pp. 575–595 Zbl 0219.57027
[4] V.A. Rokhlin, A.S. Shvarč, "The combinatorial invariance of Pontrjagin classes" Dokl. Akad. Nauk SSSR , 114 (1957) pp. 490–493 (In Russian) Zbl 0078.36803
[5a] J. Milnor, Matematika , 3 : 4 (1959) pp. 3–53 MR0339964
[5b] J. Milnor, Matematika , 9 : 4 (1965) pp. 3–40 MR0339964
[6] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604
[7] R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977) MR0645390 Zbl 0361.57004
[a1] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) MR0440554 Zbl 0298.57008
[a2] D.W. Anderson, E.H. Brown, F.P. Peterson, "Spin cobordism" Ann. of Math. (To appear) MR0219077 MR0190939 Zbl 0156.21605 Zbl 0136.44103
[a3] J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) MR0995842 Zbl 0673.55002
How to Cite This Entry:
Pontryagin class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_class&oldid=24117
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article