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Difference between revisions of "Pontryagin character"

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'' $  \mathop{\rm ph} $''
 
'' $  \mathop{\rm ph} $''
  
The [[Characteristic class|characteristic class]] defined by the equality  $  \mathop{\rm ph} ( \xi ) =  \mathop{\rm ch} ( \xi \otimes \mathbf C ) $,  
+
The [[characteristic class]] defined by the equality  $  \mathop{\rm ph} ( \xi ) =  \mathop{\rm ch} ( \xi \otimes \mathbf C ) $,  
 
where  $  \xi \otimes \mathbf C $
 
where  $  \xi \otimes \mathbf C $
 
is the complexification of the fibre bundle  $  \xi $
 
is the complexification of the fibre bundle  $  \xi $
 
and  $  \mathop{\rm ch} $
 
and  $  \mathop{\rm ch} $
 
is the [[Chern character|Chern character]]. The Pontryagin character as an element of the ring  $  H  ^ {**} (  \mathop{\rm BO} _ {n} ;  \mathbf Q ) $
 
is the [[Chern character|Chern character]]. The Pontryagin character as an element of the ring  $  H  ^ {**} (  \mathop{\rm BO} _ {n} ;  \mathbf Q ) $
is determined by the even series  $  \sum _ {i=} 1 ^ {[} n/2] ( e ^ {x _ {i} } + e ^ {- x _ {i} } ) $
+
is determined by the even series  $  \sum _ {i=1}  ^ {[ n/2]} ( e ^ {x _ {i} } + e ^ {- x _ {i} } ) $
 
and has the following properties
 
and has the following properties
  
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$$  
 
$$  
\Phi _ {H}  ^ {-} \mathop{\rm ph} ( \Phi ( 1))  =  \widehat{A}  ( - \xi ) .
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\Phi _ {H}  ^ {-1}   \mathop{\rm ph} ( \Phi ( 1))  =  \widehat{A}  ( - \xi ) .
 
$$
 
$$
  
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is the Todd class.
 
is the Todd class.
  
For references see [[Pontryagin class|Pontryagin class]].
+
For references see [[Pontryagin class]].

Latest revision as of 07:47, 7 January 2024


$ \mathop{\rm ph} $

The characteristic class defined by the equality $ \mathop{\rm ph} ( \xi ) = \mathop{\rm ch} ( \xi \otimes \mathbf C ) $, where $ \xi \otimes \mathbf C $ is the complexification of the fibre bundle $ \xi $ and $ \mathop{\rm ch} $ is the Chern character. The Pontryagin character as an element of the ring $ H ^ {**} ( \mathop{\rm BO} _ {n} ; \mathbf Q ) $ is determined by the even series $ \sum _ {i=1} ^ {[ n/2]} ( e ^ {x _ {i} } + e ^ {- x _ {i} } ) $ and has the following properties

$$ \mathop{\rm ph} ( \xi \otimes \eta ) = \mathop{\rm ph} \xi \cdot \mathop{\rm ph} \eta ,\ \ \mathop{\rm ph} ( \xi \oplus \eta ) = \mathop{\rm ph} \xi + \mathop{\rm ph} \eta . $$

The index class $ I ( \xi ) $ is defined to be equal to $ T ( \xi \otimes \mathbf C ) $, where $ T \in H ^ {**} ( \mathop{\rm BU} _ {n} ; \mathbf Q ) $ is the Todd class. The index class $ I \in H ^ {**} ( \mathop{\rm BO} _ {n} ; \mathbf Q ) $ is expressed in Wu generators (see Characteristic class) by the formula

$$ I = \prod \frac{x _ {i} }{1 - e ^ {- x _ {i} } } \prod \frac{- x _ {i} }{1 - e ^ {x _ {i} } } . $$

The following theorem on the relationship between the Pontryagin class and the $ \widehat{A} $- class holds (cf. Pontryagin class). Let $ \xi $ be a real vector bundle over the base $ B $ with a $ \mathop{\rm Spin} _ {n} $- structure, $ n = \mathop{\rm dim} \xi = 8 k $. For such bundles there is the Thom isomorphism in real $ K $- theory:

$$ \Phi : \mathop{\rm KO} ^ {*} ( B) \rightarrow \mathop{\rm KO} tilde {} ^ {*} ( B ^ \xi ) . $$

Let

$$ \Phi _ {H} : H ^ {*} ( B ; \mathbf Q ) \rightarrow \widetilde{H} {} ^ {*} ( B ^ \xi ; \ \mathbf Q ) $$

be the Thom isomorphism, which is uniquely determined by the orientation of the bundle $ \xi $. Then

$$ \Phi _ {H} ^ {-1} \mathop{\rm ph} ( \Phi ( 1)) = \widehat{A} ( - \xi ) . $$

This formula is an exact analogue of the corresponding statement on the relationship between the Chern character and the Todd class.

If $ \xi $ is a complex vector bundle, then $ T ( \xi ) = \widehat{A} ( ( \xi ) _ {\mathbf R } ) e ^ {c _ {1} ( \xi ) / 2 } $. Here $ ( \xi ) _ {\mathbf R } $ is the real part of the bundle, and $ T $ is the Todd class.

For references see Pontryagin class.

How to Cite This Entry:
Pontryagin character. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_character&oldid=48238
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article