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'' $   \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|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|Todd class]]. The index class  $  I \in H  ^ {**} (  \mathop{\rm BO} _ {n} ;  \mathbf Q ) $
is expressed in Wu generators (see [[Characteristic class|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|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| $  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]].