Pontryagin character

$ \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.