# 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.

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