cohomology

# Contents

## Idea

The Pontryagin classes are characteristic classes on the classifying space $\mathcal{B}O(n)$ of the orthogonal group and, by pullback, on the base of any bundle with structural group the orthogonal group. The latter is where they were originally defined.

The analogs for the unitary group are the Chern classes.

## Definition

The universal Pontryagin characteristic classes $P_k$ on the classifying space $B O(n)$ are, up to a sign, the pullbacks of the Chern classes $c_{2k}$ along the complexification inclusion

$B O(n) \to B U(n) \,.$

## Properties

### As generating universal characteristic classes

The cohomology ring $H^\bullet(B SO(2n+1), \mathbb{Z})$ is the polynomial ring on all Pontryagin classes $\{P_i\}_{i = 1}^n$.

The cohomology ring $H^\bullet(B SO(2n), \mathbb{Z})$ is the quotient of the polynomial ring on Pontryagin classes $P_i$ and the Euler class $\chi$ by the relation $\chi^2 = P_n$.

### Further relation to Chern classes

Under the other canonical map

$j \;\colon\; B U(n) \to BO(2n)$

one has

$j^\ast(P_k) = \sum_{a + b = 2 k} (-1)^{a+k} c_a c_b$

and

$j^\ast(\chi) = c_n \,.$

### Splitting principle and Chern roots

Under the inclusion

$i \;\colon\; U(1)^n \hookrightarrow U(n) \to O(2n)$

of the maximal torus one has that

$(B i)^\ast(P_k) = \sigma_k(x_1, \cdots, x_n)^2$

and

$(B i)^\ast(\chi) = \sigma_n(x_1, \cdots, x_n)$

where the $x_i \in H^\bullet(B U(1)^n, \mathbb{Z})$ are the “Chern roots”.

## Trivializations and structures

The twisted differential c-structures corresponding to Pontryagin class include

## References

Classical textbook references are