Contents

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 torsion-free quotient of 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 torsion is generated by Bockstein images of $H^\bullet(BSO(2n+1);\mathbb{F}_2)$, which is generated by the Stiefel-Whitney classes.

The torsion-free quotient of 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 $\chi^2 = P_n$; again the torsion is generated by Bocksteins of monomials in the Stiefel–Whitney classes.

### 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

Classical textbook references are

With an eye towards mathematical physics:

• Gerd Rudolph, Matthias Schmidt, around Def. 4.2.19 in Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)