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 on the classifying space $B O(n)$ are, up to a sign, the pullbacks of the Chern classes along the inclusion

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

Trivializations and structures

The twisted differential c-structures corresponding to Pontryagin class include

References

Classical textbook references are