Special and general types
The Pontryagin classes are characteristic classes on the classifying space 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.
The universal Pontryagin characteristic classes on the classifying space are, up to a sign, the pullbacks of the Chern classes along the complexification inclusion
As generating universal characteristic classes
The cohomology ring is the polynomial ring on all Pontryagin classes .
The cohomology ring is the quotient of the polynomial ring on Pontryagin classes and the Euler class by the relation .
Further relation to Chern classes
Under the other canonical map
Splitting principle and Chern roots
Under the inclusion
of the maximal torus one has that
where the are the “Chern roots”.
See at Chern class - Properties – Splitting principle and Chern roots and at splitting principle - Examples - Real vector bundles for more.
Trivializations and structures
The twisted differential c-structures corresponding to Pontryagin class include
Classical textbook references are
Paul Bressler, The first Pontryagin class, math.AT/0509563
Ivan Panin, Charles Walter, Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, arxiv/1011.0649
A brief introduction is in chapter 23, section 7
Revised on January 5, 2016 22:17:27
by Urs Schreiber