Special and general types
The Chern classes are the integral characteristic classes
of the classifying space of the unitary group.
Accordingly these are characteristic classes of -principal bundles and hence of complex vector bundles.
The first Chern class is the unique characteristic class of circle group-principal bundles.
The analogous classes for the orthogonal group are the Pontryagin classes.
For the Chern universal characteristic classes of the classifying space of the unitary group are characterized as follows:
and if ;
for , is the canonical generator of ;
under pullback along the inclusion we have ;
under the inclusion we have .
The cohomology ring of is the polynomial algebra on the Chern classes:
First Chern class
In Yang-Mills theory field configurations with non-vanishing second Chern-class (and minimal energy) are called instantons. The second Chern class is the instanton number .
Standard textbook references include
or chapter IX of volume II of
A brief introduction is in chapter 23, section 7
Revised on May 30, 2012 15:49:54
by Urs Schreiber