Special and general types
The ordinary Chern classes are the integral characteristic classes
of the classifying space of the unitary group.
Accordingly these are characteristic classes in ordinary cohomology of U-principal bundles and hence of complex vector bundle
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.
More generally, there are generalized Chern classes for any complex oriented cohomology theory (Adams 74, Lurie 10).
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
Splitting principle and Chern roots
Under the splitting principle all Chern classes are determnined by first Chern classes:
Write for the maximal torus inside the unitary group, which is the subgroup of diagonal unitary matrices. Then
is the polynomial ring in generators (to be thought of as the universal first Chern classes of each copy of ; equivalently as the weights of the group characters of ) which are traditionally written :
for the induced map of deloopings/classifying spaces, then the -universal Chern class is uniquely characterized by the fact that its pullbacl to is the th elementary symmetric polynomial applied to these first Chern classes:
Equivalently, for the formal sum of all the Chern classes, and using the fact that the elementary symmetric polynomials are the degree- piece in , this means that
Since here on the right the first Chern classes appear as the roots of the Chern polynomial, they are also called Chern roots.
See also at splitting principle – Examples – Complex vector bundles and their Chern roots.
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 .
Original articles include
- A. Grothendieck, La théorie des classes de Chern, Bulletin de la Société Mathématique de France 86 (1958), p. 137–154, numdam
Textbook accounts include
Werner Greub, Stephen Halperin, Ray Vanstone, chapter IX of volume II of Connections, Curvature, and Cohomology Academic Press (1973)
John Milnor, James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press (1974).
Stanley Kochmann, section 2.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
A brief introduction is in chapter 23, section 7
For Conner-Floyd Chern classes in complex oriented cohomology theory: