In the mathematical field of differential topology , secondary characteristic classes (like the Cheeger-Chern-Simons classes ) are invariants of flat bundles .
As is well known, there can be different characteristic classes
of - principal bundles are realized by means of the Chern-Weil construction by invariant polynomials , d. H. there is an invariant polynomial such that
for each principal bundle with connected form , where
the curvature of the connection , the De Rham cohomology class of
-
,
and the image of the characteristic class under canonical homomorphism
designated.
For flat bundles is
and consequently all characteristic classes defined by the Chern-Weil construction disappear, especially Chern classes and Pontryagin classes .
The Cheeger-Chern-Simons construction now defines for every such characteristic class, more precisely for every invariant polynomial
and every cohomology class
with a differential character
-
.
The cohomology group is a subgroup of and in the case of flat bundles lies in this subgroup. The cohomology class so defined
is called (the primary characteristic class associated with) secondary characteristic class.
Application of the Bockstein homomorphism maps the secondary characteristic class to the characteristic class , the image of which disappears in.
Existence and uniqueness
Let a Lie group , an invariant polynomial and a cohomology class be given . We denote the Korand map with and the Bockstein homomorphism .
Set : For every - principal bundle with connection form there is a clear differential character
With
-
,
so that naturally transformed under bundle images .
Cheeger-Chern-Simons classes
A special case is the construction of Cheeger-Chern-Simons classes.
Let the Chern polynomials be defined by the relation
for everyone . The universal Chern-Weil homomorphism
maps invariant polynomials to cohomology classes of the classifying space .
In the case of the Chern polynomials, there are the universal Chern classes and for them applies
-
.
There is now a classifying map for a principal bundle and the Chern class of is . For a form of connection one now defines
-
.
In the case of flat bundles , you get the Cheeger-Chern-Simons classes
-
.
If it is a -dimensional closed orientable manifold, one obtains the Cheeger-Chern-Simons invariant
of the shallow bundle by applying the Cheeger-Chern-Simons class to the fundamental class .
literature
- Cheeger, Simons: Differential characters and geometric invariants. Geometry and topology (College Park, Md., 1983/84), 50-80, Lecture Notes in Math., 1167, Springer, Berlin, 1985. pdf
- Dupont, Hain, Zucker: Regulators and characteristic classes of flat bundles. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 47-92, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000.
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