Secondary characteristic class

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

${\displaystyle c\in H^{k}(B;\mathbb {Z} )}$

of - principal bundles are realized by means of the Chern-Weil construction by invariant polynomials , d. H. there is an invariant polynomial such that ${\displaystyle G}$ ${\displaystyle E\to B}$ ${\displaystyle P\in I^{k}(G)}$ ${\displaystyle P}$

${\displaystyle \left[P(\Omega )\right]=c_{\mathbb {R} }}$

for each principal bundle with connected form , where ${\displaystyle G}$ ${\displaystyle \omega }$

${\displaystyle \Omega \in \Omega ^{2}(B)}$

the curvature of the connection , the De Rham cohomology class of ${\displaystyle \omega }$ ${\displaystyle \left[P(\Omega )\right]}$

${\displaystyle P(\Omega )\in \Omega ^{2k}(B)}$,

and the image of the characteristic class under canonical homomorphism ${\displaystyle c_{\mathbb {R} }}$ ${\displaystyle c}$

${\displaystyle H^{k}(B;\mathbb {Z} )\to H^{k}(B;\mathbb {R} )}$

designated.

For flat bundles is

${\displaystyle \Omega =0}$

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

${\displaystyle P\in I^{k}(G)}$

and every cohomology class

${\displaystyle c\in H^{2k}(B)}$

with a differential character ${\displaystyle \left[P(\Omega )\right]=c_{\mathbb {R} }}$

${\displaystyle {\hat {c}}\in {\widehat {H}}^{k}(B;\mathbb {R} /\mathbb {Z} )}$.

The cohomology group is a subgroup of and in the case of flat bundles lies in this subgroup. The cohomology class so defined ${\displaystyle H^{k}(B;\mathbb {R} /\mathbb {Z} )}$ ${\displaystyle {\widehat {H}}^{k}(B;\mathbb {R} /\mathbb {Z} )}$ ${\displaystyle {\hat {c}}}$

${\displaystyle {\hat {c}}\in H^{k}(B;\mathbb {R} /\mathbb {Z} )}$

is called (the primary characteristic class associated with) secondary characteristic class. ${\displaystyle c}$

Application of the Bockstein homomorphism maps the secondary characteristic class to the characteristic class , the image of which disappears in. ${\displaystyle H^{2k-1}(B;\mathbb {R} /\mathbb {Z} )\to H^{2k}(B;\mathbb {Z} )}$ ${\displaystyle {\hat {c}}}$ ${\displaystyle c}$ ${\displaystyle H^{2k}(B;\mathbb {R} )}$

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 . ${\displaystyle G}$ ${\displaystyle P\in I^{k}(G)}$ ${\displaystyle u\in H^{2k}(BG)}$ ${\displaystyle w_{k}(P)=u_{\mathbb {R} }}$ ${\displaystyle \delta \colon {\widehat {H}}^{2k-1}(B;\mathbb {R} /\mathbb {Z} )\to A_{0}^{2k}(B)}$ ${\displaystyle b\colon {\widehat {H}}^{2k-1}(B;\mathbb {R} /\mathbb {Z} )\to H^{2k}(B;\mathbb {Z} )}$

Set : For every - principal bundle with connection form there is a clear differential character ${\displaystyle G}$ ${\displaystyle p\colon E\to B}$ ${\displaystyle \omega }$

${\displaystyle S_{P,u}(p,\omega )\in {\widehat {H}}^{2k-1}(B;\mathbb {R} /\mathbb {Z} )}$

With

• ${\displaystyle \delta S_{P,u}(p,\omega )=P(\Omega )}$
• ${\displaystyle bS_{P,u}(p,\omega )=f^{*}u}$,

so that naturally transformed under bundle images . ${\displaystyle S_{P,u}(p,\omega )\in {\widehat {H}}^{2k-1}(B;\mathbb {R} /\mathbb {Z} )}$

Cheeger-Chern-Simons classes

A special case is the construction of Cheeger-Chern-Simons classes.

Let the Chern polynomials be defined by the relation ${\displaystyle C_{k}\in I^{k}(GL(n,\mathbb {C} ))}$

${\displaystyle \det(\lambda \,\mathrm {id} _{n}-{\frac {1}{2\pi i}}A)=\sum _{k=0}^{n}C_{k}(A,\ldots ,A)\lambda ^{n-k}}$

for everyone . The universal Chern-Weil homomorphism ${\displaystyle A\in GL(n,\mathbb {C} )}$

${\displaystyle w_{k}\colon I^{k}(G)\to H^{2k}(BG;\mathbb {R} )}$

maps invariant polynomials to cohomology classes of the classifying space . ${\displaystyle BG}$

In the case of the Chern polynomials, there are the universal Chern classes and for them applies ${\displaystyle c_{k}\in H^{2k}(BGL(n,\mathbb {C} );\mathbb {Z} )}$

${\displaystyle w_{k}(C_{k})=(c_{k})_{\mathbb {R} }}$.

There is now a classifying map for a principal bundle and the Chern class of is . For a form of connection one now defines ${\displaystyle GL(n,\mathbb {C} )}$ ${\displaystyle p\colon E\to B}$ ${\displaystyle f\colon B\to BGL(n,\mathbb {C} )}$ ${\displaystyle p\colon E\to B}$ ${\displaystyle f^{*}c_{k}\in H^{2k}(B;\mathbb {Z} )}$ ${\displaystyle \omega }$

${\displaystyle {\hat {c}}_{k}(p,\omega ):=S_{C_{k},c_{k}}(p,\omega )\in {\widehat {H}}^{2k-1}(B;\mathbb {C} /\mathbb {Z} )}$.

In the case of flat bundles , you get the Cheeger-Chern-Simons classes ${\displaystyle p\colon E\to B}$

${\displaystyle {\hat {c}}_{k}(p)\in H^{2k-1}(B;\mathbb {C} /\mathbb {Z} )}$.

If it is a -dimensional closed orientable manifold, one obtains the Cheeger-Chern-Simons invariant ${\displaystyle B}$ ${\displaystyle (2k-1)}$

${\displaystyle CCS(p):=\langle {\hat {c}}_{k}(p),\left[M\right]\rangle \in \mathbb {C} /\mathbb {Z} }$

of the shallow bundle by applying the Cheeger-Chern-Simons class to the fundamental class . ${\displaystyle p\colon E\to B}$ ${\displaystyle \left[M\right]\in H_{2k-1}(M;\mathbb {Z} )}$

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.

Web links

• Chapter 3.2 in Bucher: Characteristic classes and bounded cohomology