Contents

Idea

The Chern classes are the integral characteristic classes

$c_i : B U \to B^{2 i} \mathbb{Z}$

of the classifying space $B U$ of the unitary group.

Accordingly these are characteristic classes of $U$-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.

Definition

Definition

For $n \geq 1$ the Chern universal characteristic classes $c_i \in H^{2i}(B U(n), \mathbb{Z})$ of the classifying space $B U(n)$ of the unitary group are characterized as follows:

1. $c_0 = 1$ and $c_i = 0$ if $i \gt n$;

2. for $n = 1$, $c_1$ is the canonical generator of $H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z}$;

3. under pullback along the inclusion $i : B U(n) \to B U(n+1)$ we have $i^* c_i^{(n+1)} = c_i^{(n)}$;

4. under the inclusion $B U(k) \times B U(l) \to B U(k+l)$ we have $i^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}$.

Properties

General

Proposition

The cohomology ring of $B U(n)$ is the polynomial algebra on the Chern classes:

$H^\bullet(B U(n), \mathbb{Z}) \simeq \mathbb{Z}(c_1, \cdots, c_n) \,.$

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 .

References

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 (94.136.12.233)