cohomology

# Contents

## Idea

The Chern classes are the integral characteristic classes

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

of the classifying space $BU$ 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 $n\ge 1$ the Chern universal characteristic classes ${c}_{i}\in {H}^{2i}\left(BU\left(n\right),ℤ\right)$ of the classifying space $BU\left(n\right)$ of the unitary group are characterized as follows:

1. ${c}_{0}=1$ and ${c}_{i}=0$ if $i>n$;

2. for $n=1$, ${c}_{1}$ is the canonical generator of ${H}^{2}\left(BU\left(1\right),ℤ\right)\simeq ℤ$;

3. under pullback along the inclusion $i:BU\left(n\right)\to BU\left(n+1\right)$ we have ${i}^{*}{c}_{i}^{\left(n+1\right)}={c}_{i}^{\left(n\right)}$;

4. under the inclusion $BU\left(k\right)×BU\left(l\right)\to BU\left(k+l\right)$ we have ${i}^{*}{c}_{i}={\sum }_{j=0}^{i}{c}_{i}\cup {c}_{j-i}$.

## Properties

### General

###### Proposition

The cohomology ring of $BU\left(n\right)$ is the polynomial algebra on the Chern classes:

${H}^{•}\left(BU\left(n\right),ℤ\right)\simeq ℤ\left({c}_{1},\cdots ,{c}_{n}\right)\phantom{\rule{thinmathspace}{0ex}}.$H^\bullet(B U(n), \mathbb{Z}) \simeq \mathbb{Z}(c_1, \cdots, c_n) \,.

### First Chern class

• The first Chern class of a bundle $P$ is the class of its determinant line bundle $\mathrm{det}P$

${c}_{1}\left(P\right)=\left[\mathrm{det}P\right]\phantom{\rule{thinmathspace}{0ex}}.$c_1(P) = [det P] \,.

See determinant line bundle for more.

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)