cohomology

# 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:

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

### Splitting principle and Chern roots

Under the splitting principle all Chern classes are determnined by first Chern classes:

Write $i \colon T \simeq U(1)^n \hookrightarrow U(n)$ for the maximal torus inside the unitary group, which is the subgroup of diagonal unitary matrices. Then

$H^\bullet(B T, \mathbb{Z}) \simeq H^\bullet(B U(1)^n, \mathbb{Z})$

is the polynomial ring in $n$ generators (to be thought of as the universal first Chern classes $c_i$ of each copy of $B U(1)$; equivalently as the weights of the group characters of $U(n)$) which are traditionally written $x_i$:

$H^\bullet(B U(1)^n, \mathbb{Z}) \simeq \mathbb{Z}[x_1, \cdots, x_n] \,.$

Write

$B i \;\colon\; B U(1)^n \to B U(n)$

for the induced map of deloopings/classifying spaces, then the $k$-universal Chern class $c_k \in H^{2k}(B U(n), \mathbb{Z})$ is uniquely characterized by the fact that its pullbacl to $B U(1)^n$ is the $k$th elementary symmetric polynomial $\sigma_k$ applied to these first Chern classes:

$(B i)^\ast (c_k) = \sigma_k(x_1, \cdots, x_n) \,.$

Equivalently, for $c = \sum_{i = 1}^n c_k$ the formal sum of all the Chern classes, and using the fact that the elementary symmetric polynomials $\sigma_k(x_1, \cdots, k_n)$ are the degree-$k$ piece in $(1+x_1) \cdots (1+x_n)$, this means that

$(B i)^\ast (c) = (1+x_1) (1+ x_2) \cdots (1+ x_n) \,.$

Since here on the right the first Chern classes $x_i$ appear as the roots of the Chern polynomial, they are also called 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 .

## References

Standard textbook references include

or chapter IX of volume II of

A brief introduction is in chapter 23, section 7

Revised on March 29, 2014 08:28:14 by Urs Schreiber (89.204.139.127)