nLab
Chern class

Idea

The Chern classes are the integral characteristic classes

$$c_i:BU\to B^{2i}\mathbb{Z}$$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 $c_i\in H^{2i}(BU(n),\mathbb{Z})$ of the classifying space $BU(n)$ 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 $H^2(BU(1),\mathbb{Z})\simeq \mathbb{Z}$;

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

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

Properties

General

Proposition

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

$$H^{•}(BU(n),\mathbb{Z})\simeq \mathbb{Z}(c_1,\cdots ,c_n).$$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 $\det P$

  $$c_1(P)=[\det P].$$c_1(P) = [det P] \,.

  See determinant line bundle for more.

Related concepts

• Pontryagin class

• Stiefel-Whitney class

• Chern character

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

• John Milnor, James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press (1974).

or chapter IX of volume II of

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

• A. Grothendieck, La théorie des classes de Chern, Bulletin de la Société Mathématique de France 86 (1958), p. 137–154, numdam

A brief introduction is in chapter 23, section 7

• Peter May, A concise course in algebraic topology (pdf)