nLab
Chern class

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Chern classes are the integral characteristic classes

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

of the classifying space BUB U of the unitary group.

Accordingly these are characteristic classes of UU-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 n1n \geq 1 the Chern universal characteristic classes c iH 2i(BU(n),)c_i \in H^{2i}(B U(n), \mathbb{Z}) of the classifying space BU(n)B U(n) of the unitary group are characterized as follows:

  1. c 0=1c_0 = 1 and c i=0c_i = 0 if i>ni \gt n;

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

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

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

Properties

General

Proposition

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

H (BU(n),)(c 1,,c n). H^\bullet(B U(n), \mathbb{Z}) \simeq \mathbb{Z}(c_1, \cdots, c_n) \,.

First Chern class

Splitting principle and Chern roots

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

Write i:TU(1) nU(n)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 (BT,)H (BU(1) n,) H^\bullet(B T, \mathbb{Z}) \simeq H^\bullet(B U(1)^n, \mathbb{Z})

is the polynomial ring in nn generators (to be thought of as the universal first Chern classes c ic_i of each copy of BU(1)B U(1); equivalently as the weights of the group characters of U(n)U(n)) which are traditionally written x ix_i:

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

Write

Bi:BU(1) nBU(n) B i \;\colon\; B U(1)^n \to B U(n)

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

(Bi) *(c k)=σ k(x 1,,x n). (B i)^\ast (c_k) = \sigma_k(x_1, \cdots, x_n) \,.

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

(Bi) *(c)=(1+x 1)(1+x 2)(1+x n). (B i)^\ast (c) = (1+x_1) (1+ x_2) \cdots (1+ x_n) \,.

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

See also at splitting principle – Examples – Complex vector bundles and their 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)