nLab
Chern class

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The ordinary 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 in ordinary cohomology of U-principal bundles and hence of complex vector bundle

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.

More generally, there are generalized Chern classes for any complex oriented cohomology theory (Adams 74, Lurie 10).

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

Original articles include

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

Textbook accounts include

A brief introduction is in chapter 23, section 7

For Conner-Floyd Chern classes in complex oriented cohomology theory:

Revised on February 17, 2016 04:35:16 by Urs Schreiber (195.37.209.180)