group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Chern classes are the integral characteristic classes
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.
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:
$c_0 = 1$ and $c_i = 0$ if $i \gt n$;
for $n = 1$, $c_1$ is the canonical generator of $H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z}$;
under pullback along the inclusion $i : B U(n) \to B U(n+1)$ we have $i^* c_i^{(n+1)} = c_i^{(n)}$;
under the inclusion $B U(k) \times B U(l) \to B U(k+l)$ we have $i^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}$.
The cohomology ring of $B U(n)$ is the polynomial algebra on the Chern classes:
The first Chern class of a bundle $P$ is the class of its determinant line bundle $det P$
See determinant line bundle for more.
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
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 $x_i$:
Write
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:
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
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.
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 .
Standard textbook references include
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