group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The first of the Chern classes. The unique characteristic class of circle bundles / complex line bundles.
As a universal characteristic class, the first Chern class is the weak homotopy equivalence
In complex analytic geometry consider the exponential exact sequence
For any complex analytic space $X$ this induces the long exact sequence in cohomology with connecting homomorphism
This is the first Chern-class map. It sends a holomorphic line bundle ($H^1(X,\mathbb{G}^\times)$ is the Picard group of $X$) to an integral cohomology class.
If $D$ is a divisor in $X$, then $c_1(\mathcal{O}_X(D))$ is the Poincaré dual of the fundamental class of $D$ (e.g. Huybrechts 04, prop. 4.4.13).
Over a Riemann surface $\Sigma$ the evaluation of the Chern class $c_1(L)$ of a holomorphic line bundle $L$ on a fundamental class is the degree of $L$: