algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic 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 $L$:
See the references at Chern class and characteristic class.
Last revised on January 26, 2021 at 08:53:42. See the history of this page for a list of all contributions to it.