Chern class

This article defines a characteristic class

Definition

The Chern class is a characteristic class (or collection of characteristic classes in different dimensions) for the topological group $G L (n, C)$ with coefficients in $Z$ .

Axiomatically, the Chern class can be defined as associating to every complex vector bundle $p : E \to B$ a class $c (E) \in H^{*} (B; Z)$ which lives only in even degrees, such that if $c_{i} (E)$ denotes the component of $c (E)$ in the $(2 i)^{t h}$ graded component, the following hold:

$c_{i} (f^{*} (E)) = f^{*} (c_{i} (E))$ (this is the condition for being a natural transformation, part of the definition of characteristic class)
$c (E_{1} \oplus E_{2}) = c (E_{1}) ⌣ c (E_{2})$ where $⌣$ denotes the cap product. This is a Whitney sum formula
$c_{i} (E) = 0$ if $i$ is greater than the dimension of $E$
For the canonical complex line bundle $E \to C P^{\infty}$ , $c_{1} (E)$ is a pre-specified generator of $H^{2} (C P^{\infty}; Z)$

$c$ is termed the total Chern class and $c_{i}$ is termed the $i^{t h}$ Chern class.

Related notions

Stiefel-Whitney class