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 $GL(n,\mathbb {C} )$ with coefficients in $\mathbb {Z}$.

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

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

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

Related notions

• Stiefel-Whitney class