# Chern class

From Topospaces

*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 with coefficients in .

Axiomatically, the Chern class can be defined as associating to every complex vector bundle a class which lives only in even degrees, such that if denotes the component of in the graded component, the following hold:

- (this is the condition for being a natural transformation, part of the definition of characteristic class)
- where denotes the cap product. This is a Whitney sum formula
- if is greater than the dimension of
- For the canonical complex line bundle , is a pre-specified generator of

is termed the **total Chern class** and is termed the Chern class.