This article defines a characteristic class
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.