Chern 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