Cohomology operation

Definition
A cohomology operation $$\theta$$ of type $$(n,q;\pi,G)$$ where $$n,q$$ are integers and $$\pi$$ and $$G$$ are Abelian groups, is a natural transformation of the cohomology functors (treated as functors to sets):

$$\theta: H^n(-,\pi) \to H^q(-,G)$$

restricted to CW-spaces.

Note that the map $$\theta$$ is not required to be a group homomorphism, because the cohomology functors are viewed as set-valued functors.

By convention:


 * If only one Abelian group is specified, we take it to be the group for both sides
 * If no Abelian group is specified, we take both groups to be $$\mathbb{Z}$$

A cohomology operation is equivalent to specifying a group homomorphism between the Eilenberg-Maclane spaces:

$$K(\pi,n) \to K(G,q)$$

Related notions

 * Stable cohomology operation
 * Binary operation on cohomology