Cohomology operation

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This article is about a general term. A list of important particular cases (instances) is available at Category:Cohomology operations

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