nLab cohomology ring

Idea

Given any generalized (Eilenberg-Steenrod) cohomology theory $E$, then for each topological space $X$, there is, by definition, the graded abelian group

$$E^\bullet(X) \in Ab^{\mathbb{Z}} \,.$$

This is the $E$-cohomology group of $X$. Now if $E$ is a multiplicative cohomology theory, then these groups inherit the structure of rings. As such

$$E^\bullet(X) \in Ring^{\mathbb{Z}}$$

is the $E$-cohomology ring of $X$.

Analogously for various suitable generalizations of the nature of $E$ and $X$ (see at generalized cohomology).

Related concepts

• genus

• orientation in generalized cohomology