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

References

Implementation of ordinary$\;$cohomology rings in cubical agda:

• Thomas Lamiaux, Axel Ljungström, Anders Mörtberg, Computing Cohomology Rings in Cubical Agda, CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs (2023) [arxiv:2212.04182, doi:10.1145/3573105.3575677]