cup product

Let $X$ be a topological space and $R$ be a commutative ring. The diagonal map $\mathrm{\Delta }:X\to X\times X$ induces a chain map between singular cochain complexes:

$${\mathrm{\Delta }}^{*}:C^{*}(X\times X;R)\to C^{*}(X;R)$$

.

Let $h:C^{*}(X;R)\otimes C^{*}(X;R)\to C^{*}(X\times X;R)$

denote the chain homotopy equivalence associated with the Kunneth .

Given $\alpha \in C^p(X;R)$ and $\beta \in C^q(X;R)$ we define

$\alpha ⌣\beta ={\mathrm{\Delta }}^{*}h(\alpha \otimes \beta )$.

As ${\mathrm{\Delta }}^{*}$ and $h$ are chain maps, $⌣$ induces a well defined product on cohomology groups, known as the cup product. Hence the direct sum of the cohomology groups of $X$ has the structure of a ring. This is called the cohomology ring of $X$.