# cup product

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

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

.

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

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

As $\Delta^{*}$ and $h$ are chain maps, $\smile$ 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 $X$.

Title cup product CupProduct 2013-03-22 15:37:42 2013-03-22 15:37:42 whm22 (2009) whm22 (2009) 7 whm22 (2009) Definition msc 55N45