Cross product

This article describes a binary operation involving the cohomology groups of one or more topological spaces

Definition

Let $X,Y$ be topological spaces and $R$ a commutative ring. The cross product or external cup product is a bilinear map given by:

$H^{i}(X;R)\times H^{j}(Y;R)\to H^{i+j}(X\times Y;R)$

Equivalently it can be viewed as a linear map:

$H^{i}(X;R)\otimes H^{j}(Y;R)\to H^{i+j}(X\times Y;R)$

The map is defined as follows:

$a\times b=p_{1}^{*}(a)\cup p_{2}^{*}(b)$

where $p_{1},p_{2}$ are the projections from $X\times Y$ to $X$ and to $Y$, and where $\cup$ is defined as the usual cup product.

The cross product also has a relative version. Let $(X,A)$ and $(Y,B)$ be two pairs of topological spaces. The cross product then gives a map:

$H^{i}(X,A;R)\times H^{j}(Y,B;R)\to H^{i+j}(X\times Y,X\times B\cup A\times Y;R)$

again defined in the same way:

$a\times b=p_{1}^{*}(a)\cup p_{2}^{*}(b)$

where $p_{1},p_{2}$ are the projections.