Cross product

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.