Products in algebraic topology

In algebraic topology, several types of products are defined on homological and cohomological theories.

The cross product
$$H_p(X) \otimes H_q(Y) \to H_{p+q}(X\times Y)$$

The cap product

 * $$\frown\ : H_p(X;R)\times H^q(X;R) \rightarrow H_{p-q}(X;R)$$

The slant product

 * $$\backslash\ : H_p(X;R)\times H^q(X\times Y;R) \rightarrow H^{q-p}(Y;R)$$

The cup product

 * $$H^p(X) \otimes H^q(X) \to H^{p+q}(X)$$

This product can be understood as induced by the exterior product of differential forms in de Rham cohomology. It makes the singular cohomology of a connected manifold into a unitary supercommutative ring.