Product of two spheres
= Product of two spheres
{title2=$S^p\times S^q$}
The <Cartesian product> $S^p\times S^q$ has one cell in each of dimensions $0,p,q,p+q$, counting the two middle cells separately when $p=q$. For positive $p,q$, its integral <cohomology ring> has generators $a,b$ of degrees $p,q$, with $a^2=b^2=0$ and $ba=(-1)^{pq}ab$. The class $ab$ is a product orientation class. This follows from the <Künneth theorem>.