Cohomology ring of a product of two spheres
= Cohomology ring of a product of two spheres
{title2=$H^*(S^n\times S^n;\mathbb Z)$}
For positive $n$, let $x,y$ be the pullbacks of the orientation classes from the two factors. Then $x^2=y^2=0$, $xy$ generates degree $2n$, and $yx=(-1)^nxy$. When $n$ is even, every graded ring automorphism restricts on $H^n$ to a signed permutation matrix; when $n$ is odd the ring admits every matrix in $GL_2(\mathbb Z)$.