= Cohomology ring of a product of unequal-dimensional spheres
{title2=$H^*(S^p\times S^q;\mathbb Z)$}
For positive unequal dimensions $p,q$, let $u,v$ be the pullbacks of the integral orientation classes of the factors. The <Künneth theorem> gives the free abelian basis $1,u,v,uv$, with degrees $0,p,q,p+q$. The products satisfy $u^2=v^2=0$ and $vu=(-1)^{pq}uv$, by naturality and <graded commutativity of the cup product>. In particular, for $p=2,q=4$, the <cohomology ring> is $\mathbb Z[u,v]/(u^2,v^2)$, with degrees two and four.
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