= Cup square after a diagonal sphere attachment
{title2=$u^2=-2v$}
For even $n\ge2$, attach an $(n+1)$-cell to $S^n\times S^n$ along the diagonal. The <cellular chain complex> has boundary $(1,1)$ for the new cell. The degree-$n$ <cohomology> generator restricts to $a-b$, and the top generator restricts to $ab$. Naturality of the <cup product> therefore gives $u^2=-2v$, since $(a-b)^2=-2ab$. Thus the resulting <cohomology ring> is $\mathbb Z[u,v]/(u^2+2v,uv,v^2)$, with $|u|=n$ and $|v|=2n$. This distinguishes the attachment from a wedge of spheres, whose positive-degree products vanish.
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