= Mod-p cup-square obstruction to a homotopy equivalence
{title2=$u^2\ne0$}
Two finite <CW complexes> can have isomorphic integral <cohomology rings> but different cohomology rings modulo $p$. For example, attaching a three-cell of degree $p$ to the projective line in $\mathbb{CP}^2$ preserves the nonzero square of the degree-two class modulo $p$, whereas the corresponding <Moore space> wedged with $S^4$ has zero square. The spaces therefore cannot be <homotopy equivalent>.
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