Let $X$ be a finite CW complex, and $A$ an abelian group. Given a nonzero class in singular cohomology $$x\in H^k(X;A),$$ when is its cross product square $$x\times x\in H^{2k}(X\times X;A\otimes A)$$ nonzero?

*Remarks:* The tensor product is over $\mathbb{Z}$. By the Künneth theorem $x\times x$ is always nonzero if $A$ is a field, or more generally if $A$ is a domain and $H^\ast(X)$ is flat over $A$. Examples where $x\times x=0$ arise in the study of [Lusternik-Schnirelmann category][1], in particular in spaces $X$ for which $\operatorname{cat}(X\times X)<2\operatorname{cat}(X)$. 


  [1]: http://en.wikipedia.org/wiki/Lusternik%25E2%2580%2593Schnirelmann_category