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edit approved Apr 30 at 13:32
The Amplitwist
edit approved Apr 30 at 13:32
The Amplitwist
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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 categoryLusternik-Schnirelmann category, in particular in spaces $X$ for which $\operatorname{cat}(X\times X)<2\operatorname{cat}(X)$.

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, in particular in spaces $X$ for which $\operatorname{cat}(X\times X)<2\operatorname{cat}(X)$.

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, in particular in spaces $X$ for which $\operatorname{cat}(X\times X)<2\operatorname{cat}(X)$.

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Mark Grant
Mark Grant
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When is the cohomology cross product square nonzero?

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, in particular in spaces $X$ for which $\operatorname{cat}(X\times X)<2\operatorname{cat}(X)$.