= Cohomological obstruction to separately odd sphere multiplication
{title2=$n+1=2^k$}
Suppose a <continuous map> $g:S^n\times S^n\to S^n$ changes sign on negating either input. Its quotient $\bar g:\mathbb{RP}^n\times\mathbb{RP}^n\to\mathbb{RP}^n$ pulls back the <real tautological line bundle> to the <tensor product of vector bundles> $p_1^*\gamma_n\otimes p_2^*\gamma_n$. The fiber map sends $(s x)\otimes(t y)$ to $st g(x,y)$. The <First Stiefel–Whitney class of a tensor product of real line bundles> consequently gives $\bar g^*a=a_1+a_2$. The <Künneth theorem> and the <mod-two cohomology ring of real projective space> imply
$$
(a_1+a_2)^{n+1}=0\quad\text{in }\mathbb F_2[a_1,a_2]/(a_1^{n+1},a_2^{n+1}).
$$
Every interior <binomial coefficient> in row $n+1$ must be even. By <binomial coefficients with even interior terms>, $n=2^k-1$ for some $k\geq0$. This is a necessary condition, not an assertion of existence in every such dimension.
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