= Diagonal-degree obstruction to a symmetric sphere retraction
{title2=$na=1$}
For $d\geq1$ and $n>1$, no factor-permutation-invariant map $f:(S^d)^n\to S^d$ restricts to the identity on the diagonal. The <Künneth theorem> makes the degree-$d$ <cohomology> a direct sum of $n$ copies of $\mathbb Z$. Symmetry forces $f^*\alpha=a\sum_i u_i$ for one integer $a$. Pulling back to the diagonal multiplies this coefficient by $n$, so the identity restriction would give $na=1$. The contradiction uses integral coefficients and applies in both even and odd sphere dimensions.
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