 Diagonal-degree obstruction to a symmetric sphere retraction

For $d\ge 1$ 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 $f^{\ast }\alpha =a{\sum }_i{u}_i$ 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.