Symmetric bi-skew maps and symmetrized motion planning in projective spaces

This work is motivated by the question of whether there are spaces $X$ for which the Farber-Grant symmetric topological complexity $TC^S(X)$ differs from the Basabe-González-Rudyak-Tamaki symmetric topological complexity $TC^Σ(X)$. It is known that, for a projective space $RP^m$, $TC^S(RP^m)$ captures, with a few potentially exceptional cases, the Euclidean embedding dimension of $RP^m$. We now show that, for all $m\geq1$, $TC^Σ(RP^m)$ is characterized as the smallest positive integer $n$ for which there is a symmetric $\mathbb{Z}_2$-biequivariant map $S^m\times S^m\to S^n$ with a "monoidal" behavior on the diagonal. This result thus lies at the core of the efforts in the 1970's to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both $TC$ numbers in the case of $RP^{2^e}$ for $e\geq1$. In particular, this leaves the torus $S^1\times S^1$ as the only closed surface whose symmetric (symmetrized) $TC^S$ ($TC^Σ$) -invariant is currently unknown.