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 TCS(X) differs from the Basabe-Gonzalez-Rudyak-Tamalci symmetric topological complexity TC Sigma(X). For a projective space RPm, it is known that TCS (RPm) captures, with a few potential exceptional cases, the Euclidean embedding dimension of RPm. We now show that, for all m >= 1, TC Sigma (RPm) is characterized as the smallest positive integer n for which there is a symmetric Z(2)-biequivariant map S-m x S-m -> 4 S-n with a 'monoidal' behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s 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 RP2e for e >= 1. In particular, this leaves the torus S-1 x S-1 as the only closed surface whose symmetric (symmetrized) TCS (TC Sigma) invariant is currently unknown.