Symmetric Bi-Skew Maps and Symmetrized Motion Planning in Projective Spaces

Abstract

AbstractThis work is motivated by the question of whether there are spacesXfor which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space${\open R}\hbox{P}^m$, it is known that$\hbox{TC}^S ({\open R}\hbox{P}^{m})$captures, with a few potential exceptional cases, the Euclidean embedding dimension of${\open R}\hbox{P}^{m}$. We now show that, for allm≥1,$\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$is characterized as the smallest positive integernfor which there is a symmetric${\open Z}_{2}$-biequivariant mapSm×Sm→Snwith 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${\open R}\hbox{P}^{2^{e}}$fore≥1. In particular, this leaves the torusS1×S1as the only closed surface whose symmetric (symmetrized) TCS(TCΣ) invariant is currently unknown.