Estimating the higher symmetric topological complexity of spheres

Abstract

We study questions of the following type: Can one assign continuously and ${\Sigma }_m$–equivariantly to any $m$–tuple of distinct points on the sphere $S^n$ a multipath in $S^n$ spanning these points? A multipath is a continuous map of the wedge of $m$ segments to the sphere. This question is connected with the higher symmetric topological complexity of spheres, introduced and studied by I Basabe, J González, Yu B Rudyak, and D Tamaki. In all cases we can handle, the answer is negative. Our arguments are in the spirit of the definition of the Hopf invariant of a map $f:S^{2n-1}\to S^n$ by means of the mapping cone and the cup product.