We study questions of the following type: Can one assign continuously and –equivariantly to any –tuple of distinct points on the sphere a multipath in spanning these points? A multipath is a continuous map of the wedge of 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 by means of the mapping cone and the cup product.
"Estimating the higher symmetric topological complexity of spheres." Algebr. Geom. Topol. 12 (1) 75 - 94, 2012. https://doi.org/10.2140/agt.2012.12.75