Estimating the higher symmetric topological complexity of spheres

TL;DR

This paper investigates the higher symmetric topological complexity of spheres, showing that certain continuous, symmetric multipaths spanning m-tuples of points on spheres do not exist in all cases, using methods related to Hopf invariants.

Contribution

It provides new results on the non-existence of symmetric multipaths for spheres, advancing understanding of higher symmetric topological complexity.

Findings

Answer is negative in all cases handled

Uses Hopf invariant and mapping cone techniques

Contributes to the theory of symmetric topological complexity

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 \emph{multipath} is a continuous map of the wedge of $m$ segments to the sphere. This question is connected with the \emph{higher symmetric topological complexity} of spheres, introduced and studied by I. Basabe, J. Gonz\'alez, 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.