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 \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. 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 $f: S^{2n1} \to S^n$ by means of the mapping cone and the cup product.
 Publication:

arXiv eprints
 Pub Date:
 June 2011
 arXiv:
 arXiv:1106.1549
 Bibcode:
 2011arXiv1106.1549K
 Keywords:

 Mathematics  Algebraic Topology;
 55R80;
 55R91
 EPrint:
 This version has minor corrections compared to what published in AGT