Estimating the higher symmetric topological complexity of spheres

Details Estimating the higher symmetric topological complexity of spheres Estimating the higher symmetric topological complexity of spheres

2011-06-08

arxiv.org/abs/1106.1549v3

Algebraic & Geometric Topology 12:1 (2012), 75-94

Mathematics

Algebraic Topology

This version has minor corrections compared to what published in AGT

Scientific paper

10.2140/agt.2012.12.75

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.

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