Volume 12, issue 1 (2012)

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Estimating the higher symmetric topological complexity of spheres

Roman Karasev and Peter Landweber

Algebraic & Geometric Topology 12 (2012) 75–94
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.

Keywords
topological complexity, configuration spaces
Mathematical Subject Classification 2010
Primary: 55R80, 55R91