Estimating the higher symmetric topological complexity of spheres

Karasev, Roman; Landweber, Peter

doi:10.2140/agt.2012.12.75

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Estimating the higher symmetric topological complexity of spheres

Roman Karasev and Peter Landweber

Algebraic & Geometric Topology 12 (2012) 75–94

DOI: 10.2140/agt.2012.12.75

Abstract

We study questions of the following type: Can one assign continuously and $Σ_{m}$ –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 $f : S^{2 n - 1} \to S^{n}$ by means of the mapping cone and the cup product.

Keywords

topological complexity, configuration spaces

Mathematical Subject Classification 2010

Primary: 55R80, 55R91

References

Publication

Received: 22 July 2011

Accepted: 1 November 2011

Published: 8 February 2012

Authors

Roman Karasev
Department of Mathematics Moscow Institute of Physics and Technology Institutskiy per. 9 Dolgoprudny Russia 141700	Laboratory of Discrete and Computational Geometry Yaroslavl’ State University Sovetskaya st. 14 Yaroslavl’ Russia 150000
http://www.rkarasev.ru/en/
Peter Landweber
Department of Mathematics Rutgers University Piscataway NJ 08854 USA

Volume 12, issue 1 (2012)