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

We study questions of the following type: Can one assign continuously and Σm–equivariantly to any m–tuple of distinct points on the sphere Sn a multipath in Sn 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 : S2n1 Sn by means of the mapping cone and the cup product.

topological complexity, configuration spaces
Mathematical Subject Classification 2010
Primary: 55R80, 55R91
Received: 22 July 2011
Accepted: 1 November 2011
Published: 8 February 2012
Roman Karasev
Department of Mathematics
Moscow Institute of Physics and Technology
Institutskiy per. 9
Russia 141700
Laboratory of Discrete and Computational Geometry
Yaroslavl’ State University
Sovetskaya st. 14
Russia 150000
Peter Landweber
Department of Mathematics
Rutgers University
Piscataway NJ 08854