Geodesic
Estimating the higher symmetric topological complexity of spheres
Karasev, Roman1 ; Landweber, Peter2
1Department 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
2Department of Mathematics, Rutgers University, Piscataway NJ 08854, USA
Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 75-94
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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 : S2n−1 → Sn by means of the mapping cone and the cup product.

DOI : 10.2140/agt.2012.12.75
Classification : 55R80, 55R91
Keywords: topological complexity, configuration spaces

Karasev, Roman  1   ; Landweber, Peter  2

1 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
2 Department of Mathematics, Rutgers University, Piscataway NJ 08854, USA 
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Karasev, Roman; Landweber, Peter. Estimating the higher symmetric topological complexity of spheres. Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 75-94. doi: 10.2140/agt.2012.12.75

[1] I Basabe, J González, Y B Rudyak, D Tamaki, Higher topological complexity and homotopy dimension of configuration spaces on spheres

[2] H Cartan, Espaces avec groupes d’opérateurs I : Notions préliminaires, Séminaire Henri Cartan 3 (1950–1951)

[3] H Cartan, Espaces avec groupes d’opérateurs II : La suite spectrale ; applications, Séminaire Henri Cartan 3 (1950–1951)

[4] F R Cohen, L R Taylor, On the representation theory associated to the cohomology of configuration spaces, from: "Algebraic topology (Oaxtepec, 1991)", Contemp. Math. 146, Amer. Math. Soc. (1993) 91

[5] C Domínguez, J González, P Landweber, The integral cohomology of configuration spaces of pairs of points in real projective spaces, Forum Math. 24 (2012)

[6] E R Fadell, S Y Husseini, Geometry and topology of configuration spaces, , Springer (2001)

[7] M Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003) 211

[8] M Farber, Invitation to topological robotics, , European Mathematical Society (EMS), Zürich (2008)

[9] D Handel, On products in the cohomology of the dihedral groups, Tohoku Math. J. 45 (1993) 13

[10] N H V Hung, The mod 2 equivariant cohomology algebras of configuration spaces, Pacific J. Math. 143 (1990) 251

[11] I J Leary, On the integral cohomology of wreath products, J. Algebra 198 (1997) 184

[12] M Nakaoka, Homology of the infinite symmetric group, Ann. of Math. 73 (1961) 229

[13] Y B Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010) 916

[14] A S Schwarz, The genus of a fibered space, Trudy Moskov. Mat. Obšč. 10 (1961) 217

[15] G W Whitehead, Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37 (1962/1963) 239

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