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Topological complexity of motion planning in projective product spaces

Jesús González, Mark Grant, Enrique Torres-Giese and Miguel Xicoténcatl

Algebraic & Geometric Topology 13 (2013) 1027–1047

We study Farber’s topological complexity (TC) of Davis’ projective product spaces (PPS’s). We show that, in many nontrivial instances, the TC of PPS’s coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in marked contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of nonsimply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension at most 2, and now in this work for infinite families of PPS’s. We discuss general bounds for the TC (and the Lusternik–Schnirelmann category) of PPS’s, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS’s through a generalized concept of axial maps or, alternatively (in an appendix), nonsingular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS’s.

topological complexity, projective product spaces, Euclidean immersions of manifolds, generalized axial maps, equivariant motion planning
Mathematical Subject Classification 2010
Primary: 55M30, 57R42
Secondary: 68T40
Received: 8 August 2012
Revised: 18 November 2012
Accepted: 26 November 2012
Published: 8 April 2013
Jesús González
Departamento de Matemáticas
Centro de Investigación y de Estudios Avanzados del IPN
A.P. 14-740
México City 07000
Mark Grant
School of Mathematical Sciences
The University of Nottingham
University Park
Nottingham NG7 2RD
Enrique Torres-Giese
Departamento de Matemáticas
Universidad de Guanajuato
Guanajuato, Gto 36000
Miguel Xicoténcatl
Departamento de Matemáticas
Centro de Investigación y de Estudios Avanzados del IPN
México City 07000