Topological complexity of motion planning in projective product spaces

Jesus Gonzalez*, Mark Grant, Enrique Torres-Giese, Miguel Xicotencatl

*Corresponding author for this work

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1027-1047
Number of pages21
JournalAlgebraic & Geometric Topology
Volume13
Issue number2
DOIs
Publication statusPublished - 8 Apr 2013

Keywords

  • Lusternik-Schnirelmann category
  • lens spaces
  • immersions
  • robotics
  • maps
  • topological complexity
  • projective product spaces
  • Euclidean immersions of manifolds
  • generalized axial maps
  • equivariant motion planning

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