Robot motion planning, weights of cohomology classes, and cohomology operations

Michael Farber*, Mark Grant

*Corresponding author for this work

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC( X) of the configuration space X of the system. Previously known lower bounds for TC( X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC( X). As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik-Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result.

Original languageEnglish
Pages (from-to)3339-3349
Number of pages11
JournalProceedings of the American Mathematical Society
Volume136
Issue number9
Early online date25 Apr 2008
DOIs
Publication statusPublished - 2008

Keywords

  • topological complexity
  • weights of cohomology classes
  • category weight
  • cohomology operations
  • lens spaces
  • topological robotics
  • instabilities

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