Robot motion planning, weights of cohomology classes, and cohomology operations
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.
Keywords
Cite
@article{arxiv.0706.2497,
title = {Robot motion planning, weights of cohomology classes, and cohomology operations},
author = {Michael Farber and Mark Grant},
journal= {arXiv preprint arXiv:0706.2497},
year = {2007}
}
Comments
10 pages; Revised version (changes only in section 5)