2007/06/17 by Michael Färber, Farber, Michael, Mark Grant +1 · 4 citations
Computer Science · Mathematics · #55M99 #68T40 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Optimization and Control (math.OC) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.0706.2497
openalex publication_date 2007/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.