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Topological complexity of collision-free motion planning on surfaces

Part of: Special aspects of infinite or finite groups Fiber spaces and bundles Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  27 September 2010

Daniel C. Cohen  and
Michael Farber
Daniel C. Cohen
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA (email: cohen@math.lsu.edu)
Michael Farber
Affiliation:
Department of Mathematics, University of Durham, Durham DH1 3LE, UK (email: Michael.Farber@durham.ac.uk)
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Abstract

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The topological complexity is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant measures the complexity of motion planning algorithms which can be designed for the system. In this paper, we compute the topological complexity of the configuration space of n distinct ordered points on an orientable surface, for both closed and punctured surfaces. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties. For configuration spaces of punctured surfaces, this is used in conjunction with techniques from the theory of mixed Hodge structures.

Keywords

surface configuration spacetopological complexity

MSC classification

Secondary: 55R80: Discriminantal varieties, configuration spaces 20F36: Braid groups; Artin groups 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 649 - 660
Copyright
Copyright © Foundation Compositio Mathematica 2010

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