Topological complexity of configuration spaces
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- by Michael Farber and Mark Grant PDF
- Proc. Amer. Math. Soc. 137 (2009), 1841-1847 Request permission
Abstract:
The topological complexity $\mathsf {TC}(X)$ is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space $X$, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of $n$ distinct points in Euclidean $m$-space for all $m\ge 2$ and $n\ge 2$; the answer was previously known in the cases $m=2$ and $m$ odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.References
- Daniel C. Cohen and Goderdzi Pruidze, Motion planning in tori, Bull. Lond. Math. Soc. 40 (2008), no. 2, 249–262. MR 2414784, DOI 10.1112/blms/bdn005
- D. C. Cohen and G. Pruidze, Topological complexity of basis-conjugating automorphism groups, preprint. [arXiv:0804.1825]
- A. Costa and M. Farber, Motion planning in spaces with “small” fundamental groups, preprint, 2008.
- Samuel Eilenberg and Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups, Ann. of Math. (2) 65 (1957), 517–518. MR 85510, DOI 10.2307/1970062
- Edward R. Fadell and Sufian Y. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. MR 1802644, DOI 10.1007/978-3-642-56446-8
- Michael Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221. MR 1957228, DOI 10.1007/s00454-002-0760-9
- Michael Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2-3, 245–266. MR 2074919, DOI 10.1016/j.topol.2003.07.011
- Michael Farber and Sergey Yuzvinsky, Topological robotics: subspace arrangements and collision free motion planning, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 212, Amer. Math. Soc., Providence, RI, 2004, pp. 145–156. MR 2070052, DOI 10.1090/trans2/212/07
- Michael Farber, Topology of robot motion planning, Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, Dordrecht, 2006, pp. 185–230. MR 2276952, DOI 10.1007/1-4020-4266-3_{0}5
- Michael Farber and Mark Grant, Robot motion planning, weights of cohomology classes, and cohomology operations, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3339–3349. MR 2407101, DOI 10.1090/S0002-9939-08-09529-4
- M. Farber, Invitation to Topological Robotics, EMS, 2008.
- Jesús González, Topological robotics in lens spaces, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 3, 469–485. MR 2177172, DOI 10.1017/S030500410500873X
- M. Grant, Topological complexity of motion planning and Massey products, to appear in Proceedings of the M. M. Postnikov Memorial Conference 2007, Banach Centre Publications [arxiv:0709.2287].
- J.-C. Latombe, Robot Motion Planning, Kluwer, Dordrecht, 1991.
- A. S. Švarc, The genus of a fiber space, Amer. Math. Soc. Transl. (2) 55, Amer. Math. Soc., Providence, RI, 1966, 49–140.
- George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0
Additional Information
- Michael Farber
- Affiliation: Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, United Kingdom
- Email: michael.farber@durham.ac.uk
- Mark Grant
- Affiliation: School of Mathematics, The University of Edinburgh, King’s Buildings, Edinburgh, EH9 3JZ, United Kingdom
- MR Author ID: 794577
- Email: mark.grant@ed.ac.uk
- Received by editor(s): June 25, 2008
- Published electronically: December 29, 2008
- Additional Notes: This research was supported by grants from the EPSRC and from The Royal Society
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1841-1847
- MSC (2000): Primary 55M99, 55R80; Secondary 68T40
- DOI: https://doi.org/10.1090/S0002-9939-08-09808-0
- MathSciNet review: 2470845