### Abstract

The topological complexity 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 tire topological complexity of the configuration space of n distinct points in Euclidean m-space for all m >= 2 and n >= 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.

Original language | English |
---|---|

Pages (from-to) | 1841-1847 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 5 |

Early online date | 29 Dec 2008 |

DOIs | |

Publication status | Published - May 2009 |

### Keywords

- topological complexity
- configuration spaces
- robot motion

### Cite this

*Proceedings of the American Mathematical Society*,

*137*(5), 1841-1847. https://doi.org/10.1090/S0002-9939-08-09808-0

**Topological Complexity of Configuration Spaces.** / Farber, Michael; Grant, Mark.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 137, no. 5, pp. 1841-1847. https://doi.org/10.1090/S0002-9939-08-09808-0

}

TY - JOUR

T1 - Topological Complexity of Configuration Spaces

AU - Farber, Michael

AU - Grant, Mark

N1 - This research was supported by grants from the EPSRC and from The Royal Society

PY - 2009/5

Y1 - 2009/5

N2 - The topological complexity 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 tire topological complexity of the configuration space of n distinct points in Euclidean m-space for all m >= 2 and n >= 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.

AB - The topological complexity 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 tire topological complexity of the configuration space of n distinct points in Euclidean m-space for all m >= 2 and n >= 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.

KW - topological complexity

KW - configuration spaces

KW - robot motion

U2 - 10.1090/S0002-9939-08-09808-0

DO - 10.1090/S0002-9939-08-09808-0

M3 - Article

VL - 137

SP - 1841

EP - 1847

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 5

ER -