### 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.

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

Pages (from-to) | 3339-3349 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 136 |

Issue number | 9 |

Early online date | 25 Apr 2008 |

DOIs | |

Publication status | Published - 2008 |

### Keywords

- topological complexity
- weights of cohomology classes
- category weight
- cohomology operations
- lens spaces
- topological robotics
- instabilities

### Cite this

**Robot motion planning, weights of cohomology classes, and cohomology operations.** / Farber, Michael; Grant, Mark.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 136, no. 9, pp. 3339-3349. https://doi.org/10.1090/S0002-9939-08-09529-4#sthash.ulDhWxF6.dpuf

}

TY - JOUR

T1 - Robot motion planning, weights of cohomology classes, and cohomology operations

AU - Farber, Michael

AU - Grant, Mark

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - topological complexity

KW - weights of cohomology classes

KW - category weight

KW - cohomology operations

KW - lens spaces

KW - topological robotics

KW - instabilities

U2 - 10.1090/S0002-9939-08-09529-4#sthash.ulDhWxF6.dpuf

DO - 10.1090/S0002-9939-08-09529-4#sthash.ulDhWxF6.dpuf

M3 - Article

VL - 136

SP - 3339

EP - 3349

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 9

ER -