### Abstract

In this paper we study symmetric motion planning algorithms, i.e. such that the motion from one state A to another B, prescribed by the algorithm, is the time reverse of the motion from B to A. We experiment with several different notions of topological complexity of such algorithms and compare them with each other and with the usual (non-symmetric) concept of topological complexity. Using equivariant cohomology and the theory of Schwarz genus we obtain cohomological lower bounds for symmetric topological complexity. One of our main results states that, in the case of aspherical manifolds, the complexity of symmetric motion planning algorithms with fixed midpoint map exceeds twice the cup-length.

We introduce a new concept, the sectional category weight of a cohomology class, which generalises the notion of category weight developed earlier by E. Fadell and S. Husseini. We apply this notion to study the symmetric topological complexity of aspherical manifolds.

Original language | English |
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Title of host publication | Topology and Robotics |

Editors | M Farber, R Ghrist, M Burger, D Koditschek |

Place of Publication | Providence |

Publisher | American Mathematical Society |

Pages | 85-104 |

Number of pages | 20 |

ISBN (Print) | 978-0-8218-4246-1 |

Publication status | Published - 2007 |

Event | Workshop on Topology and Robotics - Zurich, Switzerland Duration: 10 Jul 2006 → 14 Jul 2006 |

### Publication series

Name | Contemporary Mathematics Series |
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Publisher | American Mathematical Society |

Volume | 438 |

ISSN (Print) | 0271-4132 |

### Conference

Conference | Workshop on Topology and Robotics |
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Country | Switzerland |

Period | 10/07/06 → 14/07/06 |

### Keywords

- Motion planning
- equivariant cohomology
- Schwarz genus

### Cite this

*Topology and Robotics*(pp. 85-104). (Contemporary Mathematics Series; Vol. 438). Providence: American Mathematical Society.