### 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.

**Symmetric Motion Planning.** / Farber, Michael; Grant, Mark.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Topology and Robotics.*Contemporary Mathematics Series, vol. 438, American Mathematical Society, Providence, pp. 85-104, Workshop on Topology and Robotics, Switzerland, 10/07/06.

}

TY - GEN

T1 - Symmetric Motion Planning

AU - Farber, Michael

AU - Grant, Mark

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

KW - Motion planning

KW - equivariant cohomology

KW - Schwarz genus

M3 - Conference contribution

SN - 978-0-8218-4246-1

T3 - Contemporary Mathematics Series

SP - 85

EP - 104

BT - Topology and Robotics

A2 - Farber, M

A2 - Ghrist, R

A2 - Burger, M

A2 - Koditschek, D

PB - American Mathematical Society

CY - Providence

ER -