### Abstract

We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces X whose higher topological complexity TCn (X) is as low as possible, namely n - 1.

Original language | English |
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Pages (from-to) | 73-81 |

Number of pages | 9 |

Journal | Homology, Homotopy and Applications |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 |

### Keywords

- Lusternik-Schnirelmann category
- topological complexity
- topological robotics
- acyclic space
- co-H-space
- homology sphere

## Cite this

Grant, M., Lupton, G., & Oprea, J. (2013). Spaces of topological complexity one.

*Homology, Homotopy and Applications*,*15*(2), 73-81. https://doi.org/10.4310/HHA.2013.v15.n2.a4