### 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 |
---|---|

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

*Homology, Homotopy and Applications*,

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

**Spaces of topological complexity one.** / Grant, Mark; Lupton, Gregory; Oprea, John.

Research output: Contribution to journal › Article

*Homology, Homotopy and Applications*, vol. 15, no. 2, pp. 73-81. https://doi.org/10.4310/HHA.2013.v15.n2.a4

}

TY - JOUR

T1 - Spaces of topological complexity one

AU - Grant, Mark

AU - Lupton, Gregory

AU - Oprea, John

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Lusternik-Schnirelmann category

KW - topological complexity

KW - topological robotics

KW - acyclic space

KW - co-H-space

KW - homology sphere

U2 - 10.4310/HHA.2013.v15.n2.a4

DO - 10.4310/HHA.2013.v15.n2.a4

M3 - Article

VL - 15

SP - 73

EP - 81

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

IS - 2

ER -