### Abstract

We show that the topological complexity of an aspherical space $X$ is bounded below by the cohomological dimension of the direct product $A\times B$, whenever $A$ and $B$ are subgroups of $\pi_1(X)$ whose conjugates intersect trivially. For instance, this assumption is satisfied whenever $A$ and $B$ are complementary subgroups of $\pi_1(X)$. This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zero-divisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group.

Original language | English |
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Pages (from-to) | 78-91 |

Number of pages | 14 |

Journal | Topology and its Applications |

Volume | 189 |

Early online date | 13 Apr 2015 |

DOIs | |

Publication status | Published - 1 Jul 2015 |

### Keywords

- math.AT
- 55M99
- 55P20 (primary)
- 55M30
- 20J06
- 68T40 (secondary)
- topological complexity
- aspherical spaces
- Lusternik–Schnirelmann category
- cohomological dimension
- topological robotics
- infinite groups

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## Profiles

### Mark Grant

- School of Natural & Computing Sciences, Mathematical Science - Senior Lecturer
- Mathematical Sciences (Research Theme)

Person: Academic

## Cite this

Grant, M., Lupton, G., & Oprea, J. (2015). New lower bounds for the topological complexity of aspherical spaces.

*Topology and its Applications*,*189*, 78-91. https://doi.org/10.1016/j.topol.2015.04.005