### Abstract

Original language | English |
---|---|

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 |

### Fingerprint

### Keywords

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

### Cite this

*Topology and its Applications*,

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

**New lower bounds for the topological complexity of aspherical spaces.** / Grant, Mark; Lupton, Gregory; Oprea, John.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - New lower bounds for the topological complexity of aspherical spaces

AU - Grant, Mark

AU - Lupton, Gregory

AU - Oprea, John

N1 - Date of Acceptance: 5/04/2015 15 pages, 4 figures

PY - 2015/7/1

Y1 - 2015/7/1

N2 - 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.

AB - 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.

KW - math.AT

KW - 55M99

KW - 55P20 (primary)

KW - 55M30

KW - 20J06

KW - 68T40 (secondary)

KW - topological complexity

KW - aspherical spaces

KW - Lusternik–Schnirelmann category

KW - cohomological dimension

KW - topological robotics

KW - infinite groups

U2 - 10.1016/j.topol.2015.04.005

DO - 10.1016/j.topol.2015.04.005

M3 - Article

VL - 189

SP - 78

EP - 91

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -