# New lower bounds for the topological complexity of aspherical spaces

Mark Grant, Gregory Lupton, John Oprea

Research output: Contribution to journalArticle

7 Citations (Scopus)

### 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 78-91 14 Topology and its Applications 189 13 Apr 2015 https://doi.org/10.1016/j.topol.2015.04.005 Published - 1 Jul 2015

### Fingerprint

Topological Complexity
Subgroup
Lower bound
Pi
Cohomological Dimension
Zero-divisor
Braid Group
Direct Product
Fundamental Group
Intersect
Generator

### 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

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

In: Topology and its Applications, Vol. 189, 01.07.2015, p. 78-91.

Research output: Contribution to journalArticle

title = "New lower bounds for the topological complexity of aspherical spaces",
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.",
keywords = "math.AT, 55M99, 55P20 (primary), 55M30, 20J06, 68T40 (secondary), topological complexity, aspherical spaces, Lusternik–Schnirelmann category, cohomological dimension, topological robotics, infinite groups",
author = "Mark Grant and Gregory Lupton and John Oprea",
note = "Date of Acceptance: 5/04/2015 15 pages, 4 figures",
year = "2015",
month = "7",
day = "1",
doi = "10.1016/j.topol.2015.04.005",
language = "English",
volume = "189",
pages = "78--91",
journal = "Topology and its Applications",
issn = "0166-8641",
publisher = "Elsevier",

}

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 -