# New lower bounds for the topological complexity of aspherical spaces

Mark Grant, Gregory Lupton, John Oprea

Research output: Contribution to journalArticlepeer-review

11 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

## 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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