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 fourgenerator group), which in some cases improve upon the standard lower bounds in terms of zerodivisors cuplength. 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 

Pages (fromto)  7891 
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