New lower bounds for the topological complexity of aspherical spaces

Mark Grant, Gregory Lupton, John Oprea

Research output: Contribution to journalArticle

8 Citations (Scopus)
7 Downloads (Pure)

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 languageEnglish
Pages (from-to)78-91
Number of pages14
JournalTopology and its Applications
Volume189
Early online date13 Apr 2015
DOIs
Publication statusPublished - 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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