Document Type

Article

Publication Date

7-1-2015

Publication Title

Topology and Its Applications

Abstract

We show that the topological complexity of an aspherical space X is bounded below by the cohomological dimension of the direct product A×BA×B, whenever A and B are subgroups of π1(X)π1(X) whose conjugates intersect trivially. For instance, this assumption is satisfied whenever A and B are complementary subgroups of π1(X)π1(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.

Comments

This work was partially supported by grants from the Simons Foundation: (#209575 to Gregory Lupton) and (#244393 to John Oprea).

DOI

10.1016/j.topol.2015.04.005

Version

Postprint

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Volume

189

Included in

Mathematics Commons

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