Pure Maths Colloquium: Lower bounds for the topological complexity of groups
16 February 2015 16:00 in CM221
Topological complexity is a numerical homotopy invariant, introduced by Michael Farber in the course of his topological study of the robot motion planning problem. It is a close relative of the Lusternik--Schnirelmann category, although the two invariants are independent.
Given a discrete group G, the topological complexity TC(BG) of its classifying space is an algebraic invariant of the group. One can therefore ask for a description of TC(BG) in terms of well-known algebraic invariants (such as cohomology). Such a description seems out of reach, even in conjectural form. By contrast, the Lusternik--Schnirelmann category of BG is equal to the cohomological dimension of G over the integers.
In this talk I will present lower bounds for TC(BG) coming from the subgroup structure of G, which in some cases improve upon the standard lower bounds in terms of the cohomology algebra of G. We illustrate that our bounds are sharp in various cases of interest, including pure braid groups and Higman's acyclic group.
This is joint work with Greg Lupton and John Oprea.
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