On the topological complexity of non-simply connected spaces

TL;DR

This paper explores the topological complexity of non-simply connected spaces, extending cohomological methods and spectral sequences to evaluate motion planning complexity in complex 3-manifolds.

Contribution

It generalizes previous cohomological bounds for topological complexity using group homomorphisms and applies these techniques to specific 3-manifolds with nonabelian fundamental groups.

Findings

Extended cohomology class bounds for non-simply connected spaces

Developed a spectral sequence for evaluating nilpotency

Determined topological complexity of certain 3-manifolds

Abstract

Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose nilpotency gives a lower bound of topological complexity. Farber and Mescher constructed a spectral sequence that evaluates this nilpotency without direct computation. We extend these results with respect to a group homomorphism. As an application, we determine the topological complexity of some 3-manifolds with nonabelian fundamental group.