On the topological complexity of non-simply connected spaces

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.