We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(\pi_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its fundamental group.
Structure of the fundamental groups of orbits of smooth functions on surfaces
The homotopy dimension of codiscrete subsets of the 2-sphere
On Certain 5-manifolds with Fundamental Group of Order 2
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