On the topological complexity of maps

We define and develop a homotopy invariant notion for the topological complexity of a map f : X -> Y, denoted TC (f), that interacts with TC (X) and TC (Y) in the same way cat (f) interacts with cat (X) and cat (Y). Furthermore, TC (f) and cat (f) satisfy the same inequalities as TC (X) and cat (X). We compare it to other invariants defined in the papers [15-18,20]. We apply TC (f) to studying group homomorphisms f : H -> G. (C) 2022 Elsevier B.V. All rights reserved.