Effectual Topological Complexity

Abstract

We introduce the effectual topological complexity (ETC) of a G-space X. This is a G-equivariant homotopy invariant sitting in between the effective topological complexity of the pair (X, G) and the (regular) topological complexity of the orbit space X/G. We study ETC for spheres and surfaces with antipodal involution, obtaining a full computation in the case of the torus. This allows us to prove the vanishing of twice the non-trivial obstruction responsible for the fact that the topological complexity of the Klein bottle is 4. In addition, this gives a counterexample to the possibility -suggested in Pavešić's work on the topological complexity of a map-that ETC of (X, G) would agree with Farber's TC(X) whenever the projection map X → X/G is finitely sheeted. We conjecture that ETC of spheres with antipodal action recasts the Hopf invariant one problem, and describe (conjecturally optimal) effectual motion planners.