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 nontrivial 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.

中文翻译：

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有效拓扑复杂度

我们介绍了一个有效拓扑复杂度（ETC）G-空间X. 这是一个G- 等变同伦不变量位于对的有效拓扑复杂度之间(X,G)以及轨道空间的（常规）拓扑复杂性X/G. 我们研究了具有对映对合的球体和表面的 ETC，在圆环的情况下获得了完整的计算。这使我们能够证明克莱因瓶的拓扑复杂性为 4. 此外，这给出了一个反例来说明这种可能性——在 Pavešić 关于地图拓扑复杂性的工作中提出——ETC(X,G)会同意法伯的吨C(X)每当投影图X → X/G是有限张的。我们推测具有对映作用的球体的 ETC 重铸了 Hopf 不变量一个问题，并描述了（推测最优的）有效运动规划器。