Effectual Topological Complexity

TL;DR

This paper introduces effectual topological complexity (ETC), a new invariant for $G$-spaces, computes it for spheres and surfaces, and explores its implications for classical topological problems and motion planning.

Contribution

It defines ETC as a novel $G$-equivariant homotopy invariant, computes it for specific spaces, and demonstrates its potential to address longstanding topological questions.

Findings

ETC computed for spheres, surfaces, and the torus.

Proves vanishing of certain obstructions in Klein bottle case.

Provides counterexamples to previous conjectures about ETC and $TC(X)$.

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 Pave\v{s}i\'c'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 $X\to X/G$ 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.