Proper topological complexity
Jose M. Garcia-Calcines, Aniceto Murillo
TL;DR
The paper defines proper topological complexity ${ m pTC}(X)$ as a motion-planning invariant that accounts for avoiding any compact 'unsafe' region, and situates it within exterior homotopy theory by showing ${ m pTC}(X)={ m eTC}(X_{cc})={ m e ext{-}secat}( abla_{X_{cc}})$ (or equivalently ${ m e ext{-}secat}( abla_X)$). It develops the exterior and proper sectional categories ${ m e ext{-}secat}$ and ${ m p ext{-}secat}$, along with cohomological lower bounds and product inequalities, to establish a robust, functorial framework for noncompact spaces. The exterior-topological-complexity bridge yields a toolkit of invariants that behave well under exterior homotopy, with precise bounds and a suite of nontrivial examples illustrating how ${ m pTC}$ can differ from the classical ${ m TC}$ in noncompact settings. The paper concludes with applications to robotics via a relative-topological-complexity notion ${ m TC}(Y,Z)$, demonstrating how constraints on paths remaining within a subregion can be captured by ${ m pTC}$ and related invariants, thus providing a principled approach to planning under safety constraints. $$${ abla}$$ and cocompact externology play central roles in making these invariants functorial and computable for a broad class of spaces.$$
Abstract
We introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe'' area. To make it a homotopy functorial invariant we characterize it as a particular instance of the exterior sectional category of an exterior map, an invariant of the exterior homotopy category which is also deeply analyzed.