The relative topological complexity of a pair
Topological complexity is a homotopy invariant introduced by Michael Farber in the early 2000s. Denoted $TC(X)$, it counts the smallest size of a continuous motion planning algorithm on $X$. In this sense, it solves optimally the problem of continuous motion planning in a given topological space. In topological robotics, a part of applied algebraic topology, several variants of $TC$ are studied. In a recent paper, I introduced the relative topological complexity of a pair of spaces $(X,Y)$ where $Y\subset X$. Denoted $TC(X,Y)$, this counts the smallest size of motion planning algorithms that plan from $X$ to $Y$.
In this talk, we will provide an overview of techniques used to study relative topological complexity and compute this invariant for several simple spaces relating to real-world robotics problems.