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.