Topological Complexity of Collision-free Motion Planning on Surfaces

The topological complexity TC(X) is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant TC(X) measures the complexity of all possible motion planning algorithms designed for the system. In this paper, we compute the topological complexity of the configuration space of n distinct ordered points on an orientable surface. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties.