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

Topological complexity of collision-free motion planning on surfaces