  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.
