Topological Complexity of Configuration Spaces

The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of tire topological complexity of the configuration space of n distinct points in Euclidean m-space for all m >= 2 and n >= 2; the answer was previously known in the cases m = 2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.