  The complexity of algorithms solving the motion planning problem is measured
by a homotopy invariant TC(X) of the configuration space X of the system.
Previously known lower bounds for TC(X) use the structure of the cohomology
algebra of X. In this paper we show how cohomology operations can be used to
sharpen these lower bounds for TC(X). As an application of this technique we
calculate explicitly the topological complexity of various lens spaces. The
results of the paper were inspired by the work of E. Fadell and S. Husseini on
weights of cohomology classes appearing in the classical lower bounds for the
Lusternik - Schnirelmann category. In the appendix to this paper we give a very
short proof of a generalized version of their result.
