  We give new lower bounds for the (higher) topological complexity of a space,
in terms of the Lusternik-Schnirelmann category of a certain auxiliary space.
We also give new lower bounds for the rational topological complexity of a
space, and more generally for the rational sectional category of a map, in
terms of the rational category of a certain auxiliary space. We use our results
to deduce consequences for the global (rational) homotopy structure of simply
connected, hyperbolic finite complexes.
