  The Baum-Connes map for finitely generated free abelian groups is a
K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We
describe this K-theoretic transform in the language of topological
correspondences, and compute its action on K-theory (of tori) described
geometrically in terms of Baum-Douglas cocycles, showing that the Fourier-Mukai
transform maps the class of a subtorus to the class of a suitably defined dual
torus. We deduce the Fourier-Mukai inversion formula. We use these results to
give a purely geometric description of the Baum-Connes assembly map for free
abelian groups.
