I give a mathematical treatment of the timelike reduction of Abelian
gauge theories (considered in their manifestly duality-covariant
formulation) on a three-manifold of arbitrary topology. In particular,
I describe the set of gauge-equivalence classes of dual Abelian Higgs
pairs in the general case when the naive reduction procedure fails to
produce an equivalence due to the presence of Wilson lines of the
reduced gauge field. This clarifies some subtleties involved in the
general construction of supergravity c-maps.