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.