For time-independent fields the Aharonov-Bohm effect has been obtained by idealizing the coordinate space as multiply-connected and using representations of its fundamental homotopy group to provide information on what is physically identified as the magnetic flux. With a time-dependent field, multiple-connectedness introduces the same degree of ambiguity; by taking into account electromagnetic fields induced by the time dependence, full physical behavior is again recovered once a representation is selected. The selection depends on a single arbitrary time (hence the so-called holonomies (...) are not unique), although no physical effects depend on the value of that particular time. These features can also be phrased in terms of the selection of self-adjoint extensions, thereby involving yet another question that has come up in this context, namely, boundary conditions for the wave function. (shrink)