  We study the role played by multiply-connectedness in the time evolution of
the energy E(t) of a radiating system that lies in static flat space-time
manifolds M_4 whose t=const spacelike sections M_3 are compact in at least one
spatial direction. The radiation reaction equation of the radiating source is
derived for the case where M_3 has any non-trivial flat topology, and an exact
solution is obtained. We also show that when the spacelike sections are
multiply-connected flat 3-manifolds the energy E(t) exhibits a reverberation
pattern with discontinuities in the derivative of E(t) and a set of relative
minima and maxima, followed by a growth of E(t). It emerges from this result
that the compactness in at least one spatial direction of Minkowski space-time
is sufficient to induce this type of topological reverberation, making clear
that our radiating system is topologically fragile. An explicit solution of the
radiation reaction equation for the case where M_3 = R^2 x S^1 is discussed,
and graphs which reveal how the energy varies with the time are presented and
analyzed.
