  We study the role played by the compactness and the degree of connectedness
in the time evolution of the energy of a radiating system in the
Friedmann-Robertson-Walker (FRW) space-times whose $t=const $ spacelike
sections are the Euclidean 3-manifold ${\cal R}^3$ and six topologically
non-equivalent flat orientable compact multiply connected Riemannian
3-manifolds. An exponential damping of the energy $E(t)$ is present in the
${\cal R}^3$ case, whereas for the six compact flat 3-spaces it is found
basically the same pattern for the evolution of the energy, namely relative
minima and maxima occurring at different times (depending on the degree of
connectedness) followed by a growth of $E(t)$. Likely reasons for this
divergent behavior of $E(t)$ in these compact flat 3-manifolds are discussed
and further developments are indicated. A misinterpretation of Wolf's results
regarding one of the six orientable compact flat 3-manifolds is also indicated
and rectified.
