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We study the role played by the compactness and the degree of connectednessin the time evolution of the energy of a radiating system in theFriedmann-Robertson-Walker (FRW) space-times whose $t=const $ spacelikesections are the Euclidean 3-manifold ${\cal R}^3$ and six topologicallynon-equivalent flat orientable compact multiply connected Riemannian3-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 foundbasically the same pattern for the evolution of the energy, namely relativeminima and maxima occurring at different times (depending on the degree ofconnectedness) followed by a growth of $E(t)$. Likely reasons for thisdivergent behavior of $E(t)$ in these compact flat 3-manifolds are discussedand further developments are indicated. A misinterpretation of Wolf's resultsregarding one of the six orientable compact flat 3-manifolds is also indicatedand rectified.Comment: 13 pages, RevTeX, 5 figures, To appear in Phys. Rev. D 15, vol. 57  (1998

Radiation damping in FRW space-times with different topologies