  We introduce class A spacetimes, i.e. compact vicious spacetimes $(M,g)$ such
that the Abelian cover $(\bar{M},\bar{g})$ is globally hyperbolic. We study the
main properties of class A spacetimes using methods similar to the one
introduced in D. Sullivan "Cycles for the dynamical study of foliated manifolds
and complex manifolds" (Invent. Math.,36, 225-255 (1976)) and D. Yu. Burago
"Periodic metrics" (Representation theory and dynamical systems (Adv. Soviet
Math.), 9, 205-210 (1992)). As a consequence we are able to characterize
manifolds admitting class A metrics completely as mapping tori. Further we show
that the notion of class A spacetime is equivalent to that of SCTP (spacially
compact time-periodic) spacetimes as introduced in Galloway "Splitting theorems
for spatially closed spacetimes" (Comm Math Phys 96:423-429, 1984). The set of
class A spacetimes is shown to be open in the $C^0$-topology on the set of
Lorentzian metrics. As an application we prove a coarse Lipschitz property for
the time separation of the Abelian cover. This coarse Lipschitz property is an
essential part in the study of Aubry-Mather theory in Lorentzian geometry.
