  Could cosmic topology imply dark energy? We use a weak field (Newtonian)
approximation of gravity and consider the gravitational effect from distant,
multiple copies of a large, collapsed (virialised) object today (i.e. a massive
galaxy cluster), taking into account the finite propagation speed of gravity,
in a flat, multiply connected universe, and assume that due to a prior epoch of
fast expansion (e.g. inflation), the gravitational effect of the distant copies
is felt locally, from beyond the naively calculated horizon. We find that for a
universe with a $T^1xR^2$ spatial section, the residual Newtonian gravitational
force (to first order) provides an anisotropic effect that repels test
particles from the cluster in the compact direction, in a way algebraically
similar to that of dark energy. For a typical test object at comoving distance
$\chi$ from the nearest dense nodes of the cosmic web of density perturbations,
the pressure-to-density ratio $w$ of the equation of state in an FLRW universe,
is w \sim - (\chi/L)^3, where $L$ is the size of the fundamental domain, i.e.
of the universe. Clearly, |w|<<1. For a T^3 spatial section of exactly equal
fundamental lengths, the effect cancels to zero. For a T^3 spatial section of
unequal fundamental lengths, the acceleration effect is anisotropic in the
sense that it will *tend to equalise the three fundamental lengths*. Provided
that at least a modest amount of inflation occurred in the early Universe, and
given some other conditions, multiple connectedness does generate an effect
similar to that of dark energy, but the amplitude of the effect at the present
epoch is too small to explain the observed dark energy density and its
anisotropy makes it an unrealistic candidate for the observed dark energy.
