  We consider the vacuum Einstein flow with a positive cosmological constant on
spatial manifolds of product form. In spatial dimension at least four we show
the existence of continuous families of recollapsing models whenever at least
one of the factors or admits a Riemannian Einstein metric with positive
Einstein constant. We moreover show that these families belong to larger
continuous families with models that have two complete time directions, i.e. do
not recollapse. Complementarily, we show that whenever no factor has positive
curvature, then any model in the product class expands in one time direction
and collapses in the other. In particular, positive curvature of one factor is
a necessary criterion for recollapse within this class. Finally, we relate our
results to the instability of the Nariai solution in three spatial dimensions
and point out why a similar construction of recollapsing models in that
dimension fails. The present results imply that there exist different classes
of initial data which exhibit fundamentally different types of long-time
behavior under the Einstein flow whenever the spatial dimension is strictly
larger than three. Moreover, this behavior is related to the spatial topology
through the existence of Riemannian Einstein metrics of positive curvature.
