  In this article, we extend a construction of [6] to obtain a large class of
vacuum cosmological spacetimes that do not contain any CMC Cauchy surfaces. The
allowed spatial topologies for these examples are of the form $M \# M$, where
$M$ is any closed, oriented, irreducible $3$-manifold which is not spherical.
This complements the recent results of [19], where, instead of initial data
methods, global spacetime gluing arguments were used. The study of such
examples is sure to yield insight into Bartnik's cosmological splitting
conjecture [1].
