  CMC (constant mean curvature) Cauchy surfaces play an important role in
mathematical relativity as finding solutions to the vacuum Einstein constraint
equations is made much simpler by assuming CMC initial data. However, in [2]
Bartnik constructed a cosmological spacetime without a CMC Cauchy surface whose
spatial topology is the connected sum of two three-dimensional tori. Similarly,
in [9], Chru\'sciel, Isenberg, and Pollack constructed a vacuum cosmological
spacetime without CMC Cauchy surfaces whose spatial topology was also the
connected sum of two tori. In this article, we enlarge the known number of
spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by
generalizing Bartnik's construction. Specifically, we show that there are
cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies
are the connected sum of any compact oriented Euclidean or hyperbolic
three-manifold with any another compact oriented Euclidean or hyperbolic
three-manifold. We work with the Tolman-Bondi class of metrics and prove gluing
results for variable marginal conditions, which allows for smooth gluing of
Schwarzschild to FLRW models.
