  Four-dimensional CDT (causal dynamical triangulations) is a lattice theory of
geometries which one might use in an attempt to define quantum gravity
non-perturbatively, following the standard procedures of lattice field theory.
Being a theory of geometries, the phase transitions which in usual lattice
field theories are used to define the continuum limit of the lattice theory
will in the CDT case be transitions between different types of geometries. This
picture is interwoven with the topology of space which is kept fixed in the
lattice theory, the reason being that "classical" geometries around which the
quantum fluctuations take place depend crucially on the imposed topology. Thus
it is possible that the topology of space can influence the phase transitions
and the corresponding critical phenomena used to extract continuum physics. In
this article we perform a systematic comparison between a CDT phase transition
where space has spherical topology and the "same" transition where space has
toroidal topology. The "classical" geometries around which the systems
fluctuate are very different it the two cases, but we find that the order of
phase transition is not affected by this.
