  We study the subdivision properties of certain lattice gauge theories based
on the groups $Z_{2}$ and $Z_{3}$, in four dimensions. The Boltzmann weights
are shown to be invariant under all type $(k,l)$ subdivision moves, at certain
discrete values of the coupling parameter. The partition function then provides
a combinatorial invariant of the underlying simplicial complex, at least when
there is no boundary. We also show how an extra phase factor arises when
comparing Boltzmann weights under the Alexander moves, where the boundary
undergoes subdivision.
