  We study a class of subdivision invariant lattice models based on the gauge
group $Z_{p}$, with particular emphasis on the four dimensional example. This
model is based upon the assignment of field variables to both the $1$- and
$2$-dimensional simplices of the simplicial complex. The property of
subdivision invariance is achieved when the coupling parameter is quantized and
the field configurations are restricted to satisfy a type of mod-$p$ flatness
condition. By explicit computation of the partition function for the manifold
$RP^{3} \times S^{1}$, we establish that the theory has a quantum Hilbert space
which differs from the classical one.
