  We explicitly construct a series of lattice models based upon the gauge group
$Z_{p}$ which have the property of subdivision invariance, when the coupling
parameter is quantized and the field configurations are restricted to satisfy a
type of mod-$p$ flatness condition. The simplest model of this type yields the
Dijkgraaf-Witten invariant of a $3$-manifold and is based upon a single link,
or $1$-simplex, field. Depending upon the manifold's dimension, other models
may have more than one species of field variable, and these may be based on
higher dimensional simplices.
