Dimensional phase transitions: Coupling of matter to the cell complex

We extend Euclidean lattice field theory from a usual D-dimensional lattice onto a cell complex. Using the extension we present a formalism to describe a change in the dimension of space-time as a phase transition. The coupling of matter to the cell complex is considered. For two-dimensional models with continuous time we give a number of exact results for the free energy of the systems. We show that for certain models two-dimensional space-time decays into one-dimensional space-time.