“Each space slice is approximated by a simplicial manifold composed by regular (d-1)-dimensional simplices” This is imposed ab-initio in CDT whereas I had not thought it necessary. It may be topologically necessary—a mere question of logic, not physics. Not an unreasonable requirement in either case. “the connection between these slices is made by a piecewise linear manifold of d-simplices” This is easier said than done, or perhaps too easily done as there are a multitude of ways to connect two slices, even when we agree on their metrical relationship. “but the edges that have a time arrow must agree in direction, wherever the edges are joined.” They do require local time direction agreement, but not globally! It is clear that you can build a 2D Klein manifold which respects time locally but not globally.

In this plan I suppose that one could build one abstract 3D simplical complex and merely use it over and over for each time-slice. The space time between the same tetrahedron in two slices can be decomposed into 4 4D simplexes in a trivial way. How dull!