  We extend Cellular Automata to time-varying discrete geometries. In other
words we formalize, and prove theorems about, the intuitive idea of a discrete
manifold which evolves in time, subject to two natural constraints: the
evolution does not propagate information too fast; and it acts everywhere the
same. For this purpose we develop a correspondence between complexes and
labeled graphs. In particular we reformulate the properties that characterize
discrete manifolds amongst complexes, solely in terms of graphs. In dimensions
$n<4$, over bounded-star graphs, it is decidable whether a Cellular Automaton
maps discrete manifolds into discrete manifolds.
