For smooth surfaces, the generic singularities of a map of one surface to another are folds and cusps (Whitney). It is a standard result in singularity theory that the generic isotopy of such a map is an isotopy of the projection plane punctuated by a sequence of local moves. The moves are one of three types in which a two cusps either appear or disappear.
Has the analogous theory for PL surfaces and maps been developed? I've been unable to find it using the usual search tools.
