Aim of the paper is to translate the homeomorphism problem for n-dimensional PL-manifolds, with or without boundary, into an equivalence problem for pseudosimplicial triangulations (i.e. a suitable generalization of simplicial triangulations, where two (curvilinear) simplices may intersect in more than one face), by means of a finite number of moves, called (geometric) dipole moves. Note that the environment of the present work is closely related to the representation method for PL-manifolds via edge-colorured graphs, since (n+1)-coloured graphs representing n-manifolds are a «discrete way» to visualize suitable pseudosimplicial triangulations. From the graphtheoretical point of view, the equivalence problem was already faced – and solved – in [FG] for closed n-manifolds and in [C2] in the general 3-dimensional setting; here, the equivalence criterion via (geometric) dipole moves is proved to hold for the whole class of PL n-manifolds; moreover, it is proved to be equivariant with respect to boundary triangulation.