Complete transversals and the classification of singularities

The classification of map-germs (up to a variety of equivalences) has many applications in differential geometry, to the study of wavefronts and caustics and to bifurcation theory. In a previous paper the first and last authors together with C T C Wall, gave some useful criteria for map-germs to be finitely determined with respect to a wide range of equivalence relations. The results presented in this paper, used in conjunction with these determinacy techniques, provide a very efficient classification procedure. Moreover, we show that the algebraic criteria involved in these calculations may be reduced to finite-dimensional symbolic problems which may be performed by a computer.