Methods for classification of singularities

Abstract

In this chapter we will recall the methods which were introduced by Thom [Le] and especially Mather [MIII, MVI] to classify germs of mappings under various equivalence groups by reducing to the induced actions of Lie groups on jet spaces. This involves using finite determinacy results and Mather’s geometric lemma for actions of Lie groups. This was considerably strengthened by the much improved order of determinacy results from the stronger method of unipotent groups due to Bruce-Du Plessis-Wall [BDW]. These results will be appropriately adapted to apply to our situation.