TOPOLOGICAL TRIVIALITY AND VERSALITY FOR SUBGROUPS OF A AND K .2. SUFFICIENT CONDITIONS AND APPLICATIONS

Many different questions involving the local structure of mappings, the varieties they define, and their local bifurcations are studied by introducing a group of germs of diffeomorphisms, so that the equivalence via elements of the group captures the desired properties. Then, the local classification of mappings, their local determination by finite parts of their Taylor expansions, and the construction of finite parameter families exhibiting all of their perturbation behaviour can be frequently carried out following the original ideas of Thom and Mather. However, the presence of moduli parametrizing continuous change in the equivalences classes forces one to substitute a corresponding topological equivalence, so the results no longer apply. In this paper we shall prove theorems which do solve these problems for the topological analogues of a wide variety of equivalence relations. These results are given by infinitesimal conditions but relative to algebraic filtration conditions on the tangent spaces to the group of diffeomorphisms and to the spaces of germs. These results are applied to a number of problems involving the classification of finite map germs, equivariant bifurcation problems, and the equivalence of germs preserving varieties.