The Relative Finite Determinacy Of Map-germs

In the present thesis, we ?rst give the de?nition of weak relative ?nite determination and relative in?nite determination of map-germs, and then give the su?cient and necessary conditions for a map-germ to be weak relative ?nitely determined and relative in?nitely determined respectively. Moreover, we obtain two ways to discriminate the relative V-su?ciency of jets and the algebraic characterization of the relative stable map-germs.In the singularity theory, every ?nitely determined map-germ is equivalent to its Taylor polynomial of some degree, Therefore, the local topology properties of ?nitely determined map-germs can be determined by the Taylor polynomial of some degree. In this way, we can convert the problem with in?nite dimension into the problem with ?nite dimension, which is an important method in singularity theory. It also makes the ?nite determinacy of map-germs becomes a very active topic in singularity theory. So, we mainly study the relative ?nite determinacy and relative in?nite determinacy of mapgerms in this thesis. For the problem of determinacy, we mainly search the discriminated methods of ?nite determination or in?nite determination, and the algebraic and geometric characterizations of ?nitely determined map-germs or in?nitely determined map-germs.This thesis is organized as follows.In chapter 1, we introduce the background and development of ?nitely determined map-germs. Moreover, we introduce the structure of the full thesis and describe the main content of this thesis.In Chapter 2, we mainly introduce some de?nitions in singularity theory and the theory of ?nite determinacy, as well as some concepts and notations in the relative case.In Chapter 3, we study the weak relative ?nite determinacy for function-germs and map-germs. In the ?rst section, we give the su?cient and necessary conditions for a function-germ to be weak relative ?nitely determined with respect to right-equivalence and relative right-equivalence respectively, and discuss the relation between them. In the second section, we give three su?cient conditions for weak relative ?nitely determined map-germs with respect to relative right-left-equivalence. In the third section, we obtain a su?cient condition for weak relative ?nitely determined map-germs with respect to relative contact-equivalence.In Chapter 4, we study the relative in?nite determinacy of function-germs and mapgerms. We study the weak relative in?nite determinacy for function-germs with respect to right-equivalence and relative right-equivalence, and relative in?nite determinacy for map-germs with respect to relative right-left-equivalence and relative contact-equivalence.We obtain su?cient and necessary conditions for the corresponding in?nite determinacy.In Chapter 5, we discuss the relative V-su?ciency of jets, and obtain two ways to determine the relative V-su?ciency.In Chapter 6, we study the relative stability of map-germs. In the ?rst section, we give the concept of relative stable map-germ and relative in?nitesimal stable map-germ,and discuss the relation between the both stability. Moreover, we also prove that a relative stable map-germ is weak relative ?nitely determined, and give the order of weak ?nite determination for relative stable map-germs. In the second section, we obtain several equivalent conditions for a relative stable unfolding.