The Classification Of High Codimensional Smooth Function Germs And Strongly Relative Stable Map Germs

The classification is one of the most important problems in mathematics.Since smooth function germs space and map germs space are all real vector spaces of infinite dimensional,so in the classification of smooth function germs and map germs,a primary idea is to transform infinite dimension spaces to finite dimension spaces.As the finite codimension of the smooth function germs or map germs are equivalent to the finite determinacy of the corresponding equivalent group action,therefore the classification problem is closely related to the finite deterministic problem.As finitely determined smooth function germs or map germs are equivalent to their Taylor polynomials of certain degree,so their local topology properties can be determined by their Taylor polynomials of certain degree.Furthermore,the finite determinacy of smooth function germs and map germs become a very active topic in the singularity theory.This thesis mainly contains two parts.In the first part,we discuss the classification of smooth function germs with high codimension under the action of the right equivalency group.In the second part,we discuss the classification of stable map germs under the strongly relative conditions.The explicit arrangement is as follows.In Chapter 1,we introduce the geometrical background of the singularity theory and the previous thoughts about the relative finitely determined map germs.In Chapter 2,we introduce some related algebraic knowledge,some preliminary theorems and some corollaries about finitely determined map germs.In Chapter 3,we study the classification of the high codimensional smooth function germs under the action of the right equivalent group,including the classification problems of the Arnold function family with codimension 5 and the Whitney function family with codimension 8.In Chapter 4,firstly,we introduce the relative Malgrange preparation theorem and its corollary and give the notions of relative stable of map germs and relative infinitesimal stable of map germs.Secondly,we also discuss the relations between relative stable of map germs and relative infinitesimal stable of map germs.In Chapter 5,by the relation between relative stability of map germs and relative infinitesimal stability of map germs,we prove the classification theorem of stable map germs under the strongly relative conditions.In Chapter 6,as an application of classification theorem of stable map germs under the strongly relative conditions,we give the Theorem 6.2.1 and the Corollary 6.2.2.