Study On The Stability Of Unfolding Of Equivariant Relative Map Germs Under The Left-right Equivalence

Based on the basic concepts of singularity theory,this paper defines the equivariant relative map germs and the equivariant relative left-right equivalence group,and calculates the tangent space of equivariant relative map germs under the left-right equivalence,gives relative Schwarz theorem under the relative condition,equivariant formula of relative Malgrange preparatory theorem and the Judgment theorem of relative trivial unfolding.Then,the sufficient and necessary condition for the existence of the relative infinitesimal stable unfolding of equivariant relative map germs under the left-right equivalence is given.And the relative infinitesimal stable unfolding of equivariant relative map germs is proved to be unique in the sense of the left-right equivalence.In addition,we obtain the relationship between the codimensional finiteness of map germs under the left-right equivalence and the codimensional finiteness of map germs under the contact equivalence.This paper consists of four chapters:In the first chapter: We introduce the research background of singularity theory and dynamic state about the stability of map germs.Then we made a brief summary of the research questions and significance of the paper.In the second chapter: We introduce the basic knowledge related to this paper,such as relative invariant function germs,equivariant relative map gems,Nakayama theorem,etc.In the third chapter: We give several important theorems under the relative condition.In the fourth chapter: We give the main conclusions and proofs.