Finite Determinacy Of Relative Map-germs

In this paper,we consider the finite determinacy of relative map-germs which map S the germ of submanifold of Rn to T the germ of submanifold of Rp.We combine the basic theory of relative maps and classical singularity theory together. We study the finite determinacy of relative map-germs with regard to the strong relative left-right equivalent group As,r and strong relative contact equivalent group Ks,t, the relation between As,t finite determinacy and relative versal unfolding is also studied. We prove theorems which can be used to decide the AS,T and KS,T finite determinacy respectively, we also discuss the relations between AS,T finite determinacy and A finite determinacy and obtain that A finite determinacy contains As,t finite determinacy; Fur-themore, making use of the theory of transversality unfolding we gain that AS,T finite determinacy and the existence of relative versal unfolding is equivalence, finally, we provid two examples on relative stable map-germs as a application of this paper.