Critieria For K-versal Deformations And A-equivalence Of Relative Maps

It is meaningful for studing relative maps beacause of requirement of studing theories and practical problems. The aim is to study the diffrences and similarities between relative maps and usally maps and find some special properties of relative maps so that new theory can be advanced on the base of old ones. It is difficult to study relative maps from a general manifold M to a general manifold N such that mapping a subset S of M to a subset T of N. As a result, problems can be simplied by restricting some conditions and definiting appropriate relations of relative equivalence.Bulajich and other scholars studied versal unfoldings of relative maps with relative strong-A equivalence. Deformations of relative maps have been studied in the paper with classical method. In order to make problems simple relative map germs which map proper subspace R~s of R~m to proper subspace R~t of R~n are considered in the paper. Versal deformations of relative maps with K equivalence have been studied and a critierion for versal deformations of relative maps is given and some properties on versal defofmations are given. Furthermore a class of maps are studied simply whose differentials are clossed in the ring of relative maps.Basic theory of relative maps and classical singularity theory are used together. To apply the critieria for A equivalence to the ring of relative maps and advance original results. A method determining A equivalence of two relative maps with the condition of K equivalence is given.