The Classification Of Z₂?Z₂-Equivariant Mapping Germs Under The Left-right Equivalence

In this paper,with the compact Lie group Z2(?)Z2 as a symmetric group,we discusses the classification and recognition of Z2(?)Z2-equi-variant map germs with codimension no more than 4 under the left-right equivalence group action,and give the corresponding normalized form.When H?(?1x,?1y),the codimension of H is 0;when H?(?1x,(?2u+m1v)y)The codimension of H is 1;when H?(?1x,(?3v+m2u2)y),the codimension of H is 2;when H?(?1x,(m3u2+m4uv+?4v2)y),the codimension of H is 4;when H?(?1x,(m5uv+?5v2+m6u3)y),the codimension of H is 4;when H?((n1u+?2v)x,?1y),the codimension of H is 1;when H?((?3u+n2v)x,(m7u+?3v)y),the codimension of H is 3;when H?((n3v+?4u2)x,(m8u+?3v))y,the codimension of H is 3.This paper consists of four chapters.In the first chapter:This paper briefly summarizes the development background and research significanceIn the second chapter:We introduce some basic concepts and concl-usion,including the definitions of left-right equivalence,codimension,tangent space and unipotent tangent space,as well as the basic conclusio-ns to be used in chapters 3 and 4.In the third chapter:We give the Hilbert basis of Z2(?)Z2-invariant function germs and Z2(?)Z2-equivariant mapping germs module as the generator of ?2(Z2(?)Z2)-module,from this,we get the unipotent tangent space and tangent space under the left-right equivalence group action.In the fourth chapter:We obtain the codimension after discussing the tangent space and give the Z2(?)Z2-equivariant mapping germs with codimension no more than 4 under the action of group A(Z2(?)Z2)are classified,and the corresponding normalized form are obtained.