Research On The Properties Of Several Kinds Of Specific Bounded Surface Sum Of Three-Dimensional Manifolds

Three-dimensional manifold is an important branch of low-dimensional topology,whether three-dimensional manifolds attach along certain surfaces has genus additivity has always been an important topic in the study of three-dimensional manifold theory.In recent decades,experts in low-dimensional topology at home and abroad have achieved remarkable progress in the study of bounded surface sum of three-dimensional manifolds.From the study of the genus additivity of amalgamation of three-dimensional manifolds,and discusses the additivity of the genus of some two to six punctured torus sum of specific thickened orientable closed surfaces.Specifically,the additivity of genus of some two to six punctured torus sum of specific thickened orientable closed surfaces is proved by using research skills and methods of three-dimensional manifold combinatorial topology,if M=(P₁×I)?_F(P₂×I),where P₁ and P₂ are connected orientable closed surfaces,and F is an incompressible two to six punctured torus in P₁×{0}and P₂×{0},and the separation of F on P₁×{0}and P₂×{0}is discussed.It is obtained that the genus of some two to six punctured torus sum of specific thickened orientable closed surfaces are additive,i.e. g(M)=g(P ₁)+g(P ₂).On this basis,it can prove that the additivity of genus of some two to six punctured torus sum of specific complicated three-dimensional manifolds.The above results have positive roles in further study the additivity of genus of bounded surface sum of some complicated three-dimensional manifolds.