加柄定理的一个推广  

A generalization of the handle addition theorem

作　　者：赵凌琪[1] 黎明[2] 

机构地区：[1]内蒙古民族大学计算机科学与技术学院,内蒙古通辽028043 [2]内蒙古民族大学基建处,内蒙古通辽028043

出　　处：《辽宁师范大学学报（自然科学版）》2011年第2期146-148,共3页Journal of Liaoning Normal University:Natural Science Edition

摘　　要：Przytycki在1983年给出沿可定向柄体边界上一条简单闭曲线添加2-把柄后所得3-流形有不可压缩边界的一个充分条件,随后在1984年,Jaco又把Przytycki的结果推广到一般的3-流形上,得到了著名的加柄定理.后来,加柄定理又被推广到更一般的形式,这些加柄定理被用来处理与不可压缩曲面、Dehn手术、Heegaard分解等有关的一些问题中,取得了巨大的成功,人们自然考虑它的进一步推广.考虑两个3-流形沿各自边界上的一个平环相粘所得的3-流形,它是加柄定理所考虑的流形的一种一般化.所得主要结果:设At是3-流形Mi上一个分离的平环,i=1,2.如果Mt-Ai在Mi中是不可压缩的,i=1,2,则M1和M2沿A1和A2相粘所得的3-流形有不可压缩的边界.主要结果一定程度上推广了已有的加柄定理.In 1983,Przytycki gave a sufficient condition for a 3-manifold incompressible boundary obtained by adding a 2-handle to a handlebody along a simple closed curve on the boundry of the handlebody.In 1984,Jaco extended Przytycki＇s theorem to general 3-manifold and got the well-known handle addition theorem.Later on,the handle addition theorem has been generalized in several ways.These theorems have been successfully applied in dealing with many problems related to incompressible surfaces,Dehn surgeries,and Heegaard splitting.It is natural to consider its further generalizations.In the present paper,we consider the 3-manifold obtained by gluing two 3-manifolds along an annulus in each boundary of the manifolds.The main result is as follows：Let At be an separating annulus in the boundary of the 3-manifold M1,i=1,2.If Mi-Ai is incompressible in Mi,i=1,2 then the 3-manifold obtained by gluing M1 and M2 along A1 and A2 has incompressible boundary.The result extends the handle addition theorem in some sense.

关 键 词：3-流形 不可压缩曲面 加柄定理 

分 类 号：O189[理学—数学]