Research On 3-manifolds With Complete Surface Systems

Topology theory of 3-manifold mainly studies topological properties and constructions of 3-manifolds.Usually,the topological properties and constructions of 3-manifolds can be effec-tively studied by using some surfaces in the manifold,such as Heegaard surfaces,incompressible surfaces,normal surfaces,etc.Handlebody is the elementary factor of 3-manifold.Each closed orientable 3-manifold can be divided into two handlebodies of the same genera.This is the Heegaard splitting theory.Heegaard splitting theory is an important method to study 3-manifolds.There is a collection of pairwise disjoint properly embedded disks in a handlebody such that we can obtain a 3-ball by cutting the handlebody along these disks.In this paper,we study a family of 3-manifolds,which can be considered as a family of generalized handlebodies.Let M be a compact orientable irreducible 3-manifold with a single boundary component S.If there exists a collection of pairwise disjoint connected orientable surfaces with a single boundary F= {F1,…,Fn} properly embedded in M such that(?)F(?)S is a complete curve system on S,and cutting M along F,we obtain a 3-manifold M0,then F is called to be a complete surface system of M,and M is called to be a 3-manifold with complete surface systems,denoted by(M,F).In this paper,we study 3-manifolds with complete surface systems,and obtain some prop-erties of this family of 3-manifolds.The main conclusions in this paper are as follows:1.For a 3-manifold with complete surface systems(M,F),we discuss the equivalence between two complete surface systems in M.For a boundary reducible 3-manifold,the relation between the complete surface system and the maximal boundary compression disc set is given.We also prove that this kind of 3-manifold maintain the boundary connected sum decomposition of the complete surface systems.2.We prove the uniqueness of the equivalent classes of the complete surface system for the 3-submanifolds of S3;discuss the relation between the boundary compression disc set of the 3-manifold and the compression discs of the handlebody in the complement of the 3-submanifold based on handle addition,re-embedding theorem and boundary compression.We study the connection between the complete surface system for the 3-submanifold and the Seifert surfaces of knots.3.We prove that the mapping class group of the 3-submanifold of S3 which admits a complete surface system is a subgroup of the handlebody subgroup of the mapping class group of the surface.4.We discuss the relation between the boundary link in S3 and the 3-submanifold of S3 which admits a complete surface system.