An Investigation Of The Incompressible Surfaces In Amalgamated 3-Manifolds

This article is a dissertation written by a candidate making a specialty of the Theory of3-Manifolds for the Master's Degree at Specialty of Pure Mathematics, Department ofApplied Mathematics, Dalian University of Technology. The subject investigated in thisarticle is mainly the incompressible surface(s) obtained from the amalgamation of two3-manifolds with toms boundaries.3-manifold theory is a branch of low-dimensional topology which is of great importance.Based on the combinatorial structures, complicated 3-manifolds can be decomposed intosome simple objects by certain surface(s) in 3-manifolds, such as Heegaard surfaces,incompressible surfaces, essential 2-spheres and normal surfaces, etc. Then we can study thetopological properties and geometric structures of 3-manifolds by analyzing those simplerpieces. It is a powerful method in the study of 3-manifolds.In this dissertation, we focus on the incompressible surface(s) obtained from theamalgamation of two 3-manifolds with torus boundaries and obtained some good results. Firstit previewed some fundamental concepts of the theory of 3-manifolds, including 3-manifolds,irreducible manifolds, prime manifolds, torus decomposition of 3-manifolds, Seifertmanifolds, incompressible manifolds, Heegaard splittings of 3-manifolds, reducible Heegaardsplittings of 3-manifolds, weakly reducible Heegaard splittings of 3-manifolds, etc. It alsogives a brief introduction about some classical results in the theory of 3-manifolds, includingLoop Theorem, Dehn's Lemma, Poincaré's Conjecture, Sphere Theorem, etc. It alsointroduces some concepts and results about boundary surfaces on incompressible surfaces,including some estimation of the genus of a 3-manifold obtained by an amalgamation of two3-manifolds. The article roughs out some result about the mapping class group of a toms,including a classical theorem about the classification of all automorphisms of a toms. Someimport results about the incompressible surfaces and small knots in 3-manifolds are alsomentioned in this dissertation. Then it proves a necessary and sufficient condition for anamalgamation of two 3-manifolds with toms boundary with no new incompressible surfaces.Lastly, with the help of the condition and the theorem about the classification of allautomorphisms of a torus, it is proveed that there exists infinitely many 3-manifolds each ofwhich contains no incompressible surfaces besides a torus.