The Genus Of A 3-manifold Which Contains An Nonseparating Incompressible Torus

Mathematics in last century focuses on manifold, which is obtained by pastingEuclidean spaces in pieces. It is a little strange that the study of manifold begins fromhigh dimensions to lower ones, which contradicts our intuition to space. The studyof 3-manifold begins at 70's of last century, it is a main research field in topology.There is something unique in it, mainly on the way of handling problems. We usethe tool of piece linear theory, that is paste-and-cut, which is different from traditionalmathematics. This is also our method to deal with the problem.The field of 3-manifold make a huge progress last century, a lot of importantproblems got resolved, the most amazing one is Poincare′conjecture. Some specialstructures on 3-manifold have been found, some important ones are weakly reducible,Heegaard distance. These special structures are backgrounds of our thesis.The study of relationship between manifolds and those ones obtained by iden-tifying their homomorphic boundaries is a popular topic in the field of 3-manifold.Let Mi be a compact, connected, orientable manifold, and Fi be a noncompressibleboundary component of Mi, satisfying: g(F_i)≥₁, i = ₁, ₂, and F₁ (?) F₂. Let : F₁→F₂ is a homomorphism, M = M₁ (?) M₂. If Vi si Wi is a Heegaardsplitting of Mi, then V₁∪S₁ W₁ and V₂∪S₂ W₂ induce a natural Heegaard splitting ofM, call it the amalgamation of V₁∪S₁ W₁ and V₂∪S₂ W₂ along F₁å’ŒF₂. Obviously,g(M)≤g(M₁) + g(M₂) ? g(F).A lot of previous work gave various conditions under which the equation holds,for example, the work of Lackenby and Schultens. A common point in their work isthat the manifolds M₁, M₂ are different. In this thesis, we will consider the relation-ship between a manifold and the one obtained by identifying its own homomorphicboundaries. If M is obtained from M by identifying its own boundaries (we considerthe situation of torus in this thesis), we will prove: after putting some restraints onthe manifold M, equation g(M ) = g(M; T₁ T₂) + ₁ holds, where g(M; T₁ T₂)denotes the minimal Heegaard genus of M among those satisfying T₁,T₂ lie on thenegative boundaries of the same compression body.