Annulus-disk Complex And Self-amalgamation Of Heegaard Splittings Of Three Manifolds

Two dimensional manifolds, together with three dimensional manifolds, play important roles in mathematics, and many topologists and geometrists are concerning about them. In this paper, we research on them, and get some results.Through the classification of finite type two dimensional manifolds are already done, a deal of advanced stuctures, such as Teichmuller spaces and mapping class groups, become main objects of low dimensioal topology. Recently, the theory of the complex of curves is growing rapidly, not only for the relation to Teichmuller spaces and mapping class groups, but also for its own interesting properties. In this paper, we define a new simple complex, which is a sub-complex of the curve complex, and we call it annulus-disk complex. We research on it, and obtain that the annulus-disk comlex is often contractible and is always quasi-convex in the curve complex.Heegaard splitting is a fundamental structure of3-manifolds, while Heegaard genus is an important invariant of3-manifolds. In this paper, we consider the Heegaard genus of a3-manifold which is obtained by gluing two homeomorphic boundary components of an irre-ducible,(?)-irreducible3-manifolds. A complexity function is defined for the gluing map, and we prove that if the value of the complexity function is sufficiently large and the original man-ifold satisfies certain conditions, then the Heegaard genus of the resulting manifold equals the Heegaard genus relative to all boundaries of original one plus1.