Research Of Some Properties Of Heegaard Splittings Based On Curve Complex

The Heegaard splitting theory is one of the most important methods in combinatorial 3-manifold theory. The surface which divides complicated 3-manifolds into two compression bodies is called the Heegaard surface. In 2000, Hempel defined the distance of a Heegaard splitting from a deep observation of curve complex. This is a generalization of reducible, weakly reducible Heegaard splittings and provides an important tool for the research of 3-manifolds, such as surfaces in 3-manifolds, amalgamations of 3-manifolds and etc.Based on the property of the Heegaard distance and the subsurface projection, this paper mainly studies the influence of the handle addition to the Heegaard distance and the stabilization of the self-amalgamation of Heegaard splitting along essential subsurfaces. The main contents are as follows:1. We investigate the properties of the compression bodies with only one essential disk up to isotopy and the annulus-busting curves on the positive boundary. From these results, we construct a strongly irreducible Heegaard splitting satisfying the property of annulus-busting which is more wide-ranging than those Heegaard splitting with distance 3.2. We generalize the definition of the self-amalgamation of Heegaard splitting. We intro-duce the definition of self-amalgamation of Heegaard splitting along essential subsurfaces or disks. And we will give a sufficient condition for self-amalgamation of Heegaard splitting along essential subsurfaces or disks to be unstabilized.3. Based on the property of subsurface projection, we will prove that the diameter of the set of distance degenerating curves of a kind of strongly irreducible Heegaard splittings is bounded. With a similar argument, we will give a sufficient condition for (?)-irreducible handle addition to a handlebody.