Untelescopings of Heegaard Splittings

In this paper, we shall use Casson Gordon Scharlemann ideas to prove three main theorems in structures on Heegaard splittings.

An Application of the Word Theorem on Heegaard Splittings

The word theorem states that x can be denoted as a rotation inserting word of A if x is in the normal closure of A in F(X). As an application of the theorem, in this note a condition that guarantees reducing the genus of Heegaard splitting of 3-manifolds is given. This leads Poincare conjecture to a new formulation.

Unstabilized Self-amalgamation of a Heegaard Splitting along Disks

In this paper, we prove that a self-amalgamation of a strongly irreducible Heegaard splitting along disks is unstabilized.

STRONGLY IRREDUCIBLE SELF-AMALGAMATION OF A HANDLEBODY

In this paper,we will give a sufficient condition for the self-amalgamation of a handlebody to be strongly irreducible.

On ■-reducible 3-manifolds Which Admit Complete Surface Systems

In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M, then there exists a complete surface system S for M, which is equivalent to S′, such that D is disjoint from S. We also obtain some properties of such 3-manifolds which can be embedded in S;.

Reducing a Representation of Homotopy 3-spheres

In the present note we show that a representation of homotopy 3-spheres can be somewhat simplified under some circumstances.

A Theorem on Combinatorial Group Theory

Let F=F(X) be a free group of rand n, be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of if x is in the normal closure of in F(X). Finally, an application of t he theorem in Heegaard splitting of 3-manifolds is given.

Some Sufficient Conditions for Tunnel Numbers of Connected Sum of Two Knots Not to Go Down

In this paper, we show the following result： Let Ki be a knot in a closed orientable 3- manifold Mi such that （Mi,Ki） is not homeomorphic to （S^2 × S^1,x0 × S^1）, i = 1,2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E（Ki） is less than the difference of one and twice of the tunnel number of Ki. Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive. We further show that if the distance of a Heegaard splitting of each knot complement is strictly bigger than twice the tunnel number of the knot （twice the sum of the tunnel number of the knot and one, respectively）, then the tunnel number of connected sum of two such knots is additive （super additive, respectively）.