The supreme aim of my research is to classify the combinatorial structures of orientable closed 3-manifolds.
It is well known that the set of closed 2-dimensional manifolds (surfaces) are completely classified, that is, we can determine a “canonical” combinatorial presentation for each homeomorphism class of surfaces. It seems to be very hard to classify all the 3-dimensional manifolds completely, but I think that we can get a combinatorial complete classification for 3-manifolds with low complexity, for example 3-manifolds with Heegaard genus 2 or 3. Our recent results related to the above aims and our research plans thought to be realizable in the near future are as follows:
1. We have developed a method for representing a 3-manifold by a finite sequence of symbols, and we have obtained a necessary and sufficient condition for two such sequences to represent mutually homeomorphic manifolds. We have also studied some relations between such a symbolic representation and other combinatorial representations, for example, Heegaard diagrams, Dehn surgery and branched coverings.
2. Using the above symbolic representation, we have defined a new complexity, which we call “block number”, for 3-manifolds, and we have shown that manifolds with block number 1 or 2 are equivalent to those with Heegaard genus 1 or 2 respectively. Corresponding to the fact that the Heegaard diagram of genus 1 for a lens space can be uniformized, we can uniformize the symbolic representation of block number 1.
3. Our present research focuses on 3-manifolds having a symbolic representation with block number 2. In order to classify such 3-manifolds, we must develop some methods for deciding that two symbolic representations give the same 3-manifold or not. For this purpose, we are studying some torsion invariants, and some quantum invariants from the viewpoint of symbolic representations of 3-manifolds. Furthermore, we are looking for topological invariants which are defined directly by a symbolic representation. For a 3-manifold, like a Seifert manifold or a connected sum of two lens spaces, whose combinatorial structure is well known, we can get a symbolic representation which can be considered to be canonical in some sense. However, for other manifolds, especially hyperbolic manifolds, there remain many open problems.