任意の閉3次元多様体Mは，閉曲面Σで切り開くことにより，ハンドル体と呼ばれる基本的なピースに分解することができる．このような分解をMのHeegaard分解とよぶ．また，曲面Σの種数をこの分解の種数とよぶ．Heegaard分解はすべての閉3次元多様体Mが許容する最も基本的な分解であり，本研究の主役であるGoeritz群は概ねこの自己同型群のことである．本研究では，組み合わせ的手法，特異点的手法双方を駆使してこの群，および関連する群の構造解明に取り組んだ．まず，田中勇輝氏(広島大学)と共同で，結び目の(1,1)-分解のGoeritz群の構造を全て決定した．また，高尾和人氏(東北大学)と共同で，Goeritz群の有限性条件をHeegaard図式を用いて記述した．さらに，Sangbum Cho氏（韓国・Hanyang Univ.），Jung Hoon Lee氏（韓国・Jeonbuk National Univ.）と共同で，3次元球面の種数3のHeegaard分解のGoeritz群に関するPowell予想(M. FreedmanとM. Scharlemannにより解決がアナウンスされている)の簡明な別照明を与えた．これらの成果は，それぞれarXivに掲載している．
関連する研究成果として，石井一平氏，石川昌治氏(慶應義塾），直江央寛(中央大)と共同で実施したフロースパインと接触構造に関する研究成果，石川昌治氏(慶應義塾），直江央寛(中央大)と共同で実施したシャドウの補空間の基本群に関する研究成果がそれぞれを査読付き国際誌に掲載された．また，古谷凌雅氏(広島大)と共同で，ディバイド絡み目の双曲構造に関する研究を実施し，得られた成果が査読付き国際誌に掲載を受理された．野崎雄太氏(横国大)，Tamás Kálmán氏(東工大)，寺垣内政一氏(広島大)らとは，大域的位相欠陥のホモトピー分類に関する成果を挙げ，arXivで公表した．
Any closed 3-manifold M can be decomposed into two basic pieces called handlebodies by cutting it open along a closed surface Σ. Such a decomposition is called a Heegaard splitting of M. The Goeritz group, which is the main subject of this research project, is defined to be, roughly, the automorphism group of a Heegaard splitting. In this research, we have studied the structures of the Goeritz groups, as well as several related groups, by using both combinatorial and singular methods. In collaboration with Y. Tanaka (Hiroshima University), we completely described the structure of the Goeritz group of (1,1)-decompositions of knots. In a joint work with K. Takao (Tohoku University), the finiteness conditions of the Goeritz group were described using the Heegaard scheme. Further, in collaborations with S. Cho (Hanyang Univ., Korea) and J. H. Lee (Jeonbuk National Univ., Korea), we gave a short alternative proof of the Powell conjecture for the Goeritz group of the genus-3 Heegaard splitting of the 3-sphere. (whose solution has been announced by M. Freedman and M. Scharlemann). Each of these results has been posted on the arXiv.
As related research results, the results on flow spines and contact structures in collaboration with I. Ishii, M. Ishikawa (Keio Univ.), and T. Naoe (Chuo Univ.), and the results on fundamental groups of complementary spaces of shadows in collaboration with M. Ishikawa (Keio Univ.) and T. Naoe (Chuo Univ.), have been published in peer-reviewed international journals, respectively. The results of a joint research with Ryoga Furuya (Hiroshima Univ.) on hyperbolic structures of divide entanglement have been accepted for publication in an international peer-reviewed journal. With Y. Nozaki (Yokohama National Univ.), T. Kálmán (Tokyo Institute of Technology), and M. Teragakiuchi (Hiroshima Univ.), we have classified global phase defects up to homotopy, and this result is also posted on the arXiv.