Project/Area Number  02302003 
Research Category 
GrantinAid for Cooperative Research (A).

Research Field 
代数学・幾何学

Research Institution  Kyushu University 
Principal Investigator 
KATO Mitsuyoshi Kyushi Univ. Fac. Sci. Professor, 理学部, 教授 (60012481)

CoInvestigator(Kenkyūbuntansha) 
FUKUDA Takuo Tokyo Inst. Tech. Fac. Sci. Professor, 理学部, 教授 (00009599)
KAWAKUBO Katsuo Osaka Univ. Fac. Sci. Professor, 理学部, 教授 (50028198)
KAWAUCHI Akio Osaka City Univ. Fac. Sci. Professor, 理学部, 教授 (00112524)
MORITA Shigeyuki Tokyo Inst. Tech. Fac. Sci. Professor, 理学部, 教授 (70011674)
MATSUMOTO Yukio Univ. of Tokyo Fac. Sci. Professor, 理学部, 教授 (20011637)

Project Fiscal Year 
1990 – 1991

Project Status 
Completed(Fiscal Year 1991)

Budget Amount *help 
¥18,000,000 (Direct Cost : ¥18,000,000)
Fiscal Year 1991 : ¥7,900,000 (Direct Cost : ¥7,900,000)
Fiscal Year 1990 : ¥10,100,000 (Direct Cost : ¥10,100,000)

Keywords  Knot Theory / 3Manifold / Mapping Class Group / Casson Invariant / Heegaard Splitting / Floer Homology / Maslov Index / Symplectis Geometry / 結び目理論 / 3次元多様体 / Heegaard分解 / 曲面の写像類群 / Casson不変量 / 量子群 / Floerホモロジ / 衛星結び目 / 衛星結び目(Satellite Knet) / コンパニオン(Companion) / Heegaard種数 / Mathieuの問題 / HassーThompson予想 / 位相的量子場理論 / Verlindeの等式 / DeligneーMumfordのコンパクト化 / 写像類群 / キャッソン(Casson)不変量 / Kー2方程式のモノドロミ表現 / Rー行列 / マスロフ(Maslov)指数 
Research Abstract 
This research project has supported 25 conferences concerning various branches of topology and brought fruitful results. These conferences were proposed by topologists in our country. In particularly, research in knot theory and 3manifold theory is distinguished and will be highly appreciated. Shigeyuki Morita studied the mapping class groups of surfaces by making use of various branches of mathematics, for example, cohomology theory, combinatorial group theory, algebraic geometry and topology, to give a new interpretation of Casson invariant and its generalization and proposed new problems. His work will be regarded as a noncommutative algebraic topology in 3manifold theory. Toshitake Kohno proved that the monodromy representation of KZ equation can be written by a Rmatrix. He also constructed a projective representation of the mapping class group to define topological invariants of closed oriented 3manifolds via their Heegaard splittings. Tomoyoshi Yoshida gave an algorithm to compute Floer homology of various homology 3spheres by proving that the relative Morse index appeared in the definition of Floer homology is considered as a Maslov index of a certain finite dimensional symplectic geometry. After these experts' works around ICM'90, Kyoto, young topologists are giving various nice results in this field as follows ; Masaharu Kohno and Kimihiko Motegi solved the sattelite knot problem in its oriented version. Tsuyoshi Kobayashi and Haruko Nishi gave a proof of conjecture of HassThompson. Akira Yasuhara answered Mathieu's problem negatively. Toshie Takada proved the Verlinde's formula, appeared in the mathematical physics, by commuting link invariants in two different manners. These results by young mathematicians after ICM'90 would symbolise the success of this research project.
