1991 Fiscal Year Final Research Report Summary
Co-operative Research of Topology
| Project/Area Number |
02302003
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| Research Category |
Grant-in-Aid for Co-operative Research (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Kyushu University |
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Principal Investigator |
KATO Mitsuyoshi Kyushi Univ. Fac. Sci. Professor, 理学部, 教授 (60012481)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1990 – 1991
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| Keywords | Knot Theory / 3-Manifold / Mapping Class Group / Casson Invariant / Heegaard Splitting / Floer Homology / Maslov Index / Symplectis Geometry |
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| 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 3-manifold 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 non-commutative algebraic topology in 3-manifold theory.
Toshitake Kohno proved that the monodromy representation of K-Z equation can be written by a R-matrix. He also constructed a projective representation of the mapping class group to define topological invariants of closed oriented 3-manifolds via their Heegaard splittings.
Tomoyoshi Yoshida gave an algorithm to compute Floer homology of various homology 3-spheres 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 Hass-Thompson.
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.
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