1989 Fiscal Year Final Research Report Summary
Co-operative Research on Topology
| Project/Area Number |
63302001
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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 | University of Tokyo |
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Principal Investigator |
MATSUMOTO Yukio University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (20011637)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Tadashi Yamaguchi University, Faculty of Education, Associate Professor, 教育学部, 助教授 (10107724)
KAWAUCHI Akio Osaka City University, Faculty of Science, Professor, 理学部, 教授 (00112524)
MATSUMOTO Takao Hiroshima University, Faculty of Science, Associate Professor, 理学部, 助教授 (50025467)
KATO Mitsuyoshi Kyushu University, Faculty of Science, Professor, 理学部, 教授 (60012481)
MORITA Shigeyuki Tokyo Institute of Technology, Professor, 理学部, 教授 (70011674)
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| Project Period (FY) |
1988 – 1989
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| Keywords | Low dimensional topology / Jones polynomials / Instanton / Mapping class groups / Floer homology / 葉層構造 / 群作用 / 特異点論 |
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| Research Abstract |
I shall firstly describe the general character of the results of this project and secondly point out some remarkable achievements in the term 1988-89. A general aspect in this term was that activities in low dimensional topology were remarkable. In particular, the formerly unexpected and newly discovered interaction between low dimensional topology and theoretical physics was extensively investigated.
One of the newest achievements in this direction is Kohno's discovery of new invariants of 3-manifolds. which was motivated by conformal field theory. Such invariants had been of "constructed" by Witten but only at the physicist's level of rigor. Kohno's construction is mathematically rigorous. His invariants are essentially related to the mapping class groups of surfaces. Cohomology of these groups were deeply studied b,y S. Morita. He also clarified the relation between the Torelli group and the Casson invariant.
It is well-known that 4-dimensional topology has a strong connection with gauge theory. In this respect, the geometry and topology of moduli spaces of instantons were deeply investigated by a group at Hiroshima Univ. and by another at Univ. of Tokyo. The gauge theory has a 3-dimensional counterpart, and it produces an interesting homology theory of homology 3-spheres, that is, the Floer homology. The structure of this homology was intensively calculated by T. Yoshida. His results suggest an interesting connection between the homology and the Maslov index in symplectic geometry.
The Jones polynomial were extensively studied by A. Kawauchi et al. Kawauchi studied to what extent the polynomial determines the types of knots by his "imitation' theory. There were many other interesting results in our project. For details, please consult with the report we are preparing.
We had more that 20 symposia, and two annual symposia at Shizuoka Univ. (1988) and at Fukushima Univ. (1989). In conclusion. we have achieved our aim to a satisfactory extent.
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