2000 Fiscal Year Final Research Report Summary
4-dimensional topology and topological field theory.
| Project/Area Number |
11440021
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | University of Tokyo (2000)
Kyoto University (1999) |
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Principal Investigator |
FURUTA Mikio University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50181459)
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| Co-Investigator(Kenkyū-buntansha) |
MINAMI Norihiko Nagoya Insitute of Technology, Associate professor, 工学部・共通講座教室, 助教授 (80166090)
KONNO Hiroshi University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (20254138)
UE Masaaki Kyoto University, Associate Professor, 総合人間学部, 助教授 (80134443)
MOCHIZUKI Takuro Osaka City University, Assistant Professor, 理学研究科, 助手 (10315971)
KOTANI Motoko Tohoku University, Associate Professor, 理学研究科, 助教授 (50230024)
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| Project Period (FY) |
1999 – 2000
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| Keywords | 4-dimensional topology / hyperkahler quotient / Seiberg-Witten theory / gauge theory / topological field theory / index theorem / monopole equation / moduli space |
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| Research Abstract |
1 The main purpose of the present research is to develope K-theoretical aspect of Seiberg-Witten theory.
(1) Kametani and Furuta extended the notions of Pin structures and e-invariants to obtain an application to the stable-homotopy Seiberg-Witten invariants. They actually showed an inequality stronger than the 11/8-inequality under some condition on the cup product on H^1.
(2) Minami showed "G-join theorem" and apply it to the stable-homotopy Seiberg-Witten invariants and improved the 10/8-theorem for spin 4-manifolds.
(3) Ue, Furuta and Y.Fukumoto investigated properties of of the w-invariant, which they defined, and obtained some applications to Seiberg-fibered homology 3-spheres.
(4) Konnno calculated the cohomology ring of the moduli spaces of polygons as examples of hyper-kahler quotients.
(5) Kotani obtained a central limit theorem for magnetic transition operators on a crystal lattice.
2 The main members of the present research regularly attended a series of conferences to discuss the following topics.
(1) Mochizuki explained his work on r-spin structure. Furuta pointed out a possibility of a construction in 2-dimensional topology which is parallel to the stable homotopy version of the Seiberg-Witten invariants.
(2) M.Morishita (Univ. Kanazawa) was invited to explain his idea about a similarity between number theory and 3-dimensional topology. We discussed Mazur's unpublished work on a similarity between general knots and fibered knots when the fundamental groups of the knot complements are completed in some way.
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