2007 Fiscal Year Final Research Report Summary
The study of Low-dimensicnal manifolds with various geometric structures
| Project/Area Number |
18540081
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Kyoto University |
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Principal Investigator |
UE Masaaki Kyoto University, Graduate School of Science, Professor (80134443)
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| Co-Investigator(Kenkyū-buntansha) |
FUJII Michihiko Kyoto University, Graduate School of Science, Associate Professor (60254231)
KATO Tsuyoshi Kyoto University, Graduate School of Science, Associate Professor (20273427)
KATO Shin'ichi Kyoto University, Graduate School of Science, Professor (90114438)
USHIKI Shigehiro Kyoto University, Graduate School of Hunan and Environmental Studies, Professor (10093197)
NISHIWADA Kimimasa Kyoto University, Graduate School of Science, Professor (60093291)
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| Project Period (FY) |
2006 – 2007
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| Keywords | 3-manifold / 4-manifold / Seiberg-Witten theory / Heegaard Floer homology / homology 3-sphere |
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| Research Abstract |
The head investigator Ue continued the research of 3 and 4-manifolds. In particular he considered Fukumoto-Furuta invariant for rational homology 3-spheres coming from Seiberg-Witten theory (which coincides with the Neumann-Siebenmann invariant in case of plumbed 3-manifolds) and Ozsvath-Szabo's d-invariant defined by Heegaard Floer homology. He showed that these two invariants coincide for spherical 3-manifolds, and also for certain plumbed 3-manifolds. This implies that under such conditions both invariants are the integral 〓 of the classical
Rochlin invariant and also homology cobordism invariants. But they are not the sane in general, and it is sill open whether there exists an invariant satisfying the above two conditions. He also gave certain constraits for the signature of 4-manifolds bounding Seifert rational homology 3-spheres in terms of the above invariants.
The investigator Tsuyoshi Kato estimated the growth of the Casson handles embedded in the K3 surface by Yang-Milis Gauge theory, and also analized the entropy of iterations by families of maps. Fujii found the confluence phenomena of singular points of ordinary differential equations induced by deformations of hyperbolic 2-cone-manifolds, and descried harmonic vector fields on hyperbola 3-cone-manifolds in terms of hypergeometric functions. Shin'ichi Kato established the relative = symmetric space version of Jaquet's theorem, which claims that every irreducible admissible representation of p-adic reductive groups is embedded to an induced representation for irreducible cusp representation of a parabolic subgroup. Ushiki considered transgression operators over the space of distributions induced by complex dynamical systems over the Riemann sphere and showed that the Fredholm determinant is represented by Artin-Mazurzeta function.
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