1998 Fiscal Year Final Research Report Summary
Study on global theory of singularities of mapping and 3-and 4-dimensional manifolds
| Project/Area Number |
09640112
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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 | HIROSHIMA UNIVERSITY |
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Principal Investigator |
KURA Takeshi Faculty of Science, HIROSHIMA UNIVERSITY Research Associate, 理学部, 助手 (10161720)
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| Co-Investigator(Kenkyū-buntansha) |
TERAGAITO Masakazu Faculty of School Education, Associate Professor, 学校教育学部, 助教授 (80236984)
KANNO Hiroaki Faculty of Science, Associate Professor, 理学部, 助教授 (90211870)
SAEKI Osamu Faculty of Science, Associate Professor, 理学部, 助教授 (30201510)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Definite fold singularities / Stein factorization / Topological Quantum Field Theory / Seiberg-Witten Theory / Dehn surgery / Lens space / Riemannian Manifold / Green function |
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| Research Abstract |
The following results (1-4) have been obtained in the term of this project.
1. (0. Saeki) About maps into R^3 with only definite fold singularities, which are the simplest ones among generic singularities, various results about the structure of the source manifolds have been obtained by using their Stein factorizations. In particular, the existence of such maps on a 4-manifold has been shown to be strongly related to its smooth structure. Furthermore, the existence of maps with only Morin singularities has been shown to have close relationship to the Hopf invariant one problem. Moreover, a formula for the number of singularities of maps of surfaces with boundary into 3-manifolds has been obtained.
2. (H.Kanno) We have constructed cohomological gauge theories on Spin(7) or SU(4) manifold in eight dimensions as first examples of topological quantum field theories in higher dimensions. A relation to ten dimensional supersymmetric gauge theory has been clarified. We also investigated the Seiberg-Witten theory. Based on the Picard-Fuchs equation for the period integrals on the Seiberg-Witten curve, an analysis was mede on an effective theory of five dimensional supersymmetric gauge theory compactified on a circle. We have computed the topological partition function (the Donaldson-Witten function) of N=2 supersymrnetric massless QCD, which is regarded as a generating function of invariantsof four manifold.
3. (M.Teragaito) Dehn surgery on knots is a fundamental important operation to create closed ori-entable 3-manifolds. We focus on genera of knots as a new standing point, and we proved that there is an estimate for the orderof the fundamental group of a lens space obtained by Dehn surgery on a hyperbolic knot by using the genus of the knot.
4. (T.Kura) By means of curvatures, we have obtained the classification conditions of Riemannian manifolds based on the existence of p-Green functions.
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