2003 Fiscal Year Final Research Report Summary
Research for low-dimensional manifolds with various geometric structures
| Project/Area Number |
14540076
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | KYOTO UNIVERSITY |
|---|
Principal Investigator |
UE Masaaki Kyoto Univ., Graduate School of Science, Ass.Professor, 大学院・理学研究科, 助教授 (80134443)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KONO Norio Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90028134)
KATO Shinichi Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90114438)
IMANISHI Hideki Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90025411)
FUJII Michihiko Kyoto Univ., Graduate School of Science, Ass.Professor, 大学院・理学研究科, 助教授 (60254231)
NISHIWADA Kimimasa Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60093291)
|
|---|
| Project Period (FY) |
2002 – 2003
|
| Keywords | 3-manifolds / 4-manifold / spin structure / Dirac operator / Dehn surgery / Seiberg-Witten theory / V-manifold / cone manifold |
|---|
| Research Abstract |
Ue studied the constraints on the diffeomorphism types of 3, 4-manifolds by certain invariants originated from Seiberg-Witten theory. The contribution of the index of the Dirac operator to the isolated singularities of V 4-manifolds previously studied by him is an integer valued invariant for the pair of a spherical 3-manifold and its spin structure, which gives an integral lift of the Rochlin invariant (determined modulo 16), which coincides with the Neumann-Siebenmann invariant. He considered the case when a certain spherical 3-manifold is obtained by surgery on a knot and gave some constraints on its type in terms of the above invariant and also gave certain relations between the invariants of the spherical 3-manifolds in the case that they are obtained by simultanious surgery on a common knot. He also extended the results to the case of general Seifert 3-manifolds and gave some constraints of them to be obtained by surgery on a knot in terms of the Neumann-Siebenmann invariants. Recently some constraints for the Seifert 3-manifolds to be obtained by surgery on a knot are given by Ozsvath-Szabo's Floer homology. So our next task is to investigate the relations between the Floer homology and the above invariants. Fujii suceeded the study of the local transfromations of 3-dimension hyperbolic cone manifolds in terms of Gaussian hypergeometric functions. Imanishi suceeded the study of the cohomology of the the group of Lipschitz homeomorphisms preserving the differentiable foliations of codimension 1 by utilizing several results about the group of Lipschitz homeomorphisms of the interval.
|
