2006 Fiscal Year Final Research Report Summary
Geometry and topology of 4-manifolds
| Project/Area Number |
16204003
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (A)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | University of Tokyo |
|---|
Principal Investigator |
MATSUMOTO Yukio University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理科学研究科, 教授 (20011637)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
ASHIKAGA Tadashi Tohoku Gakuin Univ., Dept. of Tech., Prof., 工学部, 教授 (90125203)
IMAYOSHI Yoichi Osaka City Univ., Grad. School of Sci., Prof., 大学院理学研究科, 教授 (30091656)
SAEKI Osamu Kyushu Univ., Grad. School of Math. Sci., Prof., 大学院数理学研究院, 教授 (30201510)
KAMADA Seiichi Hiroshima Univ., Grad. School of Sci., Prof., 大学院理学研究科, 教授 (60254380)
ENDO Hisaaki Osaka Univ., Grad. School of Sci., Associate Prof., 大学院理学研究科, 教授 (20323777)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | 4-manifolds / signature / Lefschetz fibration / Riemann surface |
|---|
| Research Abstract |
Since a decade ago, there have been many research activities on 4-manifolds having the structure of Lefschetz fibrations. The concept of Lefschetz pencils was introduced in algebraic geometry in 1920's, but recently the concept has become the central theme of investigations mainly due to the results of Donaldson and Gompf according to which Lefschetz fibrations are the underlying structure of the symplectic structure. The structure of Lefschetz fibrations is topologically described by their monodromy representations taking the value in the mapping class groups, while the structure is, as mentioned above, closely related to algebraic geometry and symplectic geometry. Thus the study of Lefschetz fibrations is considered to be a meeting place of many research areas, such as 4-manifolds, mapping class groups of surfaces, algebraic geometry, symplectic geometry ad so on. In the present research, we took the above mentioned situation in consideration, and put stress on communication between different research areas. For example, we held a research seminar every summer during the research period entitled "Topology and Algebraic Geometry". Also by the same idea, we held two seminars entitled "Topology and Symplectic Geometry".
We have many new results on Lefschetz fibrations, fibering structure of low dimensional manifolds, monodromy, combinatorics of charts, mapping class groups (Teichmuller modular groups), degenerations of Riemann surfaces and the construction of splitting families, topological and analytic local signature, Dedekind sums, and symplectic geometry. By these results, we have obtained some new prospects of research.
|
