2001 Fiscal Year Final Research Report Summary
The study on controlled surgery theory and its application
| Project/Area Number |
11640093
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Josai University |
|---|
Principal Investigator |
YAMASAKI Masayuki Josai Univ., Fac. Sci., Professor, 理学部, 教授 (70174646)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TSUCHIYA Takahiro Josai Univ., Fac. Sci., Lecturer, 理学部, 講師 (60316677)
TSUCHIYA Susumu Josai Univ., Fac. Sci., Associate Professor, 理学部, 助教授 (60077914)
NISHIZAWA Kiyoko Josai Univ., Fac Sci, Professor, 理学部, 教授 (90053686)
CHENG Qing-Ming Saga Univ., Fac. Sic., & Eng., Professor, 理工学部, 教授 (50274577)
NAKAMURA Toshiko Josai Univ., Fac. Sci., Lecturer, 理学部, 講師 (70316678)
|
|---|
| Project Period (FY) |
1999 – 2001
|
| Keywords | controlled surgery / stability / splitting / local simpleness |
|---|
| Research Abstract |
We studied properties of surgery groups controlled over metric spaces. Such groups are supposed to appear in controlled surgery sequences. Controlled surgery sequences have been verified to be exact in the case of trivial local fundamental groups. A key ingredient of the proof was the stability of the controlled surgery groups. Our main objective was to prove the stability of the controlled surgery groups in a more general setting.
To prove the stability, one needs a method to split controlled Poincare quadratic com-plexes. Splitting is always possible in the case of trivial local fundamental groups, because controlled Whitehead groups vanish. Unfortunately we cannot hope to acomplish splitting in general.
During the period of this research project, we introduced the notion of local simpleness for controlled Poincare quadratic complexes and used this notion to give a certain sufficient condition to splitting. Although this condition is not satisfied in general, there is a hope that the complexes appearing in the proof of stability of controlled surgery groups. In fact we have succeeded to verify the stability using our splitting in the case when the control space is a subcomplex of the unit circle.
We plan to continue this using some induction argument to establish the stability for control spaces embedded in higher dimensional spheres.
|
