2018 Fiscal Year Final Research Report
Seifert fiber spaces, L-spaces, left-orderability of fundamental groups and Dehn surgery
| Project/Area Number |
26400099
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Nihon University |
|---|
Principal Investigator |
|
|---|
| Research Collaborator |
ICHIHARA kazuhiro
TERAGAITO masakazu
MIYAZAKI katura
|
|---|
| Project Period (FY) |
2014-04-01 – 2019-03-31
|
| Keywords | 幾何学 / トポロジー / ３次元多様体 / 結び目 / デーン手術 / ザイフェルト多様体 / L-空間 / 共役ねじれ元 |
|---|
| Outline of Final Research Achievements |
We study Seifert surgeries on knots, which are exceptional Dehn surgeries, from a networking viewpoint. In particular, a joint work with Baker (University of Miami) extends a foundation of Seifert Surgery Network by generalizing a notion of seiferter. Using seiferters which have key roles in a study of Seifert Surgery Network, we study a relationship between L-space knots and twisting operation and provide a plenty of twist families of knots which contain infinitely many L-space knots. Furthermore, recently in a joint work with Baker we gave a necessary condition for a twist family of knots to contain infinitely many L-space knots.
We also study generalized torsion elements in 3-manifold groups, and a joint work with Teragaito (Hiroshima University) proves that the fundamental group of a geometric 3-manifold which is not hyperbolic has no bi-ordering if and only if it has a generalized torsion element.
|
| Free Research Field |
低次元トポロジー
|
| Academic Significance and Societal Importance of the Research Achievements |
本研究ではザイフェルト手術のネットワークの視点からの研究で導入されたSeifert Surgery Networkの連結性に関連してseiferterの概念を一般化し、研究の枠組みを拡張した。また、ネットワークの視点を現在活発に研究されているL-空間結び目の研究に応用し、豊富な例の構成をはじめ、L-空間結び目への幾何的制限を与えることに成功した。証明の過程でタイトな接触構造をサポートするファイバー結び目に関する興味深い事実を明らかにした。最近注目されている３次元多様体の基本群の共役ねじれ元についても基本群の両側不変順序と関連した研究を進展させ、今後の研究の新たな方向を与えることができた。
|
