2013 Fiscal Year Final Research Report
A research on Thurston's inequality for foliations and contact topology
| Project/Area Number |
23540106
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Chuo University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
MITSUMATSU Yoshihiko 中央大学, 理工学部, 教授 (70190725)
TAKAKURA Tatsuru 中央大学, 理工学部, 准教授 (30268974)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 葉層構造 / 接触トポロジー / Thurston の不等式 / h 原理 / 沈め込み分類理論 |
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| Research Abstract |
As a foliation is an integrable tangent plane field, in the case of 1-dimensional line fields, we treated a completely integrable vector field, which satisfies an extra integrability condition. We consider a placement problem of a closed leaf (i.e., a periodic trajectory) of the 1-dimensional foliation tangent to a completely integrable vector field on an open 3-manifold. Such a foliation is given by the inverse images of a submersion to the plane. We gave a necessary and sufficient condition for the realization for any given link, and in the case of a knot, we described the condition by the words of classical invariants.
A 2-dimensional foliation transverse to a completely integrable vector field satisfies Thurston's inequality and it gives a natural class of such foliations. Thus, we can consider such a class of foliations is a natural class on an open 3-manifold.
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