Research on foliations, contact structures and Euler class
| Project/Area Number |
18540095
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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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 |
MIYOSHI Shigeaki Chuo University, Faculty of Science and Engineering, Professor (60166212)
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| Co-Investigator(Kenkyū-buntansha) |
MITSUMATSU Yoshihiko Chuo University, Faculty of Science and Engineering, Professor (70190725)
TAKAKURA Tatsuru Chuo University, Faculty of Science and Engineering, Associate Professor (30268974)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥2,340,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | foliations / Thurston norm / open book decomposition / contact structures / confoliation |
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| Research Abstract |
W. Thurston showed that a foliation on a 3-manifold which has no Reeb component enjoys the property that the Euler class of the tangent bundle satisfies an inequality, Thurston's inequality. On the other hand, the Reeb foliation on the three sphere satisfies Thurston's inequality and a previous research followed by this research showed that there is a class of foliations each of which has Reeb components and satisfi es Thurston's inequality.
In the research in 2006, for a class of foliations which are called spinnable foliations, we obtained a sufficient condition for the foliation satisfying Thurston's inequality. Moreover, we revealed an aspect where Thurston's inequality does not hold. They are described by properties of the monodromy diffeomorphisms which determine the spinnable foliations
In view of the research with respect to the convergence of contact structures to foliations, we studied a finer inequality, the relative version of Thurston's inequality, which deepens the research until 2006. In fact, for spinnable foliations we showed that the relative version implies the absolute version. The same statement for contact structures was known however, it does not hold in general for foliations. Also in 2007, we found the method to construct a foliation which satisfies Thurston's inequality with Euler class of infinite order. Until then, all foliations which satisfies Thurston's inequality have trivial Euler class. Indeed, we can find'a spinnable foliation whose Euler class is of infinite order by the research in 2006. Then we can perform Dehn surgery along the Reeb component and with certain condition on the original spinnable foliation we can conclude that with finitely many exceptions the resultant satisfies Thurston's inequality with Euler class of infinite order by virtue of D. Gabai's sutured manifold theory.
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Report
(3 results)
Research Products
(51 results)
