2010 Fiscal Year Final Research Report
Framed cobordism invariant of V-manifolds and its application to the cobordism category of 3-manifolds
| Project/Area Number |
20740048
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Geometry
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| Research Institution | Tottori University of Environmental Studies |
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Principal Investigator |
FUKUMOTO Yoshihiro Tottori University of Environmental Studies, 理工学部, 准教授 (90341073)
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| Project Period (FY) |
2008 – 2010
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| Keywords | 同境圏 / ホモロジー同境群 / ホモロジー環 / V-多様体 / 3次元多様体 / 指数定理 / Seiberg-Witten理論 / 非結合代数 |
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| Research Abstract |
The boundary of a surface is a curve. Conversely, can any curve be realized as the boundary of a surface? In our research, we study this problem for objects called manifold which is a higher-dimensional generalization of curves and surfaces. In particular, we study what kind of 4-manifold has a given 3-dimensional manifold as boundary. In general, we can compute an invariant called homology ring of any manifold, which approximate the shape of manifolds by algebras. To study the relationship between the homology ring of 3-manifolds and that of 4-manifolds, we applied an inequality called Furuta-Kametani inequality of 4-dimensional manifolds (orbifolds), and introduced a kind of distances between 3-manifolds called Φ-Bounding genus which is a generalization of the notion of the Bounding genus introduced by Y. Matsumoto, and studied their properties.
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