2013 Fiscal Year Final Research Report
Structures of homology cobordism invariants in the cobordism category of 3-manifolds
| Project/Area Number |
23540113
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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 | Ritsumeikan University |
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Principal Investigator |
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| Project Period (FY) |
2011 – 2013
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| Keywords | ホモロジー同境 / ゲージ理論 / 結び目 / 指標多様体 / 平坦接続 / チャーン・サイモンズ不変量 / 指数定理 / オービフォルド |
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| Research Abstract |
Two 3-manifolds are called cobordant if they are realized as the boundary of a smooth 4-manifold. In particular, if it preserves the homology group which is one of the topological invariants of manifolds, then they are called homology cobordant, and their structures reflect the depth of the fundamental group which is also one of the topological invariants. In this research, by using a method called the gauge theory which originates in physics, we gave a necessary condition for several number of 3-manifolds called lens space are the boundary of a certain kind of smooth 4-manifold in terms of representations of the fundamental group of the 4-manifold giving the cobordism. Moreover, we explicitly calculated the topology of the space of all representations in a certain kind of matrices of the fundamental group of the space obtained by removing a knot from the 3-sphere, and determined the structure of a certain part.
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