1999 Fiscal Year Final Research Report Summary
Handlebody splittings of 3-manifolds with bounday
| Project/Area Number |
10640094
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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 | Waseda University |
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Principal Investigator |
SUZUKI Shinichi Waseda University, School of Education (Rigaku) , Professor, 教育学部, 教授 (10030777)
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| Project Period (FY) |
1998 – 1999
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| Keywords | 3-manifolds / Heegaard splittings / handlebodies |
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| Research Abstract |
We call a compact, connected, orientable 3-manifold M with nonempty boundary ∂M a bordered 3-manifold. A bordered 3-manifold H is said to be a handlebody of genus g iff H is the disk-sum (= the boundary connected-sum) of g copies of the solid-torus.
It is well-known that a closed (=compact, without boundary) , connected, orientable 3-manifold M is decomposed into two homeomorphic handlebodies ; and such a splitting is called a Heegaard splitting for M.
On the other hand, in 1970 J. S. Downig proved that every bordered 3-manifold can be decomposed into two homeomorphic handlebodies, and L. G. Roeling discoursed on these decompositions for bordered 3-manifolds with connected boundary. The research results are (1) to report the Downing's results in slightly modified forms, (2) to generalize the Roeling's results to borderd 3-manifolds with several boundaries, (3) to formulate a Haken type theorem for these decompositions in the way of Casson and Gordon, and (4) to discuss another Haken type theorem for these decomposition with essential proper disks.
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