The mapping class groups of Heegaard splittings

2016 Fiscal Year Final Research Report

• Project/Area Number: 26800028
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hiroshima University
• Principal Investigator: Koda Yuya 広島大学, 理学研究科, 准教授 (20525167)
• Research Collaborator: ISHIKAWA Masaharu ; OZAWA Makoto ; CHO Sangbum ; SEO Arim
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 3 次元多様体 / Heegaard 分解 / 写像類群 / 結び目 / トンネル / ヘンペル距離 / 国際情報交換 韓国

Outline of Final Research Achievements

Every closed orientable 3-manifold can be decomposed into 2 handlebodies by cutting it along a closed orientable surface of genus g. This decomposition is called a Heegaard splitting. Given a Heegaard splitting, the group of isotopy classes of orientation-preserving self-homeomorphisms of the 3-manifold that preserve the splitting is called the Goeitz group of the splitting. The aim of this project was to provide a finite presentation of the Goeritz group for every reducible genus-2 Heegaard splitting, and it has completed successfully. As applications or related topics, we obtained the following: (1) a sufficient condition that the Goeritz group of a higher genus Heegaard splitting admits a finite generationg set; (2) the "uniqueness" of (1,1)-decompositions of 2-bridge knots; (3) a characterization of "knotted" subspaces in terms of "relative" homotopy type of knots in the subspaces.

Free Research Field

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