Automorphism group of a smooth G-manifold and its applications.

• Project/Area Number: 21540074
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Shinshu University
• Principal Investigator: ABE Kojun 信州大学, 理学部, 特任教授 (30021231)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 微分同相群 / 群の完全性 / 1次元ホモロジー群 / 可微分G-多様体 / 一様完全性 / 同変微分同相群 / 可微分軌道体 / 特異点をもつ多様体 / リプシッツ同相群 / 特異点をもっ多様体

Research Abstract

(1) Let M be a smooth G-manifold. We consider the diffeomorphisms, Lipschtz homeomorphisms and homeomorphisms of M preserving the G-action which are isotopic to the identity through the isotopies with compact support. We determined the first homology of those groups when M has codimension one orbits. From those results we see that the first homology reflects the properties of those categories.
(2) We consider the perfectness and uniformly perfectness for the group of the identity component of the diffeomorphisms of a smooth manifold pair. We have determined the condition for the group to be uniformly perfect when the dimension of the submanifold is one. The result can be applied to the knot theory.

Research Products

• [Journal Article] On the structure of the first homology group of equivariant diffeomorphisms of manifolds with smooth torus actions (2011)
  • Author(s): K. Abe
  • Journal Title: 数理解析研究所講究録 Volume: Vol 12 Pages: 91-102
• [Journal Article] On the structure of the first homology group of equivariant Diffeomorphisms of manifolds with smooth torus actions. (2011)
  • Author(s): 阿部孝順
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1732 Pages: 32-41
• [Journal Article] On the first homology of the automorphism groups of G-manifolds (2010)
  • Author(s): K. Abe
  • Journal Title: Trend in Mathematics-New Series Volume: Vol 12 Pages: 91-102
• [Journal Article] On the first homology of the automorphism groups of G-manifblds. (2010)
  • Author(s): K.Abe
  • Journal Title: Trend in Mathematics-New Series Volume: 12 Pages: 91-102
• [Journal Article] 余次元1軌道をもつG-多様体の同変部分群の1次元ホモロジー (2009)
  • Author(s): 阿部孝順
  • Journal Title: 京都大学数理解析研究所講究録 Volume: Vol 1670 Pages: 91-99
• [Journal Article] Commutators of $ C^{￥infty}$-diffeomorphisms preserving a submanifold (2009)
  • Author(s): K. Abe, K. Fukui
  • Journal Title: Jour. Math. Soc. Japan Volume: Vol 61 Pages: 427-436
  • Peer Reviewed
• [Journal Article] Commutators of $C^{\infty}$-diffeomorphisms preserving a submanifold (2009)
  • Author(s): K.Abe, K.Fukui
  • Journal Title: Jour.Math.Soc.Japan 61 Pages: 427-436
  • Peer Reviewed
• [Journal Article] 余次元1軌道をもつG-多様体の同変部分群の1次元ホモロジー (2009)
  • Author(s): 阿部孝順
  • Journal Title: 京都大学数理解析研究所講究録 1670 Pages: 91-99
• [Presentation] Uniform perfectness of diffeomorphism groups (2011)
  • Author(s): K. Abe
  • Organizer: Max Planck Institute for Mathematics
  • Year and Date: 2011-11-21
• [Presentation] Uniform perfectness of diffeomorphism groups, I, II (2011)
  • Author(s): K.Abe
  • Organizer: Geometric Topology Seminar
  • Place of Presentation: Max Planck Institute for Mathematics(招待講演)
  • Year and Date: 2011-11-21
• [Presentation] On the uniform perfectness of diffeomorphism groups preserving a submanifold (2011)
  • Author(s): K. Abe and K. Fukui
  • Organizer: 多様体の平面場と微分同相群2011研究集会
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2011-11-04
• [Presentation] On the uniform perfectness of diffeomorphism groups preserving a submanifold (2011)
  • Author(s): K. Abe and K. Fukui
  • Organizer: 尾鷲微分トポロジー2011
  • Place of Presentation: 尾鷲中央公民館
  • Year and Date: 2011-08-23
• [Presentation] On the uniform perfectness of diffeomorphism preserving a submanifold (2011)
  • Author(s): K. Abe and K. Fukui
  • Organizer: 変換群の幾何と組合せ論
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-06-13
• [Presentation] On the uniform perfectness of diffeomorphisms preserving a submanifold (2011)
  • Author(s): 阿部孝順
  • Organizer: 変換群の幾何と組合せ論
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-06-13
• [Presentation] On the first homology of the group of equivariant diffeomorphismsof smooth S^1-manifolds (2010)
  • Author(s): K. Abe
  • Organizer: The Fourth Group Action Forum Conference
  • Place of Presentation: University of Milano-Bicocca
  • Year and Date: 2010-09-16
• [Presentation] On the first homology of the group of equivariant diffeomorphisms of smooth S^<1>-manifolds (2010)
  • Author(s): 阿部孝順
  • Organizer: Group Actions in Topology and Analysis The Fourth Group Action Forum Conference
  • Place of Presentation: Universit of Milano-Bicocca
  • Year and Date: 2010-09-16
• [Presentation] On the structure of the first homology of the group of equivariant diffeomorphisms of manifolds with smooth torus actions (2010)
  • Author(s): K. Abe
  • Organizer: 変換群と手術理論
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-08-11
• [Presentation] On the first homology group of diffeomorphisms of smooth orbifolds and its applications (2010)
  • Author(s): K. Abe
  • Organizer: Workshop on Toric Topology and Related Topics
  • Place of Presentation: Fudan University
  • Year and Date: 2010-05-05
• [Presentation] On the first homology of automorphism groups of G-manifolds (2010)
  • Author(s): K. Abe
  • Organizer: KAIST Toric Topology Workshop
  • Place of Presentation: KAIST
  • Year and Date: 2010-02-26
• [Presentation] On the first homology of automorphism groups of G-manifolds (2010)
  • Author(s): K.Abe
  • Organizer: KAIST Toric Topology Workshop 2010
  • Place of Presentation: KAIST(韓国)
  • Year and Date: 2010-02-26
• [Presentation] The first homology of automorphism group of G-manifolds (2009)
  • Author(s): K. Abe
  • Organizer: 多様体の微分同相群
  • Place of Presentation: 東京大学玉原セミナーハウス
  • Year and Date: 2009-10-28
• [Presentation] The first homology of automorphism group of $G$-manifolds (2009)
  • Author(s): K.Abe
  • Organizer: 多様体の微分同相群
  • Place of Presentation: 東京大学玉原セミナーハウス
  • Year and Date: 2009-10-28
• [Presentation] The first homology of the group of equivariant diffeomorphisms and its application (2009)
  • Author(s): K. Abe
  • Organizer: Hajos Geometria szeninariuum
  • Place of Presentation: Eotovos Lorand Univrsity
  • Year and Date: 2009-09-11
• [Presentation] The first homology of the group of equivariant diffeomorphisms and its application (2009)
  • Author(s): K.Abe
  • Organizer: Hajos Geometria szeninariuum
  • Place of Presentation: Eotovos Lorand Univrsity(ハンガリー)
  • Year and Date: 2009-09-11
• [Presentation] On the first homology of the automorphism groups of G-manifolds (2009)
  • Author(s): K. Abe
  • Organizer: Group Actions and Homogeneous Spaces
  • Place of Presentation: コメニウス大学
  • Year and Date: 2009-09-07
• [Presentation] On the first homology of the automorphism groups of G-manifolds. (2009)
  • Author(s): K.Abe
  • Organizer: Group Actions and Homogeneous Spaces.
  • Place of Presentation: コメニウス大学(スロバキア)
  • Year and Date: 2009-09-07
• [Presentation] 同変同相群の構造,変換群の幾何とその周辺 (2009)
  • Author(s): 阿部孝順
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-08-19
• [Presentation] 同変同相群の構造 (2009)
  • Author(s): 阿部孝順
  • Organizer: 変換群も幾何とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-08-19
• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~kabe/index-j.html
• [Remarks]
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• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~kabe/index-j.html
• [Remarks]
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