Topology of the group of diffeomorphisms and the spaces of embeddings of manifolds

• Project/Area Number: 21840002
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Hokkaido University
• Principal Investigator: WATANABE Tadayuki Hokkaido University, 大学院・理学研究院, 助教 (70467447)
• Project Period (FY): 2009 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥2,015,000 (Direct Cost: ¥1,550,000、Indirect Cost: ¥465,000); Fiscal Year 2010: ¥1,105,000 (Direct Cost: ¥850,000、Indirect Cost: ¥255,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: ファイバー束 / 微分同相 / 埋め込み / Morse / 理論 / 代数的K理論 / 微分同相群 / Morse理論 / 有理ホモトピー群 / 擬アイソトピー

Research Abstract

The author studied the classification problem of smooth fiber bundles with fiber diffeomorphic to the sphere. He defined "algebraic obstructions" for fiber bundles over 2-sphere with fiber diffeomorphic to odd dimensional sphere and showed that if the fiber dimension is at least 7 and if the obstructions vanish then the fiber bundle is in fact trivial. He also showed that if the fiber dimension is 5 and if the obstructions vanish then the fiber bundle is presentable by a family of embeddings of 2-spheres and an element of some discrete abelian group. Thus the classification of sphere bundles is reduced to an algebraic problem and the classification of families of embeddings.

Research Products

• [Journal Article] 1-loop graphs and configuration space integralfor embedding spaces (2011)
  • Author(s): K.Sakai, T.Watanabe
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society
  • Peer Reviewed
• [Journal Article] 1-loop graphs and configuration space integral for embedding spaces (2011)
  • Author(s): Keiichi Sakai, Tadayuki Watanabe
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: (印刷中)
  • NAID: 120005228644
  • Peer Reviewed
• [Journal Article] 1-loop graphs and configuration space integral for embedding spaces (2010)
  • Author(s): Keiichi Sakai, Tadayuki Watanabe
  • Journal Title: ArXiv(preprint) ArXiv : 1002. 4696 Pages: 1-33
  • NAID: 120005228644
• [Journal Article] On Kontsevich' s characteristic classes for higher dimensional sphere bundles. II. Higher classes (2009)
  • Author(s): T.Watanabe
  • Journal Title: Journal of Topology 2 Pages: 624-660
  • Peer Reviewed
• [Journal Article] On Kontsevich's characteristic classes for higher dimensional sphere bundles. I. The simplest class (2009)
  • Author(s): T.Watanabe
  • Journal Title: Mathematische Zeitschrift 262 Pages: 683-712
  • Peer Reviewed
• [Presentation] S^2上の可微分なD^5-bundleの表示について (2010)
  • Author(s): 渡邉忠之
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-23
• [Presentation] S^2上の可微分なD^5-bundleの表示について (2010)
  • Author(s): 渡邉忠之
  • Organizer: 日本数学会秋季分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-23
• [Presentation] Sphere bundles, pictures and families of embeddings (2010)
  • Author(s): 渡邉忠之
  • Organizer: 信州トポロジーセミナー
  • Place of Presentation: 信州大学理学部
  • Year and Date: 2010-07-15
• [Presentation] Sphere bundles, pictures and families of embeddings (2010)
  • Author(s): T.Watanabe
  • Organizer: Lefschetz fibration and category theory
  • Place of Presentation: 大阪大学数学教室
  • Year and Date: 2010-06-10
• [Presentation] Sphere bundles, pictures and families of embeddings (2010)
  • Author(s): Tadayuki Watanabe
  • Organizer: Lefschetz fibration and category theory
  • Place of Presentation: 大阪大学 招待講演
  • Year and Date: 2010-06-10
• [Presentation] Sphere bundles, pictures and families of embeddings (2010)
  • Author(s): 渡邉忠之
  • Organizer: 北海道大学特異点論セミナー
  • Place of Presentation: 北海道大学数学教室
  • Year and Date: 2010-05-14
• [Presentation] Sphere bundles, pictures and families of embeddings (2010)
  • Author(s): T.Watanabe,
  • Organizer: mini-workshop on the topology of the space of knots
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2010-02-17
• [Presentation] Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds (2009)
  • Author(s): 渡邉忠之
  • Organizer: 北海道大学幾何学コロキウム
  • Place of Presentation: 北海道大学数学教室
  • Year and Date: 2009-12-22
• [Presentation] グラフホモロジーと特性類 (2009)
  • Author(s): 渡邉忠之
  • Organizer: 北海道大学談話会
  • Place of Presentation: 北海道大学数学教室
  • Year and Date: 2009-05-07
• [Remarks] ホームページ等
  • URL: http://www.math.sci.hokudai.ac.jp/~tadayuki/publications.php
• [Remarks]
  • URL: http://www.math.sci.hokudai.ac.jp/^tadayuki/publications.php