Symmetry from the homotopical viewpoint

• Project/Area Number: 21540068
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Chiba University
• Principal Investigator: MARUYAMA Kenichi 千葉大学, 教育学部, 教授 (70173961)
• Co-Investigator(Renkei-kenkyūsha): IWASE Norio 九州大学, 大学院・数理学研究院, 教授 (60213287)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 位相幾何学 / ホモトピー論 / ホモトピー同値 / 自己同型 / 群論 / 自己同型群

Research Abstract

Study of symmetry has been one of the major subjects in geometry for centuries. In our study we investigated symmetry from the homotopy theoretical viewpoint. As in the other geometrical theories, the symmetry of a space is described by a certain set of automorphisms. Our main result in this study is that the algebraic structure of the automorphisms is closely related to the linear groups in algebra. This result is valid for quite general spaces and hence it is thought as a fundamental result of the field.

Research Products

• [Journal Article] The groups of self-homotopy equivalences and the Tits alternative (2011)
  • Author(s): 丸山研一
  • Journal Title: Bulletin of London Mathematical Society Volume: Vol.43 Pages: 1191-1197
  • Peer Reviewed
• [Journal Article] On the rank of the group of self-homotopy equivalences (2011)
  • Author(s): 丸山研一
  • Journal Title: 千葉大学教育学部研究紀要 Volume: 59巻 Pages: 275-277
  • NAID: 120007065383
• [Journal Article] The groups of self-homotopy equivalences and the Tits alternative (2011)
  • Author(s): Ken-ichi Maruyama
  • Journal Title: Bulletin of London Mathematical Society Volume: 43 Issue: 6 Pages: 1191-1197
  • DOI: 10.1112/blms/bdr053
  • Peer Reviewed
• [Journal Article] On the rank of the group of self-homotopy equivalences (2011)
  • Author(s): 丸山研一
  • Journal Title: 千葉大学教育学部紀要 Volume: 59 Pages: 275-277
  • NAID: 120007065383
• [Journal Article] Topological complexity is a fibrewise L-S category (2010)
  • Author(s): 岩瀬則夫, 酒井道宏
  • Journal Title: Topology Appl. Volume: Vol.157 Pages: 10-21
  • NAID: 120005289486
  • Peer Reviewed
• [Journal Article] Categorical length,relative L-S category and higher Hopf invariants、 Algebraic topology old and new、 Banach Center Publ 、Polish Acad. Sci (2009)
  • Author(s): 岩瀬則夫
  • Journal Title: Inst. Math. Volume: Vol.85 Pages: 205-224
  • Peer Reviewed
• [Presentation] The groups of self-homotopy equivalences and the Tits alternative (2011)
  • Author(s): 丸山研一
  • Organizer: トポロジーセミナー
  • Place of Presentation: マラガ大学(スペイン)
  • Year and Date: 2011-09-21
• [Presentation] The groups of self-homotopy equivalences and the Tits alternative (2010)
  • Author(s): 丸山研一
  • Organizer: トポロジーセミナー
  • Place of Presentation: ルーバンカトリック大学(ベルギー)
  • Year and Date: 2010-09-23
• [Presentation] 対称性をめぐって (2009)
  • Author(s): 丸山研一
  • Organizer: 茨城大学理学部談話会(兼先端科学トピックス)
  • Place of Presentation: 茨城大学
  • Year and Date: 2009-11-10
• [Presentation] 対称性をめぐって (2009)
  • Author(s): 丸山研一
  • Organizer: 茨城大学理学部談話会(兼先端科学トピックス)
  • Place of Presentation: 茨城大学(水戸市)
  • Year and Date: 2009-11-10
• [Remarks]
  • URL: http://www.e.chiba-u.jp/~maruyama