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The study on controlled surgery theory and its application

Research Project

Project/Area Number 11640093
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionJosai University

Principal Investigator

YAMASAKI Masayuki  Josai Univ., Fac. Sci., Professor, 理学部, 教授 (70174646)

Co-Investigator(Kenkyū-buntansha) TSUCHIYA Takahiro  Josai Univ., Fac. Sci., Lecturer, 理学部, 講師 (60316677)
TSUCHIYA Susumu  Josai Univ., Fac. Sci., Associate Professor, 理学部, 助教授 (60077914)
NISHIZAWA Kiyoko  Josai Univ., Fac Sci, Professor, 理学部, 教授 (90053686)
CHENG Qing-Ming  Saga Univ., Fac. Sic., & Eng., Professor, 理工学部, 教授 (50274577)
NAKAMURA Toshiko  Josai Univ., Fac. Sci., Lecturer, 理学部, 講師 (70316678)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1999: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordscontrolled surgery / stability / splitting / local simpleness / ホワイトヘッドのねじれ / マイヤー・ビートリス完全系列
Research Abstract

We studied properties of surgery groups controlled over metric spaces. Such groups are supposed to appear in controlled surgery sequences. Controlled surgery sequences have been verified to be exact in the case of trivial local fundamental groups. A key ingredient of the proof was the stability of the controlled surgery groups. Our main objective was to prove the stability of the controlled surgery groups in a more general setting.
To prove the stability, one needs a method to split controlled Poincare quadratic com-plexes. Splitting is always possible in the case of trivial local fundamental groups, because controlled Whitehead groups vanish. Unfortunately we cannot hope to acomplish splitting in general.
During the period of this research project, we introduced the notion of local simpleness for controlled Poincare quadratic complexes and used this notion to give a certain sufficient condition to splitting. Although this condition is not satisfied in general, there is a hope that the complexes appearing in the proof of stability of controlled surgery groups. In fact we have succeeded to verify the stability using our splitting in the case when the control space is a subcomplex of the unit circle.
We plan to continue this using some induction argument to establish the stability for control spaces embedded in higher dimensional spheres.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (21 results)

All Other

All Publications (21 results)

  • [Publications] Masayuki YAMASAKI: "Controlled surgery theory"Sugaku Expositions. 13. 113-124 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 西沢 清子: "Moduli space of the polynomials with degree n"数理明析研究所講実録. 1187. 221-227 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Takahiro TSUCHIYA: "General saddlepoint approximations to distributions under anelliptional population"Communications in Statistics. 28. 727-754 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Toshihiko NAKAMURA: "Spiral traveling wave solutions of some parabolic equations on annuli"NLA99 Computer AlgebraJosai Mathematical Monographs. 2. 15-34 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Qing-Ming CHENG: "Hypersurfaces in aunt sphere S^<n+1>(1)with constant scalar curvature"Journal of the London Nath.Soc.. 64. 775-768 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Qing-Ming CHENG: "Submanifolds with constant scalar curvature"Proc.Royal Soc.Edinburgh. 131.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Masayuki YAMASAKI: "Controlled surgery theory"Sugaku Expositions. 13. 113-124 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kiyoko NISHIZAWA: "Moduli space of polynomials with degree n"Suuri-Kaiseki Kenkyusho Kokyuyoku. 1187. 221-227 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Takahiro TSUCHIYA: "General saddlepoint approximations to distributions under an ellip-tical population"Communications in Statdstics. 28. 727-754 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Toshiko NAKAMURA (OGIWARA): "Spiral travelling wave solutions of some parabolic equations on annuli, NLA99 Computer Algebra"Josai Mathematical Monographs. 2. 15-34 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Qing-Ming CHENG: "Hypersurfaces in a unit sphere S^<n+1>(1) with constant scalar curvature"Journal of the London Mathematical Society. 64. 755-768 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Qing-Ming CHENG: "Sumanifolds with constant scalar curvature"(to appear) in Proceedings of the Royal Society of Edinburgh. 131.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 西澤 清子: "Paramentrization by fixed points multipliers of the polynomials with degree n"数理解析研究所講究禄. 1199. 127-131 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Qing-Ming Cheng: "Hypersurfaces in a unit sphere S^<n+1>(1) with constant scalar curvature"Journal of the London Mathematical Society. 64. 755-768 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Qing-Ming Cheng: "Submanifolds with constant scalar curvature"Proceedings of the Royal Society of Edinburgh. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Qing-Ming Cheng: "Complete hypersurfaces in a Euclidean space R^<n+1> with constant scalar curvature"Indiana University Mathematics Journal. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Kiyoko Nishizawa, et al.: "Chaotic bifurcations along algebraic curves"Communications in difference equations (Proceedings,Poznan,1998). 273-282 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Toshiko Ogiwara(Nakamura), et al.: "Spiral traveling wave solutions of some parabolic equations on annuli"Josai Mathematical Monographs. vol.2. 15-34 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Qing-Ming Cheng: "Compact locally conformally flat Riemannian manifolds"Bull.London Math.Soc.. vol.33. 1-7 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Masayuki Yamasaki: "Controlled surgery theory"Sugaku Expositions. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Qing-Ming Cheng,et. al: "Conformally flat 3-manifolds with constant scalar curvature II"Japanese J. Math.. 26. (2000)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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