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Study on Transformation Group Theory and Equivariant K-theory

Research Project

Project/Area Number 12640075
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KOMIYA Katsuhiro  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (00034744)

Co-Investigator(Kenkyū-buntansha) MIYAZAWA Yasuyuki  Yamaguchi University, Faculty of Science, Assistant, 理学部, 助手 (60263761)
NAITOH Hiroo  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10127772)
ANDO Yoshifumi  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (80001840)
SATO Yoshihisa  Yamaguchi University, Faculty of Education, Lecturer, 教育学部, 講師 (90231349)
WATANABE Tadashi  Yamaguchi University, Faculty of Education, Professor, 教育学部, 教授 (10107724)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsBorsuk-Ulam theorem / linear representation / equivariant map / equivariant K-theory / cutting and pasting / SK-equivalence / SK-group / Euler characteristic / cutting / pasting / Z_2多様体 / double / 境界をもつ多様体 / Borelコホモロジー理論 / 写像度 / 表現環
Research Abstract

[1] The classical Borsuk-Ulam theorem is only concerned with equivariant maps between spheres with the antipodal action of the cyclic group of order 2. In our study we generalize this to equivariant maps between unit spheres of unitary representations of a compact abelian group. We obtain a necessary and sufficient condition for the existence of such equivariant maps in terms of the Euler classes of unit spheres, and also obtain the result concerning with the degrees of equivariant maps. The method depends heavily on the equivariant K-theory.
[II] Cuttings and pastings of manifolds lead to the so-called SK-groups of manifolds. Various kinds of such groups are found in the literature. In our study we investigate the relation among them, and show that the following three SK-groups are essentially isomorphic to each other : (i) the SK-group of pairs of manifolds, (ii) the SK-group of manifolds with boundary, and (iii) the SK-group of manifolds with involution.

Report

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

    (11 results)

All Other

All Publications (11 results)

  • [Publications] Katsuhiro Komiya: "Equivariant K-theoretic Euler classes and maps of representation spheres"Osaka Journal of Mathematics. 38. 239-249 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Komiya Katsuhiro: "Cutting, pasting and the doubles of manifolds with boundary"Kyushu Journal of Mathematics. 55. 321-328 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小宮克弘: "位相幾何入門"裳華房. 152 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Katsuhiro Komiya: "Equivariant K-theoretic Euler classes and maps of representation spheres"Osaka J. Math. 38. 239-249 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Katsuhiro Komiya: "Cutting pasting and the doubles of manifolds with boundary"Kyushu J. Math. 55. 321-328 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Katsuhiro Komiya: "Introduction to Topology"Shokabo. 152 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Katsuhiro Komiya: "Equivariant K-theoretic Euler classes and maps of representation spheres"Osaka Journal of Mathematics. 38. 239-249 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Katsuhiro Komiya: "Cutting, pasting and the doubles of manifolds with boundary"Kyushu Journal of Mathematics. 55. 321-328 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小宮克弘: "位相幾何入門"裳華房. 152 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Katsuhiro Komiya: "Equivariant K-theoretic Euler classes and maps of representation spheres"Osaka Journal of Mathematics. 38-1. 1-11 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Katsuhiro Komiya: "Cutting-pasting and doubles of manifolds with boundary"Kyushu Journal of Mathematics. 55-1(or-2)(近刊). (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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