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Homotopy Theory and its Aplications

Research Project

Project/Area Number 13440020
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNagoya Institute of Technology

Principal Investigator

MINAMI Norihiko  Nagoya Institute of Technology, Faculty of Technology, Professor, 工学研究科, 教授 (80166090)

Co-Investigator(Kenkyū-buntansha) NISHIDA Goro  Kyoto University, Faculty of Science, Professor, 大学院・理学研究科, 教授 (00027377)
YOSIMURA Zen-ichi  Nagoya Institute of Technology, Faculty of Technology, Professor, 工学研究科, 教授 (70047330)
FURUTA Mikio  Tokyo University, Faculty of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50181459)
NATSUME Toshikazu  Nagoya Institute of Technology, Faculty of Technology, Professor, 工学研究科, 教授 (00125890)
SHIMAKAWA Kazuhisa  Okayama University, Faculty of Science, Professor, 理学部, 教授 (70109081)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥11,500,000 (Direct Cost: ¥11,500,000)
Fiscal Year 2003: ¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2002: ¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥4,400,000 (Direct Cost: ¥4,400,000)
KeywordsStable Homotopy Category / Chromatic Tower / Obstruction Theory / Characteristic Classes / Equivariant Homotopy Theory / 4 dimensional Manifolds / Seiberg-Witten Invariants / Fredholm Universe / adjunction inequality / charomotic tower / Seiber-Witten不変量
Research Abstract

When G is a topological group and X, Y be G-spaces, I defined an Euler class e(X,Y)∈[X_+,S^0*Y]^G_* as the very universal obstraction class for [X,Y]^G, the set of G-maps from X to Y, to be non-empty. I obstained some sufficient results when e(X,Y) becomes a faithful obstruction class. Concerning this problem, I also developed a more computable, but generally non faithful, obstruction class with Yasuhiro Hara.
This sort of idea has been applied to consider Farber's topological complexity which concerns motion plannings such as robot arms, and, through the model categorical point of view, A^1-homotopy theory of Voevodsky and Morel. Also, some advance has been obtained in generalizing the Thorn polynomials in the singularity theory through HELP (=Homotopy Extension Lifting Property).
Mikio Furuta pointed out that the stable homotopy Seiverg-Witten invariants for 4-manifolds with boundary are described by pro-spectra defined by Fredholm Universe, which is not a traditional universe in algebraic topology invented by May and his collaborators. Actually, this Fredholm universe is sort of "twisted" and appears to be very interesting.
Also, during this period, we organized many meetings and lots of research interactions with mathematicians in other areas have been achieved :
(a) International Conference on Algebraic Topology (July 27 -August 1, 2003)
(b) International Workshop on Algebraic Topology at Nagoya Institute of Technology (August 3 -August 8, 2003)
(c) Homotopy Nagoya Institute of Technology Homotopy Theory Meeting 01, 02(4 times!), 03, 04.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (37 results)

All Other

All Publications (37 results)

  • [Publications] Norihiko Minami: "On the chromatic tower"Amer.J.Math.. 125. 449-473 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yasuhiro Hara, N.Minami: "Borsuk-Ulam type theorems for compact Lie group actions"Proc.Amer.Math.Soc.. 132. 903-909 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Norihiko Minami: "An unbased G-Freudenthal theorem, homotopy representations, and a Borsuk-Ulam theorem"Trans.Amer.Math.Soc.. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Norihiko Minami: "A possible hierarchy of Morava K-theories"Isaac Newton Institute Proceedings. (掲載決定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Stefan Bauer, Mikio Furuta: "A stable cohomotopy refinement of Seiberg-Witten invariants."Invent.Math.. 155. 1-19 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yoshiyuki Kuramoto, Tsutomu Yasui: "On Haefliger's obstructions to embeddings and transfer maps"Osaka Journal of Mathematics. 40. 69-79 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Mikio Furuta, Yukio Kametani, N.Minami: "Stable-homotopy Seiberg-Witten invariants for Rational Cohomology K3#K3's"J.Math.Sci. (Univ.Tokyo). 8. 157-176 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Minami: "From K(n+1)^*(X) to K(n)^*(X)"Proc.Amer.Math.Soc.. 130-5. 1557-1562 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Minami: "Some mathematical influences of Stewart Priddy. Recent progress in homotopy theory (Baltimore, MD, 2000)"Contemp.Math., 293, Amer.Math.Soc., Providence, RI. 15-31 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Minami: "On the chromatic tower"Amer.J.Math.. 125. 449-473 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yasuhiro Hara, N.Minami: "Borsuk-Ulam type theorems for compact Lie group actions"Proc.Amer.Math.Soc.. 132. 903-909 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Minami: "An unbased G-Freudenthal theorem, homotopy representations, and a Borsuk-Ulam theorem"Trans.Amer.Math.Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Minami: "A possible hierarchy of Morava K-theories"Isaac Newton Institute Proceedings. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Miko Furuta: "Monopole equation and the (11)/8-conjecture"Mathematical Research Letters. 8-3. 279-291 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Fukumoto, M.Ue, M.Furuta: "W-invariants and Neumann-Siebenmann invariants for Seifert homology 3-spheres"Topology and its Applications. 116-3. 333-369 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Furuta: "Finite dimensional approximations in geometry"Proceedings of the International Congress of Mathematicians, Vol.II (Beijing, 2002), Higher (Ed.Press, Beijing). 395-403 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Stefan Bauer, Mikio Furuta: "A stable cohomotopy refinement of Seiberg-Witten invariants. I."Invent.Math.. 155. 1-19 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Yoshiyuki Kuramoto, Tsutomu Yasui: "On Haefliger's obstructions to embeddings and transfer maps"Osaka Journal of Math.. 40. 69-79 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Norihiko Minami: "On the chromatic tower"Amer.J.Math.. 125. 449-473 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Yasuhiro Hara, N.Minami: "Borsuk-Ulam type theorems for compact Lie group actions"Proc.Amer.Math.Soc.. 132. 903-909 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Norihiko Minami: "An unbased G-Freudenthal theorem, homotopy representations, and a Borsuk-Ulam theerem"Trans.Amer.Math.Soc.. (掲載決定).

    • Related Report
      2003 Annual Research Report
  • [Publications] Norihiko Minami: "A possible hierarchy of Morava K-theories"Isaac Newton Institute Proceedings. (掲載決定).

    • Related Report
      2003 Annual Research Report
  • [Publications] Stefan Bauer, Mikio Furuta: "A stable cohomotopy refinement of Seiberg-Witten invariants"Invent.Math.. 155. 1-19 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Yoshiyuki Kuramoto, Tsutomu Yasui: "On Haefliger's obstructions to embeddings and transfer maps"Osaka Journal of Mathematics. 40. 69-79 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Norihiko Minami: "From K(n+1)^*(X) to K(n)^*(X)"Proc.Amer.Math.Soc.. 130. 1557-1562 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Norihiko Minami: "Some mathematical influences of Stewart Priddy"Contemp.Math.. 293. 15-31 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Norihiko Minami: "On the chromatic tower"Amer.J.Math.. 125(印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yasuhiro Hara, N.Minami: "Borsuk-Ulam type theorems for compact Lie group actions"Proc.Amer.Math.Soc.. (印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Norihiko Minami: "An unbased G-Freudenthal theorem, homotopy representations, and a Borsuk-Ulam theorem"Trans.Amer.Math.Soc.. (掲載決定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshiyuki Kuramoto, Tsutomu Yasui: "On Haefliger's obstructions to embeddings and transfer maps"Osaka Journal of Mathematics. 40(印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 古田 幹雄: "指数定理2(岩波講座 現代数学の展開)"岩波書店. 327 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N. Minami: "Some mathematical influences of Stewart Priddy"Contemp. Math.. 293(印刷中). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] N. Minami: "Fron K(n+1)_*X to K(n)_*X"Proc. Amr. Math. Soc.. 130. 1557-1562 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Furuta, Y. Kanetani, N. Minami: "Stable-homotopy Seiberg-Witten invariants for Rational Cohomology K3#K3's"J. Math. Sci. Univ. Tokyo. 8. 157-176 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Furuta: "Monopole equation and the (11)/8-conjecture"Math. Res. Lett.. 8. 279-291 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Z. Yosimura: "The Quasi KO_*types of CW-spectra X with KU_0X【similar or equal】Z/2^m and KU_1X【similar or equal】Z/2^*"Mem. Fac. Sci. Kochi Univ.(Math.). 22. 67-91 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] G. Nishida, Y. Yang: "On a p-local stable splitting of U(n)"J. Math. Kyoto Univ.. 41. 387-401 (2001)

    • Related Report
      2001 Annual Research Report

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

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