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2003 Fiscal Year Final Research Report Summary

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
KeywordsStable Homotopy Category / Chromatic Tower / Obstruction Theory / Characteristic Classes / Equivariant Homotopy Theory / 4 dimensional Manifolds / Seiberg-Witten Invariants / Fredholm Universe
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.

  • Research Products

    (18 results)

All Other

All Publications (18 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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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

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Published: 2005-04-19  

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