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Study on global theory of singularities of mapping and 3-and 4-dimensional manifolds

Research Project

Project/Area Number 09640112
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionHIROSHIMA UNIVERSITY

Principal Investigator

KURA Takeshi  Faculty of Science, HIROSHIMA UNIVERSITY Research Associate, 理学部, 助手 (10161720)

Co-Investigator(Kenkyū-buntansha) TERAGAITO Masakazu  Faculty of School Education, Associate Professor, 学校教育学部, 助教授 (80236984)
KANNO Hiroaki  Faculty of Science, Associate Professor, 理学部, 助教授 (90211870)
SAEKI Osamu  Faculty of Science, Associate Professor, 理学部, 助教授 (30201510)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1997: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsDefinite fold singularities / Stein factorization / Topological Quantum Field Theory / Seiberg-Witten Theory / Dehn surgery / Lens space / Riemannian Manifold / Green function / 複素解析的曲面 / 安定写像 / Supersymmetric Gauge Theory / Donaldson-Witten Invariants / トンネル数 / 結び目 / 種数 / ザイフェルト多様体 / 高次元Wess-Zumino-Wilten模型 / トロイダル・リー代数 / 位相的場の量子論 / 位相的ゲージ理論
Research Abstract

The following results (1-4) have been obtained in the term of this project.
1. (0. Saeki) About maps into R^3 with only definite fold singularities, which are the simplest ones among generic singularities, various results about the structure of the source manifolds have been obtained by using their Stein factorizations. In particular, the existence of such maps on a 4-manifold has been shown to be strongly related to its smooth structure. Furthermore, the existence of maps with only Morin singularities has been shown to have close relationship to the Hopf invariant one problem. Moreover, a formula for the number of singularities of maps of surfaces with boundary into 3-manifolds has been obtained.
2. (H.Kanno) We have constructed cohomological gauge theories on Spin(7) or SU(4) manifold in eight dimensions as first examples of topological quantum field theories in higher dimensions. A relation to ten dimensional supersymmetric gauge theory has been clarified. We also investigated the Seiberg-Witten theory. Based on the Picard-Fuchs equation for the period integrals on the Seiberg-Witten curve, an analysis was mede on an effective theory of five dimensional supersymmetric gauge theory compactified on a circle. We have computed the topological partition function (the Donaldson-Witten function) of N=2 supersymrnetric massless QCD, which is regarded as a generating function of invariantsof four manifold.
3. (M.Teragaito) Dehn surgery on knots is a fundamental important operation to create closed ori-entable 3-manifolds. We focus on genera of knots as a new standing point, and we proved that there is an estimate for the orderof the fundamental group of a lens space obtained by Dehn surgery on a hyperbolic knot by using the genus of the knot.
4. (T.Kura) By means of curvatures, we have obtained the classification conditions of Riemannian manifolds based on the existence of p-Green functions.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (45 results)

All Other

All Publications (45 results)

  • [Publications] C.Kearton: "Finiteness results for knots and singularities" Kobe Journal of Mathematics. 14・2. 197-206 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 佐伯 修: "On topological invariance of weights for quasihomogeneous polynomials" Pitman Research Notes in Mathematics Series. 381. 207-214 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] J.J.Nuno Ballesteros: "On the number of singularities of a generic surface with boundary in a 3-manifold" Hokkaido Math.J.27・3. 517-544 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 佐伯 修: "Maps with only Morin singularities and the Hopf invariant one problem" Math.Proc.Camb.Phil.Soc.124. 501-511 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 佐伯 修: "On special generic maps into R^3" Pacific J.Math.184・1. 175-193 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] C.Biasi: "Transrersality with deficiency and a conjecture of Sard" Trans.Amer.Math.Soc.350. 5111-5122 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Katanaga: "Embeddings of quaternion space in S^4" J.Austral.Math.Soc.Ser.A. 65. 313-325 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] L.Banlieu: "Cohomological Yang-Mills Theory in Eight Dimensions" Dualities in Gauge and String Theories,World Scientific. 365-373 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] L.Banlieu: "Special Quantum Field Theories in Eight and Other Dimensions" Commun.Math.Phys.194. 149-175 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 菅野 浩明: "Picard-Fuchs Equation and Prepotential of Five Dimensional SUSY Gauge Theory Compactified on a Circle" Nucl.Phys.13530. 73-98 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 菅野 浩明: "Donaldson Witten Function of Massless N=2 Supersymmetric QCD" Nucl.Phys.13535. 512-530 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 寺垣内 政一: "Dehn surgery on genus one knots and Seifert fibered manifolds" Proc.Appl.Math.Workshop. 8. 315-325 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 倉 猛: "On the Green function of the p-Laplace equation for Riemannian manifolds" Proc.Japan.Acad.Ser.A. 75. 37-38 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] C.Kearton: "Finiteness results for knots and singularities, Kobe Journal of Mathematics" 14 No.2. 197-206 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] O.Saeki: "On topological invariance of weights for quasihomogeneous polynomials" Pitman Research Notes in Mathematics Series. 381. 207-214 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] J.J.Nuno Ballesteros: "On the number of singularities of a generic surface with boundary in a 3-manifold" Hokkaido Math.J.27, no.3. 517-544 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] O.Saeki: "Maps with onry Morin singularities and the Hopf invariant one problem" Math.Proc.Camd.Phil.Sci.124. 501-511 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] O.Saeki: "On special generic maps into R^3" Pacific J.Math.184.no.1. 175-193 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] C.Biasi: "Transversality with deficiency and a conjecture of Sard" Trans.Amer.Math.Soc.350. 5111-5122 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Katanaga: "Embeddings of quaternion space in S^4" J.Austral.Math.Soc.Ser.A 65. 313-325 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] L.Baulieu: "Cohomological Yang-Mills Theory in Eight Dimensions" In : Duaities in Gauge and String Theories, Eds.Y.M.Cho and S.Nam, World Scientific. 365-373 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] L.Baulieu: "Special Quantum Field Theories in Eight and Other Dimensions" Commun.Math.Phys.194. 149-175 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kanno: "Picard-Fuchs Equation and Prepotential of Five Dimensional SUSY Gauge Theory Compact-ified on a Circle" Nucl.Phys.B530. 73-98 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kanno: "Donaldson-Witten Function of Massless N=2 Supersymmetric QCD" Nucl.Phys.B535. 512-530 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Masakazu Teragaito: "Dehn surgery on genus one knots and Seifert fibered manifolds" Proc.Appl.Math., Workshop. 8. 315-325 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kura: "On the Green function of the P-Laplace equation for Reimannian manifolds" Proc.Japan Acad.Ser.A. 75. 37-38 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 菅野浩明: "Picard-Fuchs Equation and Prepotential of Five Dimensional SUSY Gauge Theory Compactified on a Circle" Nucl.Phys.B530. 73-98 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 菅野浩明: "Donaldson-Witten Function of Massless N=2 Supersymmetric QCD" Nucl.Phys.B535. 512-530 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 佐伯 修: "Stable maps between 4-manifolds and elimination of their singularities" J.London Math.Soc.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 佐伯 修: "Elimination of singularities : Thom polynomial and beyond" Proceedings in Honour of C.T.C.Wall's 60th birthday, Cambridge University Press (発表予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] 佐伯 修: "On algebraic unknotting members of Knots" Tokyo J.Math.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 佐伯 修: "Special generic maps of 4-manifolds and compact complex analytic sutfaces" Math.Ann.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 佐伯 修: "Real Selfert form determines the spactrum for samtquasihomogeneous hyporsurface singularities in C^3" J.Math.Soc.Japan. 発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] C.Biasi: "On the Betti number of the union of two generic map images" Topology Appl.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] W.Motta: "On ∽-generalized Fibonacci sequences" Fibonacci Quarterly.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] W.Motta: "A two color theorem and the fundamental class of a polyhedron" Proceedings of the 5th Workshop on Real and Complex Singularities, Brazil(発表予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] 合田 洋: "Tunnel number one genus one non-simple knots" Tokyo j.Math.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 合田 洋: "On tangle decompositions of tunnel number one links" J.Knot Theory and its Rami.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 寺垣内 政一: "Dehn surgeries on composite knots creating Kleinbottles" J.Knot Theory and its Rami.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 寺垣内 政一: "Creating Klein bottles by surgery on knots" J.Knot Theory and its Rami.発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 倉 猛: "On the Green function of the p-Laplace equation for Riemannian manifolds" Proc.Japan Acad.SerA. 発表予定.

    • Related Report
      1998 Annual Research Report
  • [Publications] 寺垣内 政一: "Cyclic surgery on genus one knots" Osaka Journal of Mathematics. 34・1. 145-150 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 寺垣内 政一: "Dehn surgery on genus one knots and Seifert fibered manifolds" Prcceedings of the 5th Korea-Japan School of Knots and Links. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Inami: "Higher-dimensional WZW model on Kahler manifold and toroidal Lie algebra" Mod.Phys.Lett.A12. 2757-2764 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] L.Baulieu: "Special quantum field theories in eight and other dimensions" Commun.Math.Phys.(発表予定).

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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