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2003 Fiscal Year Final Research Report Summary

De Rham Theory of Loop Spaces and Quantum Topological Invariants

Research Project

Project/Area Number 12440014
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionThe University of Tokyo

Principal Investigator

KOHNO Toshitake  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (80144111)

Co-Investigator(Kenkyū-buntansha) MURAKAMI Jun  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90157751)
MORITA Shigeyuki  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
Project Period (FY) 2000 – 2003
Keywordsloop space / configuration space / iterated integral / braid group / mapping class group / moduli space / Poisson structure / conformal field theory
Research Abstract

We investigated the algebraic structure of the homology of the loop spaces of configuration spaces and clarified its relation to finite type topological invariants for braids. Especially, we studied the homology of the loop spaces of configuration spaces and finite type topological invariants. We focused on the homology of the loop spaces of orbit configuration spaces associated with the action of Fuchsian groups on the complex upper half plane. We showed that the total homology of such loop space is isomorphic to the algebra of horizontal chord diagrams on the quotient surface. We introduced a structure of a Poisson algebra for the homology of the iterated loop space of the orbit configuration space based on the Browder operation.
We gave a complete description of the space of conformal blocks for the conformal field theory on the Riemann sphere in terms of hypergeometric integrals. In particular, we clarified the integration cycles as the regularizable cycles in the homology of locally finite chains with coefficients in a certain local system defined over the complement a disrciminantal arrangement.
Morita investigated the structure of various moduli spaces as well as their associated modular groups, such as the moduli space of Riemann surface -mapping class groups and the moduli space of graphs -outer automorphism group of free groups. Murakami gave a new point of view on a conjecture concerning the asymptotics of the Jones invariants for knots, the hyperbolic volume of the knot complement, and the geometric structure of 3-manifolds.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Toshitake KOHNO: "Vassiliev invariants of braids and iterated integrals"Advanced Studuies in Pure Math.. 27. 157-168 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Toshitake KOHNO: "Bar complex on the Qrlik-Solomon algebra"Topology and its Applications. 118. 147-157 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Toshitake KOHNO: "Loop spaces of configuration spaces and finite type invariants"Geometry and Topology Monographs. 4. 143-160 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigeyuki MORITA: "Generators for the tautological algebra of the moduli space of curves"Topology. 42. 787-819 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hitoshi MURAKAMI, Jun MURAKAMI: "The colored Jones polynomials and the simplicial volume of a knot"Acta Math.. 186. 85-104 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Toshitake KOHNO: "Conformal field theory and topology"Translations of Mathematical Monographs, Volume 210 American Mathematical Society. 181 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Toshitake Kohno: "Vassiliev invariants of braids and iterated integrals"Advanced Studies in Pure Math.. 27. 157-168 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshitake Kohno: "Bar complex on the Orlik-Solomon algebra"Topology and its Applications. 118. 147-157 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshitake Kohno: "Loop spaces of configuration spaces and finite type in variants"Geometry and Topology Monographs. 4. 143-160 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeyuki Morita: "Generators for the tautological algebra of the moduli space of curves"Topology. 42. 787-819 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hitoshi Murakami, Jun Murakami: "The colored Jones polynomials and the simplicial volume of a knot"Acta Math.. 186. 85-104 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshitake Kohno: "Conformal field theory and topology,"Translations of Mathematical Monographs (American Mathematical Society). Volume 210. (2002)

    • 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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