Operads and Moduli Spaces of Genus 0 Riemann Surfaces

  • E. Getzler
Part of the Progress in Mathematics book series (PM, volume 129)


In this paper, we study two dg (differential graded) operads related to the homology of moduli spaces of pointed algebraic curves of genus 0. These two operads are dual to each other, in the sense of Kontsevich [21] and Ginzburg and Kapranov [14].


Modulus Space Spectral Sequence Chain Complex Hodge Structure Cohomology Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V.I. Arnold, The cohomology ring of the colored braid group, Mat. Zametki 5 (1969), 227–231.Google Scholar
  2. 2.
    A. Beilinson, V. Ginzburg, Infinitesimal structure of moduli spaces of G-bundles, Internat. Math. Res. Notices (appendix to Duke Math. J.) 66 (1992), 63–74.MathSciNetCrossRefGoogle Scholar
  3. 3.
    J.M. Boardman, R.M. Vogt, “Homotopy invariant algebraic structures on topological spaces,” Lecture Notes in Math. 347, 1973.Google Scholar
  4. 4.
    J.L. Brylinski, S. Zucker, An overview of recent advances in Hodge theory, in “Several complex variables, VI,” 39–142, Encyclopaedia Math. Sci. 69, Springer Verlag, Berlin, 1990.Google Scholar
  5. 5.
    F.R. Cohen, The homology of C n+1-spaces, n ≥ 0, in “The homology of iterated loop spaces,” Lecture Notes in Math. 533, 1976, 207–351.Google Scholar
  6. 6.
    P. Deligne, Théorie de Hodge II, Publ. Math. IHES 40 (1971), 5–58.Google Scholar
  7. 7.
    R. Dijkgraaf, E., H. Verlinde, Topological strings in d xxx I, Nucl. Phys. B352 (1991), 59–80.MathSciNetCrossRefGoogle Scholar
  8. 8.
    W. Fulton, R. MacPherson, A compactification of configuration spaces, Ann. Math., 139 (1994), 183–225.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Commun. Math. Phys. 159 (1994), 265–285.Google Scholar
  10. 10.
    E. Getzler, Equivariant cohomology and topological gravity, Commun. Math. Phys. 163 (1994), 473–490.Google Scholar
  11. 11.
    E. Getzler, J.D.S. Jones, Operads, homotopy algebra, and iterated integrals for double loop spaces, (hep-th/9403055).Google Scholar
  12. 12.
    E. Getzler, M. Kapranov, Cyclic operads and cyclic homology, to appear in “Geometry, Topology, and Physics for Raoul,” ed. S.T. Yau, International Press, Cambridge, MA, 1994.Google Scholar
  13. 13.
    E. Getzler, M. Kapranov, Modular operads, MPIM-Bonn preprint 94/78, (dg-ga/9408003).Google Scholar
  14. 14.
    V.A. Ginzburg, M.M. Kapranov, Koszul duality for operads, Duke. Math. J. 76 (1994), 203–272.Google Scholar
  15. 15.
    M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.Google Scholar
  16. 16.
    A. Joyal, Foncteurs analytiques et especes de structures, Lecture Notes in Math. 1234 (1986) 126–159.Google Scholar
  17. 17.
    M.M. Kapranov, Permuto-associahedron, MacLane’s coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Algebra, 85 (1993), 119–142Google Scholar
  18. 18.
    S. Keel, Intersection theory of moduli spaces of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545–574.Google Scholar
  19. 19.
    A.A. Klyachko, Lie elements in the tensor algebra, Siberian Math. J., 15 (1974), 914–920.Google Scholar
  20. 20.
    F.F. Knudsen, The projectivity of the moduli space of stable curves II. The stacks M g, n, Math. Scand. 52 (1983), 161–189.Google Scholar
  21. 21.
    M. Kontsevich, Formal (non)-commutative symplectic geometry, in “The Gelfand mathematics seminars, 1990–1992,” eds. L. Corwin, I. Gelfand, J. Lepowsky, Birkhäuser, Boston, 1993.Google Scholar
  22. 22.
    M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys. 164 (1994), 525–562.Google Scholar
  23. 23.
    I.G. Macdonald, “Symmetric Functions and Hall Polynomials,” Clarendon Press, Oxford, 1979.Google Scholar
  24. 24.
    D. McDuff, D. Salamon, “J-holomorphic curves and quantum cohomology,”Amer. Math. Soc., Providence, 1994.Google Scholar
  25. 25.
    S. Mac Lane, “Categories for the working mathematician,” Graduate Texts in Math. 5, 1971.Google Scholar
  26. 26.
    M. Markl, Models for operads, preprint, 1994.Google Scholar
  27. 27.
    D. Quillen, Rational homotopy theory, Ann. Math. 90 (1969), 205–295.Google Scholar
  28. 28.
    Y. Ruan, G. Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. (1994), 269–278.Google Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • E. Getzler
    • 1
  1. 1.Department of MathematicsMITCambridgeUSA

Personalised recommendations