Mathematische Annalen

, Volume 340, Issue 1, pp 97–142 | Cite as

Equivariant operads, string topology, and Tate cohomology

  • Craig Westerland


From an operad \(\fancyscript {C}\) with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of \(\fancyscript {C}\) is trivial. Applying this theory to the little disk operad \(\fancyscript {C}_2\) (which is an S 1-operad) we obtain variations on Getzler’s gravity operad, which we show governs the Chas–Sullivan string bracket.

Mathematics Subject Classification (2000)

55N91 55P43 55P92 55R12 14D22 18D50 


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  1. 1.
    Adem, A., Cohen, R.L., Dwyer, W.G.: Generalized Tate homology, homotopy fixed points and the transfer. Algebraic topology (Evanston, IL, 1988), Contemp. Math., vol. 96, pp 1–13. Amer. Math. Soc., Providence (1989)Google Scholar
  2. 2.
    Abbaspour, H., Cohen, R.L., Gruher, K.: String topology of Poincare duality groups. Preprint: math.AT/ 0511181 (2005)Google Scholar
  3. 3.
    Bruner, R.R., Rognes, J.: Differentials in the homological homotopy fixed point spectral sequence. Algebr. Geom. Topol. 5, 653–690 (2005) (electronic)Google Scholar
  4. 4.
    Boardman J.M. and Vogt R.M. (1973). Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347. Springer-Verlag, Berlin Google Scholar
  5. 5.
    Carlsson G. (1991). On the homotopy fixed point problem for free loop spaces and other function complexes. K-Theory 4(4): 339–361 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ching, M.: Bar constructions for topological operads and the Goodwillie derivatives of the identity. Geom. Topol. 9, 833–933 (2005) (electronic)Google Scholar
  7. 7.
    Cohen, R.L., Hess, K., Voronov, A.A.: String topology and cyclic homology. Advanced courses in mathematics. M Barcelona, Birkhäuser Verlag, Basel, 2006, Lectures from the Summer School held in Almerí a, September 16–20 (2003)Google Scholar
  8. 8.
    Cohen R.L. and Jones J.D.S. (2002). A homotopy theoretic realization of string topology. Math. Ann. 324(4): 773–798 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cohen, F.R., Lada, T.J., May, J.P.: The homology of iterated loop spaces. Lecture Notes in Mathematics, vol. 533. Springer-Verlag, Berlin (1976)Google Scholar
  10. 10.
    Chas, M., Sullivan, D.: String topology. Preprint: math.GT/9911159 (2001)Google Scholar
  11. 11.
    Devadoss, S.L.: Tessellations of moduli spaces and the mosaic operad. Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., vol. 239, pp 91–114. Amer. Math. Soc., Providence (1999)Google Scholar
  12. 12.
    Etingof, P., Henriques, A., Kamnitzer, J., Rains, E.: The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points. Preprint: math.AT/0507514 (2005)Google Scholar
  13. 13.
    Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in stable homotopy theory. Mathematical Surveys and Monographs, vol. 47. American Mathematical Society, Providence (1997). With an appendix by M. ColeGoogle Scholar
  14. 14.
    Elmendorf, A.D., May, J.P.: Algebras over equivariant sphere spectra. J. Pure Appl. Algebra 116(1–3), 139–149 (1997). Special volume on the occasion of the 60th birthday of Professor Peter J. FreydGoogle Scholar
  15. 15.
    Fiedorowicz, Z.: Constructions of E n operads. Preprint: math.AT/9808089 (1998)Google Scholar
  16. 16.
    Getzler E. (1994). Two-dimensional topological gravity and equivariant cohomology. Comm. Math. Phys. 163(3): 473–489 zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Getzler, E.: Operads and moduli spaces of genus 0 Riemann surfaces. The moduli space of curves (Texel Island, 1994). Progr. Math., vol. 129, pp 199–230. Birkhäuser, Boston (1995)Google Scholar
  18. 18.
    Getzler E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint: hep-th/9403055 (1994)Google Scholar
  19. 19.
    Ginzburg V. and Kapranov M. (1994). Koszul duality for operads. Duke Math. J. 76(1): 203–272 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Greenlees, J.P.C., May, J.P.: Generalized Tate cohomology. Mem. Amer. Math. Soc. 113(543), (1995) viii+178Google Scholar
  21. 21.
    Gruher, K., Salvatore, P.: Generalized string topology operations. Preprint: math.AT/0602210 (2006)Google Scholar
  22. 22.
    Gruher, K., Westerland, C.: String topology prospectra and Hochschild cohomology (2007) (in preparation)Google Scholar
  23. 23.
    Kaufmann, R.M.: On several varieties of cacti and their relations. Algebr. Geom. Topol. 5, 237–300 (2005) (electronic)Google Scholar
  24. 24.
    Kelly, G.M.: On the operads of J. P. May. Repr. Theory Appl. Categ. (13), 1–13 (2005) (electronic)Google Scholar
  25. 25.
    Klein J.R. (2001). The dualizing spectrum of a topological group. Math. Ann. 319(3): 421–456 zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kontsevich M. and Manin Yu. (1994). Gromov–Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164(3): 525–562 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lewis, Jr. L.G., May, J.P., Steinberger, M., McClure, J.E.: Equivariant stable homotopy theory. Lecture Notes in Mathematics, vol. 1213. Springer-Verlag, Berlin (1986). With contributions by J. E. McClureGoogle Scholar
  28. 28.
    May, J.P.: The geometry of iterated loop spaces. Lecture Notes in Mathematics, vol. 271. Springer Verlag, BerlinGoogle Scholar
  29. 29.
    May, J.P.: Definitions: operads, algebras and modules. Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) (Providence, RI), Contemp. Math., vol. 202. Amer. Math. Soc., 1997, pp 1–7Google Scholar
  30. 30.
    Mandell, M.A., May, J.P.: Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159(755), (2002) x+108Google Scholar
  31. 31.
    Madsen, I., Schlichtkrull, C.: The circle transfer and K-theory. Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, pp 307–328. Amer. Math. Soc., Providence, (2000)Google Scholar
  32. 32.
    Rognes, J.: Stably dualizable groups. Preprint: math.AT/0502184 (2005)Google Scholar
  33. 33.
    Strickland N.P. (2000). K(N)-local duality for finite groups and groupoids. Topology 39(4): 733–772 zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Salvatore P. and Wahl N. (2003). Framed discs operads and Batalin–Vilkovisky algebras. Q. J. Math. 54(2): 213–231 zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Voronov, A.A.: Notes on universal algebra. Graphs and patterns in mathematics and theoretical physics. In: Proc. Sympos. Pure Math., vol. 73, pp 81–103. Amer. Math. Soc., Providence, (2005)Google Scholar
  36. 36.
    Westerland, C.: String homology of spheres and projective spaces. Algebr. Geom. Topol. 7, 309–325 (2007) (electronic)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Wisconsin-MadisonMadisonUSA

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