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Equivariant operads, string topology, and Tate cohomology

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Abstract

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.

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References

  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)

  2. Abbaspour, H., Cohen, R.L., Gruher, K.: String topology of Poincare duality groups. Preprint: math.AT/ 0511181 (2005)

  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. 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. Carlsson G. (1991). On the homotopy fixed point problem for free loop spaces and other function complexes. K-Theory 4(4): 339–361

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

  8. Cohen R.L. and Jones J.D.S. (2002). A homotopy theoretic realization of string topology. Math. Ann. 324(4): 773–798

    Article  MATH  MathSciNet  Google Scholar 

  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)

  10. Chas, M., Sullivan, D.: String topology. Preprint: math.GT/9911159 (2001)

  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)

  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)

  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. Cole

  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. Freyd

    Google Scholar 

  15. Fiedorowicz, Z.: Constructions of E n operads. Preprint: math.AT/9808089 (1998)

  16. Getzler E. (1994). Two-dimensional topological gravity and equivariant cohomology. Comm. Math. Phys. 163(3): 473–489

    Article  MATH  MathSciNet  Google Scholar 

  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)

  18. Getzler E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint: hep-th/9403055 (1994)

  19. Ginzburg V. and Kapranov M. (1994). Koszul duality for operads. Duke Math. J. 76(1): 203–272

    Article  MATH  MathSciNet  Google Scholar 

  20. Greenlees, J.P.C., May, J.P.: Generalized Tate cohomology. Mem. Amer. Math. Soc. 113(543), (1995) viii+178

    Google Scholar 

  21. Gruher, K., Salvatore, P.: Generalized string topology operations. Preprint: math.AT/0602210 (2006)

  22. Gruher, K., Westerland, C.: String topology prospectra and Hochschild cohomology (2007) (in preparation)

  23. Kaufmann, R.M.: On several varieties of cacti and their relations. Algebr. Geom. Topol. 5, 237–300 (2005) (electronic)

    Google Scholar 

  24. Kelly, G.M.: On the operads of J. P. May. Repr. Theory Appl. Categ. (13), 1–13 (2005) (electronic)

  25. Klein J.R. (2001). The dualizing spectrum of a topological group. Math. Ann. 319(3): 421–456

    Article  MATH  MathSciNet  Google Scholar 

  26. Kontsevich M. and Manin Yu. (1994). Gromov–Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164(3): 525–562

    Article  MATH  MathSciNet  Google Scholar 

  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. McClure

  28. May, J.P.: The geometry of iterated loop spaces. Lecture Notes in Mathematics, vol. 271. Springer Verlag, Berlin

  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 1–7

  30. Mandell, M.A., May, J.P.: Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159(755), (2002) x+108

    Google Scholar 

  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)

  32. Rognes, J.: Stably dualizable groups. Preprint: math.AT/0502184 (2005)

  33. Strickland N.P. (2000). K(N)-local duality for finite groups and groupoids. Topology 39(4): 733–772

    Article  MATH  MathSciNet  Google Scholar 

  34. Salvatore P. and Wahl N. (2003). Framed discs operads and Batalin–Vilkovisky algebras. Q. J. Math. 54(2): 213–231

    Article  MATH  MathSciNet  Google Scholar 

  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)

  36. Westerland, C.: String homology of spheres and projective spaces. Algebr. Geom. Topol. 7, 309–325 (2007) (electronic)

    Google Scholar 

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Correspondence to Craig Westerland.

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Westerland, C. Equivariant operads, string topology, and Tate cohomology. Math. Ann. 340, 97–142 (2008). https://doi.org/10.1007/s00208-007-0140-0

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