The Segal conjecture for topological Hochschild homology of Ravenel spectra


In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing the algebraic K-theory of X(n) using trace methods, which approximates the algebraic K-theory of the sphere spectrum in a precise sense. We solve the homotopy limit problem for topological Hochschild homology of T(n) under the assumption that the canonical map \(T(n)\rightarrow BP\) of homotopy commutative ring spectra can be rigidified to map of \(E_2\) ring spectra. We show that the obstruction to our assumption holding can be described in terms of an explicit class in an Atiyah-Hirzebruch spectral sequence.

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  1. 1.

    The notation we use for the Ravenel spectra T(n) is the notation from [32]. We warn the reader that the same notation is used for the \(v_n\) telescope of a type n spectrum in [21].

  2. 2.

    This is not exactly the same as the filtration as defined by Greenlees in [16], but the difference between the two just amounts to a different choice of model for \(EC_p\) as a \(C_p\)-CW complex.

  3. 3.

    See [26][Sec. 2.2] for a survey of continuous \(\mathcal {A}_*\) comodules.

  4. 4.

    In fact, this result has recently been extended by Nikolaus–Scholze [30][Theorem III.1.7] who show that the map \(X\rightarrow (X^{\wedge p})^{tC_p}\) exhibits \((X^{\wedge p})^{tC_p}\) as the p-completion of X for all bounded below spectra without the finite type hypothesis.


  1. 1.

    Adams, J. F.: Graeme Segal’s Burnside ring conjecture. In Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), volume 12 of Contemp. Math., pages 9–18. Amer. Math. Soc., Providence, R.I. (1982)

  2. 2.

    Frank Adams, J., Gunawardena, J.H., Miller, H.: The Segal conjecture for elementary abelian p-groups. Topology 24(4), 435–460 (1985)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ausoni, C., Rognes, J.: Algebraic \(K\)-theory of topological \(K\)-theory. Acta Math. 188(1), 1–39 (2002)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bökstedt, M., Bruner, R.R., Nielsen, S.L., Rognes, J.: On cyclic fixed points of spectra. Math. Z. 276((1–2)), 81–91 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Basterra, M., Mandell, M.A.: The multiplication on BP. J. Topol. 6(2), 285–310 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: \(H_\infty \) ring spectra and their applications. Lecture Notes in Mathematics. Springer, Berlin (1986)

    Google Scholar 

  7. 7.

    Bökstedt, M.: Topological Hochschild homology of the \(\mathbb{Z} \) and \(\mathbb{Z} /p\). Universität Bielefeld, Bielefeld (1986)

    Google Scholar 

  8. 8.

    Bruner, R.R., Rognes, J.: Differentials in the homological homotopy fixed point spectral sequence. Algebra Geom Topol 5(2), 653–690 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Carlsson, G.: Equivariant stable homotopy and Segal’s Burnside ring conjecture. Ann. Math. (2) 120(2), 189–224 (1984)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chadwick, S.G., Mandell, A.M.: \( {E}_n \) genera. Geometr. Topol. 19(6), 3193–3232 (2015)

    Article  Google Scholar 

  11. 11.

    Dundas, B.I., Goodwillie, G.T., McCarthy, R.: The local structure of algebraic K-theory, volume 18 of Algebra and Applications. Springer, London (2013)

  12. 12.

    Devinatz, S.E., Hopkins, J.M., Smith, H.J.: Nilpotence and stable homotopy theory I. Ann. Math. 128(2), 207–241 (1988)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Dundas, B.I., Rognes, J.: Cubical and cosimplicial descent. J. Lond. Math. Soc. (2) 98(2), 439–460 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Dundas, B.I.: Relative \(K\)-theory and topological cyclic homology. Acta Math. 179(2), 223–242 (1997)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Elmendorf, D.A., Kriz, I., Mandell, A.M., Peter May, J.: Rings, Modules, and Algebras in Stable Homotopy Theory, vol. 47. American Mathematical Soc., New York (2007)

    Google Scholar 

  16. 16.

    Greenlees, J.P.C.: Representing Tate cohomology of \(G\)-spaces. Proc. Edinburgh Math. Soc. (2) 30(3), 435–443 (1987)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hesselholt, L., Madsen, I.: On the \(K\)-theory of finite algebras over Witt vectors of perfect fields. Topology 36(1), 29–101 (1997)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hopkins, M.J.: Stable decompositions of certain loop spaces. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Northwestern University (1984)

  19. 19.

    Hopkins, J.M., Smith, H.J.: Nilpotence and stable homotopy theory. II. Ann. Math. (2) 148(1), 1–49 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Joyal, A.: The theory of quasicategories and its applications. Lecture notes from CRM in Barcelona, (2008)

  21. 21.

    Kuhn, J.N.: Goodwillie towers and chromatic homotopy: an overview. In Proceedings of the Nishida Fest (Kinosaki 2003), volume 10 of Geom. Topol. Monogr., pp. 245–279. Geom. Topol. Publ., Coventry (2007)

  22. 22.

    Lawson, T.: E\(_n\) ring spectra and Dyer–Lashof operations. In: Handbook of Homotopy Theory, chapter 19. CRC Press, Boca Raton (2019)

  23. 23.

    Lin, W.H., Davis, D.M., Mahowald, M.E., and Adams, J.F.: Calculation of Lin’s Ext groups. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 87, pages 459–469. Cambridge Univ Press, Cambridge (1980)

  24. 24.

    Lewis, L. G., Jr., May, J. P., Steinberger, M., McClure, J. E.: Equivariant Stable Homotopy theory, volume 1213 of Lecture Notes in Mathematics. Springer, Berlin (1986) With contributions by J. E. McClure

  25. 25.

    Lunøe-Nielsen, S., Rognes, J.: The Segal conjecture for topological Hochschild homology of complex cobordism. J. Topol. 4(3), 591–622 (2011)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Lunøe-Nielsen, S., Rognes, J.: The topological Singer construction. Doc. Math. 17, 861–909 (2012)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Milnor, J.: The Steenrod algebra and its dual. Ann. Math. 2(67), 150–171 (1958)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Mandell, M.A., May, J.P., Schwede, S., Shipley, B.: Model categories of diagram spectra. Proc. Lond. Math. Soc. (3) 82(2), 441–512 (2001)

    MathSciNet  Article  Google Scholar 

  29. 29.

    McClure, J.E., Staffeldt, R.E.: On the topological Hochschild homology of \(b{\rm u}\). I. Am. J. Math. 115(1), 1–45 (1993)

    Article  Google Scholar 

  30. 30.

    Nikolaus, T., Scholze, P.: On topological cyclic homology. Acta Math. 221(2), 203–409 (2018)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Ravenel, C.D.: Localization with respect to certain periodic homology theories. Am. J. Math. 106(2), 351–414 (1984)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Ravenel, D.C.: Complex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics. Academic Press Inc, Orlando, FL (1986)

    Google Scholar 

  33. 33.

    Tsalidis, S.: Topological Hochschild homology and the homotopy descent problem. Topology 37(4), 913–934 (1998)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Waldhausen, F.: An outline of how manifolds relate to algebraic \(K\)-theory. In: Homotopy theory (Durham, 1985), volume 117 of London Math. Soc. Lecture Note Ser., pages 239–247. Cambridge Univ. Press, Cambridge (1987)

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The authors would like to thank Mark Behrens, John Rognes, and Andrew Salch for their comments on earlier versions of this paper and an anonymous referee helpful comments. The second author was partially supported by NSF Grant DMS-1547292.

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Angelini-Knoll, G., Quigley, J.D. The Segal conjecture for topological Hochschild homology of Ravenel spectra. J. Homotopy Relat. Struct. 16, 41–60 (2021).

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  • Trace methods
  • topological Hochschild homology
  • Ravenel spectra
  • Segal Conjecture

Mathematics Subject Classification

  • 55P42
  • 18D55