The Segal conjecture for topological Hochschild homology of Ravenel spectra

Abstract

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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Notes

  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.

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Acknowledgements

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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Correspondence to J. D. Quigley.

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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). https://doi.org/10.1007/s40062-021-00275-7

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Keywords

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

Mathematics Subject Classification

  • 55P42
  • 18D55