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The Segal Conjecture for topological Hochschild homology of the 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 $T(n)\to BP$ 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.