The Segal Conjecture for topological Hochschild homology of the Ravenel spectra

TL;DR

This paper proves a version of the Segal Conjecture for topological Hochschild homology of Ravenel spectra, advancing the understanding of their algebraic K-theory and homotopy limits in stable homotopy theory.

Contribution

It solves the homotopy limit problem for THH of Ravenel spectra X(n) and T(n), extending the Segal Conjecture to these spectra and linking to algebraic K-theory computations.

Findings

Homotopy limit problem for THH of X(n) solved

Homotopy limit problem for THH of T(n) addressed under certain assumptions

Obstruction to assumption characterized via Atiyah–Hirzebruch spectral sequence

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 $T(n)\to BP$ 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.