Whitehead filtrations for computations in topological Hochschild homology

Hyslop, Logan

doi:10.2140/agt.2026.26.2443

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Whitehead filtrations for computations in topological Hochschild homology

Logan Hyslop

Algebraic & Geometric Topology 26 (2026) 2443–2474

DOI: 10.2140/agt.2026.26.2443

Abstract

We discuss spectral sequences coming from Whitehead filtrations in the computation of topological Hochschild homology of ring spectra. Using cyclic invariance, this makes for simple computations of $THH$ of connective ring spectra $R$ with coefficients in discrete ring spectra (where discrete means Eilenberg–Mac Lane). In particular, we show how to use this to compute $THH (tmf, 𝔽_{2})$ and $THH (tmf, ℤ_{(2)})$ , where $tmf$ denotes the $𝔼_{\infty}$ ring spectrum of topological modular forms. Then, we obtain a description of $THH (ℓ ∕ v_{1}^{n})$ in terms of $THH (ℓ, ℓ ∕ v_{1}^{n})$ , where the latter can be computed by results of Angeltveit, Hill and Lawson (2010). We next explain how the methods of this computation generalize to give us information about $THH (cofib (x^{k} : Σ^{k | x |} R \to R))$ for $R$ and $cofib (x^{k})$ suitably structured connective ring spectra, $k > 1$ , and $x \in π_{*} (R)$ an arbitrary element in positive degree. Finally, we examine the general framework to describe the topological Hochschild homology of 2-local connective self-conjugate $K$ -theory, ${ksc}_{2}$ .

Keywords

topological Hochschild homology, trace methods, cyclic invariance, topological modular forms, Adams summand

Mathematical Subject Classification

Primary: 55R20, 55T25

Secondary: 19D55

References

Publication

Received: 29 April 2024

Revised: 5 June 2025

Accepted: 21 July 2025

Published: 1 September 2026

Authors

Logan Hyslop
Department of Mathematics Harvard University Cambridge, MA United States

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Volume 26, issue 7 (2026)