A computation of $THH_*(ku)$ using a gathered spectral sequence

TL;DR

This paper develops a new spectral sequence technique to compute the topological Hochschild homology of the connective complex K-theory spectrum $ku$, extending previous results from the Adams summand $ ell$.

Contribution

The authors introduce a gathered spectral sequence method that relates Bockstein spectral sequences for different multiplications, enabling a complete computation of $THH_*(ku)$.

Findings

Complete computation of $THH_*(ku)$ achieved.

The gathered spectral sequence technique is broadly applicable.

Relation between $THH_*( ell)$ and $THH_*(ku)$ clarified.

Abstract

In this article, we extend the computation of topological Hochschild homology (THH) of the Adams summand $\ell$ of $p$-local connective complex topological K-theory ($ku$) to $ku$ itself. We leverage the relation $u^{p-1} = v_1$, where $u$ is a generator of $ku_*$ and $v_1$ is a generator of $\ell_*$, and we consider the cofiber of the multiplication by $v_1$ in $ku$, denoted $ku/v_1$. We use the morphism between the Bockstein spectral sequences of the multiplication by $v_1$ computing $THH_*(\ell)$ and $THH_*(ku)$; we develop a general technique using what we term a gathered spectral sequence that allows us to explore the relationship between the Bockstein spectral sequences for the multiplications by $v_1$ and $u$, from which we derive a complete computation of $THH_*(ku)$. Our method is not only applicable to this specific problem but may also prove useful in other computations.