On topological Hochschild homology of the $K(1)$-local sphere

TL;DR

This paper computes the topological Hochschild homology of the connective cover of the $K(1)$-local sphere spectrum at all primes $p eq 2$, using a spectral sequence derived from a filtration of a commutative ring spectrum.

Contribution

It introduces a May-type spectral sequence approach to compute topological Hochschild homology for the $K(1)$-local sphere spectrum, extending previous methods.

Findings

Computed THH mod p and v_1 for the $K(1)$-local sphere spectrum

Developed a spectral sequence from a filtration of a commutative ring spectrum

Provided explicit calculations for all primes p ≥ 3

Abstract

We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$-local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.