Relative Loday constructions and applications to higher THH-calculations

TL;DR

This paper introduces a relative Loday construction for commutative S-algebras, enabling new spectral sequences that facilitate higher topological Hochschild homology calculations, including previously intractable examples.

Contribution

It develops a relative Loday construction for sequences of commutative S-algebras and applies it to create spectral sequences for higher THH calculations.

Findings

New spectral sequences for higher THH calculations

Successful computation of previously intractable examples

Extension of Loday construction to relative settings

Abstract

We define a relative version of the Loday construction for a sequence of commutative S-algebras $A \rightarrow B \rightarrow C$ and a pointed simplicial subset $Y \subset X$. We use this to construct several spectral sequences for the calculation of higher topological Hochschild homology and apply those for calculations in some examples that could not be treated before.