Splittings and calculational techniques for higher THH

TL;DR

This paper develops new structural and splitting techniques for higher topological Hochschild homology, enabling explicit calculations of THH for various algebraic structures.

Contribution

It introduces novel methods to analyze and compute higher THH, including splitting results and relations between constructions, advancing the understanding of these homology theories.

Findings

Established structural properties relating $X ensor (-)$ to $\Sigma X ensor (-)$

Proved splitting results for higher THH constructions

Computed $THH^{[n]}_*( ext{Z}/p^m; ext{Z}/p)$ for all $n \,\geq 1$ and $m \,\geq 2$

Abstract

Tensoring finite pointed simplicial sets with commutative ring spectra yields important homology theories such as (higher) topological Hochschild homology and torus homology. We prove several structural properties of these constructions relating $X \otimes (-)$ to $\Sigma X \otimes (-)$ and we establish splitting results. This allows us, among other important examples, to determine $THH^{[n]}_*(\mathbb{Z}/p^m; \mathbb{Z}/p)$ for all $n \geq 1$ and for all $m \geq 2$.