Simplicial Structures Over the 3-Sphere and Generalized Higher Order Hochschild Homology

TL;DR

This paper explores the simplicial structures of higher order Hochschild homology over the 3-sphere, introduces tertiary Hochschild homology, and generalizes these concepts to multiple simplicial sets with geometric realizations.

Contribution

It introduces tertiary Hochschild homology and generalizes higher order Hochschild homology over multiple simplicial sets, providing new algebraic and geometric insights.

Findings

Defined tertiary Hochschild homology for specific algebraic structures

Established bar-like resolutions in simplicial modules

Generalized Hochschild homology over multiple simplicial sets

Abstract

In this paper we investigate the simplicial structure of a chain complex associated to the higher order Hochschild homology over the $3$-sphere. We also introduce the tertiary Hochschild homology corresponding to a quintuple $(A,B,C,\varepsilon,\theta)$, which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a bar-like resolution in the context of simplicial modules. Finally, we generalize the higher order Hochschild homology over a trio of simplicial sets, which also grants natural geometric realizations.