# Deformation Theories Controlled by Hochschild Cohomologies

## TL;DR

This paper investigates how higher order Hochschild cohomologies govern deformation theories, extending the framework to spheres of any dimension and exploring tertiary Hochschild cohomology's role in this context.

## Contribution

It generalizes deformation theories controlled by Hochschild cohomologies to higher-dimensional spheres and introduces the concept of tertiary Hochschild cohomology in this setting.

## Key findings

- Controlled deformation theories for spheres of any dimension.
- Introduction of tertiary Hochschild cohomology in deformation control.
- Reduction to secondary and usual Hochschild cohomologies under specific conditions.

## Abstract

We explore how the higher order Hochschild cohomology controls a deformation theory when the simplicial set models the 3-sphere. Besides generalizing to the $d$-sphere for any $d\geq1$, we also investigate a deformation theory corresponding to the tertiary Hochschild cohomology, which naturally reduces to those studied for the secondary and usual Hochschild cohomologies under certain conditions.