# The Hochschild cohomology of the group $G^2_3$

## TL;DR

This paper uses discrete algebraic Morse theory to compute the Hochschild cohomology of the group algebra of $G_3^2$, revealing mostly trivial cohomology with specific non-trivial cases.

## Contribution

It introduces a novel application of algebraic Morse theory to compute Hochschild cohomology for the group $G_3^2$, providing explicit calculations.

## Key findings

- Most Hochschild cohomology groups are trivial.
- Two specific 1-dimensional bimodules have non-trivial $H^2$.
- The method simplifies cohomology calculations for this class of groups.

## Abstract

We apply discrete algebraic Morse theory to calculate the Anick resolution of the group algebra of the group $G_3^2$. As a corollary, we evaluate Hochschild cohomologies of $G_3^2$ with coefficients in all 1-dimensional bimodules. Almost all these groups are trivial, the only exceptions are 1-dimensional $H^2$ for two particular 1-dimensional bimodules.