# Homotopy Gerstenhaber formality of Davis-Januszkiewicz spaces

## TL;DR

This paper proves that the normalized singular cochain algebra of Davis-Januszkiewicz spaces is formal as a homotopy Gerstenhaber algebra, extending previous results and enabling the computation of loop space cohomology.

## Contribution

It establishes the homotopy Gerstenhaber formality of Davis-Januszkiewicz spaces' cochain algebra for any coefficient ring, generalizing prior formality results.

## Key findings

- Normalized singular cochain algebra is formal as a homotopy Gerstenhaber algebra.
- Cohomology rings of free and based loop spaces of Davis-Januszkiewicz spaces are determined.
- Generalizes previous formality results for classifying spaces of tori.

## Abstract

A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal as a homotopy Gerstenhaber algebra, for any coefficient ring. This generalizes a recent result by the author about classifying spaces of tori and also strengthens the well-known dga formality result for Davis-Januszkiewicz spaces due to the author and Notbohm-Ray. As an application, we determine the cohomology rings of free and based loop spaces of Davis-Januszkiewicz spaces.