Homology, Homotopy and Applications

Volume 23 (2021)

Number 2

Homotopy Gerstenhaber formality of Davis–Januszkiewicz spaces

Pages: 325 – 347

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a17


Matthias Franz (Department of Mathematics, University of Western Ontario, London, Ontario, Canada)


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.


Davis–Januszkiewicz space, homotopy Gerstenhaber algebra, formality, loop space

2010 Mathematics Subject Classification

Primary 57Sxx. Secondary 16E45, 55P35.

Copyright © 2021, Matthias Franz. Permission to copy for private use granted.

The author was supported by an NSERC Discovery Grant.

Received 7 February 2020

Accepted 13 November 2020

Published 18 August 2021