Non-realizability of a triple Massey product for the algebra $\mathbb{F}_2[a,b,c]/(ab,bc)$

TL;DR

This paper proves that a specific cohomology algebra with a non-zero triple Massey product cannot be realized as the cohomology of a differential graded algebra with a non-zero triple Massey product, using Hochschild cohomology computations.

Contribution

It demonstrates the intrinsic A_3-formality of a common example and establishes its non-realizability as a differential graded algebra with a non-zero triple Massey product.

Findings

The algebra is intrinsically A_3-formal.

The algebra cannot be realized as a DGA with a non-zero triple Massey product.

Hochschild cohomology computations reveal the obstruction.

Abstract

We show that an often used example of a cohomology algebra with non-vanishing triple Massey product is intrinsically A_3-formal and therefore, in fact, cannot be realized as the cohomology of a differential graded algebra with non-vanishing triple Massey product. We prove this result by computing the graded Hochschild cohomology group which contains the potential obstruction to the vanishing.