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Non-realizability of a triple Massey product

Eivind Xu Djurhuus, Gereon Quick

TL;DR

This paper proves that the standard cohomology algebra with a non-vanishing triple Massey product cannot be realized by a DGA with a non-vanishing triple Massey product. The obstruction to realizability is encoded by the canonical class $[m_3] \in \mathrm{HH}^{3,-1}(H^\bullet)$, and the authors show that $[m_3]=0$ for the Koszul algebra $A=\mathbb{F}_2[a,b,c]/(ab,bc)$. Using Koszulness, they compute the Hochschild cohomology by the Koszul complex and verify $\mathrm{HH}^{3,-1}(A)=0$, implying $A_3$-formality. Consequently, all triple Massey products vanish in this setting, proving non-realizability of nontrivial Massey products in DGAs with this cohomology. The work relates to Hopkins–Wickelgren's investigations into Massey products in Galois cohomology and Tate cohomology obstructions.

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.

Non-realizability of a triple Massey product

TL;DR

This paper proves that the standard cohomology algebra with a non-vanishing triple Massey product cannot be realized by a DGA with a non-vanishing triple Massey product. The obstruction to realizability is encoded by the canonical class , and the authors show that for the Koszul algebra . Using Koszulness, they compute the Hochschild cohomology by the Koszul complex and verify , implying -formality. Consequently, all triple Massey products vanish in this setting, proving non-realizability of nontrivial Massey products in DGAs with this cohomology. The work relates to Hopkins–Wickelgren's investigations into Massey products in Galois cohomology and Tate cohomology obstructions.

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.
Paper Structure (4 sections, 10 theorems, 37 equations)

This paper contains 4 sections, 10 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Hochschild cohomology and Massey products
  3. Koszul algebras
  4. Proof of the main result

Key Result

Theorem 1.1

Let $\mathcal{C}^\bullet$ be a differential graded algebra over $\mathbb{F}_2$ with cohomology algebra isomorphic to $\mathbb{F}_2[a, b, c]/(ab, bc)$. Then $\mathcal{C}^\bullet$ is $A_3$-formal. Thus, all triple Massey products for $\mathcal{C}^\bullet$ vanish. In particular, the triple Massey produ

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • ...and 15 more