DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Triple Massey product with a fixed cohomological sign convention

Definition

Let $A$ and its cohomology groups be as in Triple Massey data in a cohomological differential graded algebra, and let $α \in H^{p} (A)$ , $β \in H^{q} (A)$ and $γ \in H^{r} (A)$ be cohomology classes. Their triple Massey product in that item's sign convention is the subset

$⟨ α, β, γ ⟩ = {[ω (a, b, c; u, v)] \in H^{p + q + r - 1} (A) : (a, b, c; u, v) is a triple Massey datum representing (α, β, γ)} .$

The expression is a cohomology class because it is closed by The triple Massey expression is a cocycle. The set includes every choice of cocycle representatives and null-homotopies for which the adjacent products are boundaries; if there is no such datum, it is empty. No claim that the set is a singleton is part of the definition.

Depends on

Used by

A minimal A∞-model whose ternary operation realizes a triple Massey product Example

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bernhard Keller, Introduction to A-infinity algebras and modules, §3.3 (standard reference, not scraped)
Jean-Louis Loday and Bruno Vallette, Algebraic Operads, §9.2.5 (standard reference, not scraped)