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Ordinary cohomology products can vanish while leaving a trace of how they vanished. A Massey product records that secondary information. One chooses cochains that witness the lower products as boundaries, combines them into a new cocycle and then varies every legitimate choice. The result is generally a set of cohomology classes with controlled indeterminacy, not a single canonical multiplication.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Begin with vanishing adjacent products
Let A be a differential graded algebra and let α, β and γ be cohomology classes represented by closed homogeneous cochains a, b and c. A triple Massey product requires αβ=0 and βγ=0 in cohomology.
These conditions mean the cochain products ab and bc are exact, not necessarily zero as cochains. Choosing primitives for them is the first secondary information. Sign and degree conventions vary, so every calculation must declare one convention before formulas are compared.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Build a defining system
Choose cochains u and v with du equal, up to the declared graded signs, to ab and dv to bc. The representatives a,b,c together with u,v form a defining system for the triple product.
A signed combination m=a·v±u·c is then closed: the two terms in dm cancel by the graded Leibniz rule and the equations for du and dv. Its cohomology class is one value of ⟨α,β,γ⟩.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a cochain ledger
Use one fixed convention and suppose ab=du and bc=dv. Compute the two differential contributions to m with the convention’s signs; the ab·c terms cancel. The important sequence is representative, boundary witness, secondary cocycle, cohomology class.
| Stage | Data | Check | Output |
|---|---|---|---|
| Represent classes | a,b,c closed | da=db=dc=0 | α,β,γ |
| Witness αβ=0 | choose u | du=ab | first null product |
| Witness βγ=0 | choose v | dv=bc | second null product |
| Combine choices | m=a·v±u·c | dm=0 | one Massey value [m] |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Describe the indeterminacy
Changing u by a closed cochain can shift [m] by a class in H multiplied by γ. Changing v can shift it by α multiplied by a class in H. In a common convention the triple-product indeterminacy is α·H plus H·γ in the required total degree.
Thus ⟨α,β,γ⟩ is a subset or coset in cohomology. Saying it vanishes means that zero belongs to this set. It does not mean that one arbitrarily selected representative is literally the zero cochain.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change the defining system
Replace u by u+z where dz=0, and v by v+w where dw=0. The new secondary cocycle differs from m by a·w±z·c. Passing to cohomology produces exactly the left and right multiplication terms in the indeterminacy.
| Change | Closed difference | Shift in [m] | Interpretation |
|---|---|---|---|
| u only | z | ±[z]γ | right product term |
| v only | w | α[w] | left product term |
| u and v | z,w | α[w]±[z]γ | same Massey set |
| a,b,c representatives | exact corrections | equivalent full set | operation is cohomological |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Connect to Toda brackets
The Toda-bracket guide builds a secondary composition from null-homotopies of two adjacent composites. Massey products are the differential-graded cohomology analogue: boundary witnesses replace null-homotopies.
The analogy is powerful but not automatic. Relating a specific Massey product to a Toda bracket requires a model, grading and comparison theorem; signs and indeterminacy may change under translation.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Use products to detect non-formality
A formal differential graded algebra is connected to its cohomology algebra with zero differential by quasi-isomorphisms. A nontrivial Massey product can obstruct formality because ordinary multiplication alone does not reproduce the secondary class.
The converse is unsafe: the vanishing of every triple Massey product does not by itself prove formality. Higher products or other operations may still carry hidden information, and a product containing zero can also contain nonzero values.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Place the operation in obstruction theory
The obstruction-cocycle guide explains why a vanished primary obstruction leaves choices forming a torsor. A defining system makes those choices explicit at cochain level.
The descent spectral-sequence guide follows higher coherence through filtrations, while the Moore–Postnikov guide organises successive lifting obstructions.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
The triple product is not defined merely because αβγ=0. Both adjacent products must vanish in cohomology so the defining cochains exist. Higher Massey products need an entire compatible triangular defining system, not only adjacent pairwise vanishings.
A chosen value is not invariant without its indeterminacy. Degree, cohomological versus homological grading, differential sign, coefficient ring and commutativity convention all affect the written formula.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use a proof and source routine
Record the differential graded algebra, coefficient ring, degrees, sign convention, representative cocycles, lower products, chosen primitives, closure calculation, target degree, resulting class, indeterminacy subgroup and the precise meaning of zero or nontriviality.
Buijs, Moreno-Fernández and Murillo state defining systems, triple and higher Massey products, and the classical indeterminacy subgroup. Kraines’ AMS paper develops higher Massey products as cohomology operations. May’s paper on matric Massey products studies defining systems, indeterminacy and linearity.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Can a Massey product be empty?
Yes. If the required lower products or compatible higher defining system do not vanish, no defining system exists.
What does “nontrivial” mean?
Usually that the product is defined and does not contain zero, but authors should state their convention because a set may contain both zero and nonzero classes.
Are Massey products homotopy invariants?
Appropriately transported full products are invariant under suitable quasi-isomorphisms; a lone chosen representative without indeterminacy is not.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a defining-system transfer check
Start from ab=du and bc=dv. Replace u by u+z and v by v+w for closed z,w. Write the resulting change a·w±z·c, identify its cohomology class in the indeterminacy subgroup, and explain why the full set remains the same even though one representative changes.
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