---
tag: Z761N3D19J
kind: concept
name: Massey product
msc: [55S30]
sources:
  - url: https://en.wikipedia.org/wiki/Massey_product
same_as:
  wikidata: Q6784627
  wikipedia: Massey product
---

## Summary

A Massey product is a secondary operation on cohomology.
It takes three classes $u, v, w$ with $uv = 0$ and $vw = 0$ and produces a set of classes of degree one less than that of $u v w$.
It is defined from the [[cup product|JX47MS1R1H]] on cochains, not just on cohomology.
Massey products detect structure that the [[cohomology ring|RDT4785M4D]] alone does not see, such as the linking of the Borromean rings.

## Explanations

### L1 human

If $uv = 0$ in cohomology, then on cochains the product $a b$ is the coboundary of some cochain $s$.
The same holds for $vw$ with a cochain $t$.
The two ways of filling in, $s$ against $c$ and $a$ against $t$, combine into a new cocycle.
Its class is the Massey product $\langle u, v, w \rangle$, defined up to some choices.
In the Borromean rings, any two of the three rings are unlinked, so in the complement all cup products of the degree-one classes vanish.
But a triple Massey product is nonzero, and it shows that the three rings are linked together.