Massey product

In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
Massey triple product.
Let formula_1 be elements of the cohomology algebra formula_2 of a differential graded algebra formula_3. If formula_4, the Massey product formula_5 is a subset of formula_6, where formula_7.
The Massey product is defined algebraically, by lifting the elements formula_1 to equivalence classes of elements formula_9 of formula_3, taking the Massey products of these, and then pushing down to cohomology. This may result in a well-defined cohomology class, or may result in indeterminacy.
Define formula_11 to be formula_12. The cohomology class of an element formula_13 of formula_3 will be denoted by formula_15. The Massey triple product of three cohomology classes is defined by
The Massey product of three cohomology classes is not an element of formula_2, but a set of elements of formula_2, possibly empty and possibly containing more than one element. If formula_19 have degrees formula_20, then the Massey product has degree formula_21, with the formula_22 coming from the differential formula_23.
The Massey product is nonempty if the products formula_24 and formula_25 are both exact, in which case all its elements are in the same element of the quotient group
So the Massey product can be regarded as a function defined on triples of classes such that the product of the first or last two is zero, taking values in the above quotient group.
More casually, if the two pairwise products formula_27 and formula_28 both vanish in homology (formula_29), i.e., formula_30 and formula_31 for some chains formula_32 and formula_33, then the triple product formula_34 vanishes "for two different reasons" — it is the boundary of formula_35 and formula_36 (since formula_37 and formula_38 because elements of homology are cycles). The bounding chains formula_32 and formula_33 have indeterminacy, which disappears when one moves to homology, and since formula_35 and formula_36 have the same boundary, subtracting them (the sign convention is to correctly handle the grading) gives a cocycle (the boundary of the difference vanishes), and one thus obtains a well-defined element of cohomology — this step is analogous to defining the formula_43st homotopy or homology group in terms of indeterminacy in null-homotopies/null-homologies of "n"-dimensional maps/chains.
Geometrically, in singular cohomology of a manifold, one can interpret the product dually in terms of bounding manifolds and intersections, following Poincaré duality: dual to cocycles are cycles, often representable as closed manifolds (without boundary), dual to product is intersection, and dual to the subtraction of the bounding products is gluing the two bounding manifolds together along the boundary, obtaining a closed manifold which represents the homology class dual of the Massey product. In reality homology classes of manifolds cannot always be represented by manifolds – a representing cycle may have singularities – but with this caveat the dual picture is correct.
Higher order Massey products.
More generally, the "n"-fold Massey product formula_44 of "n" elements of formula_2 is defined to be the set of elements of the form
for all solutions of the equations
with formula_48 and formula_49, where formula_11 denotes formula_51.
The higher order Massey product formula_44 can be thought of as the obstruction to solving the latter system of equations for all formula_48, in the sense that it contains the 0 cohomology class if and only if these equations are solvable. This "n"-fold Massey product is an formula_54 order cohomology operation, meaning that for it to be nonempty many lower order Massey operations have to contain 0, and moreover the cohomology classes it represents all differ by terms involving lower order operations. The 2-fold Massey product is just the usual cup product and is a first order cohomology operation, and the 3-fold Massey product is the same as the triple Massey product defined above and is a secondary cohomology operation.
 described a further generalization called Matric Massey products, which can be used to describe the differentials of the Eilenberg–Moore spectral sequence.
Applications.
The complement of the Borromean rings gives an example where the triple Massey product is defined and non-zero. Note the cohomology of the complement can be computed using Alexander duality. If "u", "v", and "w" are 1-cochains dual to the 3 rings, then the product of any two is a multiple of the corresponding linking number and is therefore zero, while the Massey product of all three elements is non-zero, showing that the Borromean rings are linked. The algebra reflects the geometry: the rings are pairwise unlinked, corresponding to the pairwise (2-fold) products vanishing, but are overall linked, corresponding to the 3-fold product not vanishing.
More generally, "n"-component Brunnian links – links such that any formula_55-component sublink is unlinked, but the overall "n"-component link is non-trivially linked – correspond to "n"-fold Massey products, with the unlinking of the formula_55-component sublink corresponding to the vanishing of the formula_55-fold Massey products, and the overall "n"-component linking corresponding to the non-vanishing of the "n"-fold Massey product.
 used the Massey triple product to prove that the Whitehead product satisfies the Jacobi identity.
Massey products of higher order appear when computing twisted K-theory by means of the Atiyah–Hirzebruch spectral sequence (AHSS). In particular, if "H" is the twist 3-class, showed that, rationally, the higher order differentials formula_58 in the AHSS acting on a class "x" are given by the Massey product of "p" copies of "H" with a single copy of "x".
If a manifold is (in the sense of Dennis Sullivan), then all Massey products on the space must vanish; thus, one strategy for showing that a given manifold is formal is to exhibit a non-trivial Massey product. Here a formal manifold is one whose rational homotopy type can be deduced ("formally") from a finite-dimensional "minimal model" of its de Rham complex. showed that compact Kähler manifolds are formal.
 use a Massey product to show that the homotopy type of the configuration space of two points in a lens space depends non-trivially on the simple homotopy type of the lens space.