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.
Let
a,b,c
H*(\Gamma)
\Gamma
ab=bc=0
\langlea,b,c\rangle
Hn(\Gamma)
n=\deg(a)+\deg(b)+\deg(c)-1
The Massey product is defined algebraically, by lifting the elements
a,b,c
u,v,w
\Gamma
Define
\baru
(-1)\deg(u)+1u
u
\Gamma
[u]
\langle[u],[v],[w]\rangle=\{[\barsw+\barut]\midds=\baruv,dt=\barvw\}.
The Massey product of three cohomology classes is not an element of
H*(\Gamma)
H*(\Gamma)
u,v,w
i,j,k
i+j+k-1
-1
d
The Massey product is nonempty if the products
uv
vw
\displaystyle H*(\Gamma)/([u]H*(\Gamma)+H*(\Gamma)[w]).
More casually, if the two pairwise products
[u][v]
[v][w]
[u][v]=[v][w]=0
uv=ds
vw=dt
s
t
[u][v][w]
sw
ut
d(sw)=ds ⋅ w+s ⋅ dw,
[dw]=0
s
t
sw
ut
n+1
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.
More generally, the n-fold Massey product
\langlea1,1,a2,2,\ldots,an,n\rangle
H*(\Gamma)
\bara1,1a2,n+\bara1,2a3,n+ … +\bara1,n-1an,n
dai,j=\barai,iai+1,j+\barai,i+1ai+2,j+ … +\barai,j-1aj,j
with
1\lei\lej\len
(i,j)\ne(1,n)
\baru
(-1)\deg(u)u
The higher order Massey product
\langlea1,1,a2,2,\ldots,an,n\rangle
1\lei\lej\len
n-1
described a further generalization called Matric Massey products, which can be used to describe the differentials of the Eilenberg–Moore spectral sequence.
The complement of the Borromean rings[1] 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
(n-1)
(n-1)
(n-1)
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
d2p+1
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.
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