Massey Product - Higher Order Massey Products

Higher Order Massey Products

More generally the n-fold Massey product 〈a_1,1, a_2,2, ...,a_n,n〉 of n elements of H(Γ) is defined to be the set of elements of the form

for all solutions of the equations

, 1 ≤ i ≤ j ≤ n, (i,j) ≠ (1,n).

In other words it can be thought of as the obstruction to solving the latter equations for all 1≤i≤j≤n, 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 n−1 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.

May (1969) described a further generalization called Matric Massey products, which can be used to describe the differentials of the Eilenberg–Moore spectral sequence.