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 of n elements of is defined to be the set of elements of the form for all solutions of the equations , with and , where denotes . The higher order Massey product can be thought of as the obstruction to solving the latter system of equations for all , 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 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 -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 -component sublink corresponding to the vanishing of the -fold Massey products, and the overall n-component linking corresponding to the non-vanishing of the n-fold Massey product. used the Massey