Applications
The complement of the Borromean rings gives an example where the triple Massey product is defined and non-zero. 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)-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 (n − 1)-component sublink corresponding to the vanishing of the (n − 1)-fold Massey products, and the overall n-component linking corresponding to the non-vanishing of the n-fold Massey product.
Uehara & Massey (1957) 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, Atiyah & Segal (2008) showed that, rationally, the higher order differentials
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.
A manifold on which all Massey products vanish is a formal manifold: its real homotopy type follows ("formally") from its real cohomology ring. (Deligne et al. 1975) showed Kähler manifolds are formal.
Salvatore & Longoni (2005) 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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