Massey product
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In mathematics, particularly in algebraic topology but also in geometric topology and differential topology, the Massey product is a cohomology operation of higher order. It extends the range of the cup product.
On differential forms the triple product is formally defined as
leading to a product on de Rham cohomology classes.
[edit] The Massey product is a secondary operation
The exterior derivative is not in fact invertible in the space of differential forms. Instead the inverse is only well-defined modulo the addition of a closed form.
Therefore the Massey product
- PMP(ω1,ω2,ω3)
on differential forms is only well-defined modulo products of ω1 and ω3 with closed forms. These closed forms may represent nontrivial cohomology classes, and so the Massey product in de Rham cohomology is only well-defined modulo elements which may be written as a product of the class of a linear combination of ω1 and ω3 with an arbitrary cohomology element. For triple Massey product to be in de-Rham cohomology group one should have 
It is for this reason that the Massey product is a secondary and not a primary cohomology operation.
[edit] The Massey product in the AHSS
Massey products appear in the Atiyah-Hirzebruch spectral sequence (AHSS), which computes twisted K-theory with twist given by a 3-class H. Michael Atiyah and Graeme Segal have shown, in Twisted K-theory and cohomology, that rationally the higher order differentials
- d2p+1
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.
