In algebraic topology, the Massey product is a cohomology operation of higher order.

Definition

In a differential graded algebra the Massey triple product is defined as

${\displaystyle \langle \omega _{1},\omega _{2},\omega _{3}\rangle \ {\stackrel {\mathrm {def} }{=))\ \omega _{1}\wedge d^{-1}(\omega _{2}\wedge \omega _{3})-(-1)^{\deg(\omega _{1})}d^{-1}(\omega _{1}\wedge \omega _{2})\wedge \omega _{3))$

whenever ω₁ ∧ ω₂ and ω₂ ∧ ω₃ are exact (in other words in the image of d). The exterior derivative is not in fact invertible. Instead the inverse is only well-defined modulo the addition of certain closed elements (elements on which d vanishes).

Therefore the Massey product

${\displaystyle \langle \omega _{1},\omega _{2},\omega _{3}\rangle }$

is only well-defined modulo products of ${\displaystyle \omega _{1))$ and ${\displaystyle \omega _{3))$ with closed elements. These closed elements may represent nontrivial cohomology classes, and so the Massey product in cohomology is only well-defined modulo elements which may be written as a product of the class of a linear combination of ${\displaystyle \omega _{1))$ and ${\displaystyle \omega _{3))$ with an arbitrary cohomology class. For triple Massey product to be in cohomology group one should have ${\displaystyle \omega _{1))$ and ${\displaystyle \omega _{3))$ both closed.

Applications

Massey products appear in the Atiyah-Hirzebruch spectral sequence (AHSS), which computes twisted K-theory with twist given by a 3-class H. Atiyah & Segal (2008) harvtxt error: no target: CITEREFAtiyahSegal2008 (help) showed that rationally the higher order differentials

d_2p+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.