Massey product

In algebraic topology, the Massey product is a cohomology operation of higher order introduced in (Massey 1958), which generalizes the cup product.

The Massey product is defined algebraically at the level of chains (at the level of a differential graded algebra, or DGA); the Massey product of elements of cohomology is obtained by lifting the elements to equivalence classes of chains, taking the Massey products of these, and then pushing down to cohomology. This may result in a well-defined cohomology class, or may result in indeterminacy.

In a DGA ${\displaystyle \Gamma }$ with differential ${\displaystyle d}$, the cohomology ${\displaystyle H^{*}(\Gamma )}$ is an algebra. Define ${\displaystyle {\bar {u}}}$ to be ${\displaystyle (-1)^{deg(u)+1}u}$. The cohomology class of an element ${\displaystyle u}$ of ${\displaystyle \Gamma }$ will be denoted by [u]. The Massey triple product of three cohomology classes is defined by

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