Massey product

In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist. Let be elements of the cohomology algebra of a differential graded algebra . If , the Massey product is a subset of , where . The Massey product is defined algebraically, by lifting the elements to equivalence classes of elements of , 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. Define to be . The cohomology class of an element of will be denoted by . The Massey triple product of three cohomology classes is defined by The Massey product of three cohomology classes is not an element of , but a set of elements of , possibly empty and possibly containing more than one element. If have degrees , then the Massey product has degree , with the coming from the differential . The Massey product is nonempty if the products and are both exact, in which case all its elements are in the same element of the quotient group So the Massey product can be regarded as a function defined on triples of classes such that the product of the first or last two is zero, taking values in the above quotient group. More casually, if the two pairwise products and both vanish in homology (), i.e., and for some chains and , then the triple product vanishes "for two different reasons" — it is the boundary of and (since and because elements of homology are cycles). The bounding chains and have indeterminacy, which disappears when one moves to homology, and since and have the same boundary, subtracting them (the sign convention is to correctly handle the grading) gives a cocycle (the boundary of the difference vanishes), and one thus obtains a well-defined element of cohomology — this step is analogous to defining the st homotopy or homology group in terms of indeterminacy in null-homotopies/null-homologies of n-dimensional maps/chains.