In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the Massey product is a
cohomology operation In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if ''F'' is a functor defining a cohomology theory, then a coho ...
of higher order introduced in , which generalizes the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
. The Massey product was created by
William S. Massey
William Schumacher Massey (August 23, 1920 – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact ...
, an American algebraic topologist.
Massey triple product
Let
be elements of the cohomology algebra
of a
differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
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Definition
A differential graded a ...
. 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