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 was created by William S. Massey, an American algebraic topologist.
Massey triple product Let a , b , c {\displaystyle a,b,c} be elements of the cohomology algebra H ∗ ( Γ ) {\displaystyle H^{*}(\Gamma )} of a differential graded algebra Γ {\displaystyle \Gamma } . If a b = b c = 0 {\displaystyle ab=bc=0} , the Massey product ⟨ a , b , c ⟩ {\displaystyle \langle a,b,c\rangle } is a subset of H n ( Γ ) {\displaystyle H^{n}(\Gamma )} , where n = deg ( a ) + deg ( b ) + deg ( c ) − 1 {\displaystyle n=\deg(a)+\deg(b)+\deg(c)-1} . The Massey product is defined algebraically, by lifting the elements a , b , c {\displaystyle a,b,c} to equivalence classes of elements u , v , w {\displaystyle u,v,w} of Γ {\displaystyle \Gamma } , 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 u ¯ {\displaystyle {\bar {u}}} to be ( − 1 ) deg ( u ) + 1 u {\displaystyle (-1)^{\deg(u)+1}u} . The cohomology class of an element u {\displaystyle u} of Γ {\displaystyle \Gamma } will be denoted by [ u ] {\displaystyle [u]} . The Massey triple product of three cohomology classes is defined by
⟨ [ u ] , [ v ] , [ w ] ⟩ = { [ s ¯ w + u ¯ t ] ∣ d s = u ¯ v , d t = v ¯ w } . {\displaystyle \langle [u],[v],[w]\rangle =\{[{\bar {s}}w+{\bar {u}}t]\mid ds={\bar {u}}v,dt={\bar {v}}w\}.}
The Massey product of three cohomology classes is not an element of H ∗ ( Γ ) {\displaystyle H^{*}(\Gamma )} , but a set of elements of H ∗ ( Γ ) {\displaystyle H^{*}(\Gamma )} , possibly empty and possibly containing more than one element. If u , v , w {\displaystyle u,v,w} have degrees i , j , k {\displaystyle i,j,k} , then the Massey product has degree i + j + k − 1 {\displaystyle i+j+k-1} , with the − 1 {\displaystyle -1} coming from the differential d {\displaystyle d} . The Massey product is nonempty if the products u v {\displaystyle uv} and v w {\displaystyle vw} are both exact, in which case all its elements are in the same element of the quotient group
H ∗ ( Γ ) / ( [ u ] H ∗ ( Γ ) + H ∗ ( Γ ) [ w ] ) . {\displaystyle \displaystyle H^{*}(\Gamma )/([u]H^{*}(\Gamma )+H^{*}(\Gamma )[w]).}
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