Special and general types
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The Massey product of length is a certain -ary products on the cohomology ring of an A-infinity algebra (in particular a dg-algebra).
Roughly, Massey Products are to cohomology as Toda Brackets are to homotopy.
Somewhat more fully, while Toda brackets are relations between mapping space groups and chains of maps , and generalizing nullhomotopy of composition, Massey products are a relation between cohomology groups and , generalizing the vanishing of pairwise cup products.
The case is straight-forward enough: given three homogeneous classes such that , there are (various) choices of cochains with and . The Massey triple product is the set of sums , where the sign is chosen for cocyclicity.
Relation to -algebra
For a dg-algebra, its chain homology inherits an A-infinity algebra structure by Kadeishvili's theorem. Then for every the -ary -product on elements is given, up to a sign, by the Massey product .
For this is due to (Stasheff). For general this appears as (LPWZ, theorem 3.1).
- Edward J. O’Neill, On Massey products, Pacific J. Math. Volume 76, Number 1 (1978), 123-127. (EUCLID)
Relation to -algebra
The relation of Massey products to A-infinity algebra structures is in Chapter 12 of
for , and for general in Theorem 3.1 and Corollary A.5 of
- D.-M. Lu, J. H. Palmieri, Q.-S. Wu, J. J. Zhang, -structures in Ext algebras, J. Pure Appl. Alg. 213 (2009), 2017–2037 (Theorem 3.1 and Corollary A.5) (arXiv:math/0606144)
as well as from item 1.4 on in
and sections 9.4.10 to 9.4.12 of
Notice that the definition of Massey product on top of p.282 of Valette-Loday, depends on choices of which don’t appear in the notation. Then lemma 9.4.11 talks about a particular choice of which is made in the body of the proof. The actual statement of the lemma only can be deduced after reading the proof. It then says that for these particular choices of a,b the said equality holds. (See this MO discussion).
In ordinary differential cohomology
Massey products in ordinary differential cohomology/Deligne cohomology are discussed in
Wenger, Massey products in Deligne cohomology
C. Deninger, Higher order operations in Deligne cohomology, Inventiones Math. 122 N1 (1995).
Alexander Schwarzhaupt, Massey products in Deligne-Beilinson cohomology (web, pdf)