nLab Massey product





Special and general types

Special notions


Extra structure



Algebraic topology



The Massey product of length nn is a certain nn-ary products on the cohomology ring of an A-infinity algebra (in particular a dg-algebra).

In components

Roughly, Massey Products are to cohomology as Toda Brackets are to homotopy.

Somewhat more fully, while Toda brackets are relations between mapping space groups Map *(Σ nA 0,A n+2)Map_* (\Sigma^n A_0, A_{n+2}) and chains of maps A 0A n+2 A_0 \to \cdots \to A_{n+2} , and generalizing nullhomotopy of composition, Massey products are a relation between cohomology groups H p 0++p kk+1(X) H^{p_0 + \cdots + p_k - k + 1}(X) and H p 0(X)H p k(X) H^{p_0} (X) \otimes \cdots \otimes H^{p_k}(X) , generalizing the vanishing of pairwise cup products.

The case k=2k=2 is straight-forward enough: given three homogeneous classes [u],[v],[w] [u],[v],[w] such that [u][v]=[v][w]=0 [u]\smile[v] = [v]\smile[w] = 0, there are (various) choices of cochains s,t s , t with ds=uv d s = u \cdot v and dt=vw d t = v \cdot w . The Massey triple product is the set of sums [ut±sw] [ u \cdot t \pm s \cdot w ] , where the sign is chosen for cocyclicity.


Relation to Steenrod squares

Let ω,ω 1,ω 2H (X,/2)\omega, \omega_1, \omega_2 \in H^\bullet(X,\mathbb{Z}/2) such that their triple Massey product exists. Then the cup product of ω\omega with the triple Massey product is independent of the ambiguity in the Massey products and equals the cup product of ω 1\omega_1 with ω 2\omega_2 and with the Steenrod square of ω\omega of degree deg(ω)1deg(\omega)-1:

ωω 1,ω,ω 2=ω 1ω 2Sq deg(ω)1(ω). \omega \cup \left\langle \omega_1, \omega, \omega_2 \right\rangle \;=\; \omega_1 \cup \omega_2 \cup Sq^{ deg(\omega) -1 }( \omega ) \,.

(Taylor 11, slide 10, following Milgram 68)

Relation to A A_\infty-algebra

For AA a dg-algebra, its chain homology H (A)H_\bullet(A) inherits an A-infinity algebra structure by Kadeishvili's theorem. Then for every nn \in \mathbb{N} the nn-ary A A_\infty-product on elements (a 1,,a n)H (A) n (a_1, \cdots, a_n) \in H_\bullet(A)^n is given, up to a sign, by the Massey product a 1,,a n\langle a_1, \cdots, a_n\rangle.

For n=3n = 3 this is due to (Stasheff). For general nn this appears as (LPWZ, theorem 3.1).



See also

Relation to A A_\infty-algebra

The relation of Massey products to A-infinity algebra structures is in Chapter 12 of

for n=3n = 3, and for general nn in Theorem 3.1 and Corollary A.5 of

  • D.-M. Lu, J. H. Palmieri, Q.-S. Wu, J. J. Zhang, A A_\infty-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 Vallette-Loday, x,y,z\langle x,y,z\rangle depends on choices of a,ba,b which don’t appear in the notation. Then lemma 9.4.11 talks about a particular choice of a,ba,b 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).

  • Daniel Grady, Hisham Sati, Massey products in differential cohomology via stacks, J. Homotopy Relat. Struct. 13 (2017) 169-223 (arXiv:1510.06366, doi:10.1007/s40062-017-0178-y).

Last revised on February 27, 2021 at 11:11:47. See the history of this page for a list of all contributions to it.