nLab Massey product

Idea

The Massey product of length $n$ is a certain $n$-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 $Map_* (\Sigma^n A_0, A_{n+2})$ and chains of maps $A_0 \to \cdots \to A_{n+2}$, and generalizing nullhomotopy of composition, Massey products are a relation between cohomology groups $H^{p_0 + \cdots + p_k - k + 1}(X)$ and $H^{p_0} (X) \otimes \cdots \otimes H^{p_k}(X)$, generalizing the vanishing of pairwise cup products.

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

Properties

Relation to Steenrod squares

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

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

(Taylor 11, slide 10, following Milgram 68)

Relation to $A_\infty$-algebra

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

For $n = 3$ this is due to Stasheff, where all $A_\infty$-products are Massey products. For general $n$ the situation is discussed in BMM 2020 (where it is also shown that the earlier claim in LPWZ, theorem 3.1, that still all $A_\infty$-products are Massey products, is wrong for $n \gt 3$).

Related concepts

• cup product

• cohomology operation

References

General

• David Kraines, Massey higher products, Transactions of the American Mathematical Society Vol. 124, No. 3 (Sep., 1966), pp. 431-449 (jstor)

• Edward J. O’Neill, On Massey products, Pacific J. Math. Volume 76, Number 1 (1978), 123-127. (EUCLID)

• Stanley Kochmann, section 5.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• R. James Milgram, Steenrod squares and higher Massey products, Bol. Soc. Mat. Mexicana (2) 13 (1968), 32–57.MR0263074 (web)

• Laurence Taylor, Massey Triple Products, Princeton 2011 (pdf)

See also

• Wikipedia, Massey product

Relation to $A_\infty$-algebra

On the relation of Massey products to A-infinity algebra structures, for $n = 3$:

• Jim Stasheff, Chapter 12 of: H-spaces from a homotopy point of view, Lecture Notes in Mathematics 161, Springer (1970) [doi:10.1007/BFb0065896]

A statement for higher $n$ was claimed in

• D.-M. Lu, J. H. Palmieri, Q.-S. Wu, J. J. Zhang, Theorem 3.1 and Corollary A.5 of, $A_\infty$-structures in Ext algebras, J. Pure Appl. Alg. 213 (2009) 2017–2037 [arXiv:math/0606144]

but the situation is more subtle, as explained in:

• Urtzi Buijs, José Manuel Moreno-Fernández, Aniceto Murillo: A-infinity structures and Massey products, Mediterr. J. Math. 17 31 (2020) [arXiv:1801.03408, doi:10.1007/s00009-019-1464-1]

See also:

• Bruno Vallette, item 1.4 in: Algebra+Homotopy=Operad [arXiv:1202.3245]

• Bruno Vallette, Jean-Louis Loday, sections 9.4.10 to 9.4.12 of: Algebraic Operads [pdf]

Notice that the definition of Massey product on top of p.282 of Vallette-Loday, $\langle x,y,z\rangle$ depends on choices of $a,b$ which don’t appear in the notation. Then lemma 9.4.11 talks about a particular choice of $a,b$ 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).