2. The cohomology of ${M_d^{(k)}(n)}$ : additive structure

In this section we recall the geometric-combinatorial description of the additive structure of the cohomology of ${M_d^{(k)}(n)}$ , as given in [ Reference Dobrinskaya and Turchin5 ]. We also shed additional light on some points in Dobrinskaya-Turchin’s constructions. All homology and cohomology groups will be taken with either integer ( $\mathbb{Z}$ ) or mod-2 ( $\mathbb{Z}_2$ ) coefficients. Assertions made without specifying coefficients are meant to hold for both options. While $\mathbb{Z}$ coefficients are needed to set descriptions correctly, Massey product computations in Section 4 will use exclusively mod 2 coefficients for the sake of simplicity, so orientations and sign specifications below can and will safely be ignored.

In pictures like the one above, we agree that elements inside a square vertex are written increasingly, from left to right, following ingredient (b·2) of the intended orientation. Note that, in the orientation set, the transposition of a square vertex and an oriented edge produces a positive Koszul sign since $d(k-2)(d-1)$ is even. Thus, the ordering in the orientation set is really a pair of orderings, one for square vertices and another for oriented edges.

Orientation and three-term relations can be used to express any oriented k-forest as a linear combination of oriented linear k-forests, that is, of oriented k-forests whose non-trivial components, i.e. those different from an isolated round vertex, are trees with square vertices lying along an embedded arc, as in Figure 2. Similarly, orientation and generalised Jacobi relations can be used to express any oriented k-forest as a linear combination of oriented ordered k-forests, i.e., those satisfying that the largest of the integers inside round vertices attached to a given square vertex A is larger than any of the integers inside A. The two rewriting processes can then be coordinated so to yield basis elements:

Fig. 1.

A 3-forest on $\text{ 20}=\{1,2,\ldots,20\}$ with four connected components and its orientation ingredients. The small bold numbers determine the orientation set.

Fig. 2.

A non-trivial component of a linear k-forest.

Theorem 2·2 is proved in [ Reference Dobrinskaya and Turchin5 ] in two stages. First, a set of cohomology classes parametrised by oriented k-forests is constructed as Borel-Moore Poincaré duals of fundamental classes of suitably chosen oriented submanifolds of ${M_d^{(k)}(n)}$ . See the revision below. It is then checked that the cohomology classes resulting from basic k-forests give the identity matrix when paired with an explicit basis for the homology of ${M_d^{(k)}(n)}$ . For the purposes of this work, the rest of the section is devoted to recalling and illustrating the connection between oriented k-forests and their corresponding Poincaré-dual fundamental classes in the Borel-Moore homology of ${M_d^{(k)}(n)}$ .

Consider the projection $p_1:\mathbb{R}^d \to \mathbb{R}^{d-1}$ onto the last $d-1$ coordinates, i.e., $p_1(x)=(x^{(2)},\ldots,x^{(d)})$ , for $x=(x^{(1)},x^{(2)},\ldots,x^{(d)})$ . An oriented k-forest T determines a convex domain $c_T$ of a vector subspace $C_T$ of $(\mathbb{R}^d)^n$ . Explicitly, $C_T$ consists of all tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ satisfying:

1. (i) if i and j in $\mathbf{n}$ lie in the same square vertex, then $x_i=x_j$ ;

2. (ii) if two vertices A and B of T are connected by an edge oriented from A to B, then for all $i\in A$ , $j\in B$ , one has $p_1(x_i) = p_1(x_j)$ .

Note that, if i and j lie in the same connected component of T, then the condition $p_1(x_i)=p_1(x_j)$ holds true for the points $(x_1,\ldots,x_n)$ in $C_T$ . The domain $c_T$ , also referred to as a linear cell, is defined by the equalities above together with the inequalities

(2·1) \begin{equation}x_i^{(1)} \leq x_j^{(1)}\end{equation}

in the case of (ii). Note that the degree of T, $\deg(T)$ , in Definition 2·1 is the codimension of both $C_T$ and $c_T$ in $(\mathbb{R}^d)^n$ .

Recall from the introduction that ${A_d^{(k)}(n)}$ is the subset of $(\mathbb{R}^d)^n$ given by union of diagonals $x_{i_1} = \cdots = x_{i_k}$ over all k-element subsets of $\{1,\ldots,n\}$ . The locally compact linear cell $c_T$ has boundary contained in ${A_d^{(k)}(n)}$ and, thus, represents the Borel-Moore fundamental class in

$${H}_{dn-\deg(T)}^{\text{BM}}\big({M_d^{(k)}(n)}\big)$$

of the submanifold $\text{Int}(c_T)$ of ${M_d^{(k)}(n)}$ given by the interior of $c_T$ . We say that the submanifold $\text{Int}(c_T)$ is encoded by T.

The orientation ingredients of T determine (as illustrated below) a co-orientation of $C_T$ in $(\mathbb{R}^d)^n$ and, thus, of $\text{Int}(c_T)$ in ${M_d^{(k)}(n)}$ . We thus get a $\deg(T)$ -dimensional cohomology class in ${M_d^{(k)}(n)}$ which is Poincaré dual to the Borel-Moore fundamental class of $\text{Int}(c_T)$ in ${M_d^{(k)}(n)}$ . As suggested by Theorem 2·2, the resulting cohomology class is denoted by T. Note that, while oriented k-forests can be considered as elements in the cohomology of ${M_d^{(k)}(n)}$ , two distinct forests might represent the same class, see for instance Remark 4·2. Of course, such a faithfulness problem does not hold in the case of basic k-forests.

For example, the basic 4-forest $T\in H^8(M_{3}^{(4)}(7))$ given by

corresponds to the linear cell $c_T$ consisting of all tuples $(x_1,\ldots,x_7)\in(\mathbb{R}^3)^7$ such that $x_1 = x_2 = x_4$ , $p_1(x_1)=p_1(x_6)$ and $x_1^{(1)} \leq x_6^{(1)}$ . The co-orientation of $C_T$ , i.e., the orientation of the normal bundle of ${C_T}\hookrightarrow(\mathbb{R}^3)^7$ , is induced through the surjection $\pi_T\colon(\mathbb{R}^3)^7\to\mathbb{R}^{\deg(T)}=(\mathbb{R}^3)^2\times\mathbb{R}^2$ with components $\pi_{\mbox{}}\colon(\mathbb{R}^3)^7\to(\mathbb{R}^3)^2$ and $\pi_{\mbox{}}\colon(\mathbb{R}^3)^7\to\mathbb{R}^2$ given by

Note that $C_T$ is the kernel of $\pi_T$ and, hence, the tangent space of $c_T$ . Moreover, following the orientation of T, the first and second components of $\pi_T$ account, respectively, for the square vertex and the edge in T.

In such a setting, sums of (co)homology classes correspond to unions of representing linear cells, while signs arise from a consistent management of orientations. For example, consider the three-term relation

which, under the sign conventions can be written as

The point is that the term on the left-hand side encodes the linear cell given as the union of the two linear cells encoded by the summands on the right-hand side. To illustrate the phenomenon, consider the sum

which encodes the linear submanifold of $M_d^{(3)}(9)$ corresponding to the interior of the union of the co-oriented linear cells $c_1$ and $c_2$ with common defining inequalities

(2·2) \begin{equation}x_1^{(1)}=x_2^{(1)} \leq x_3^{(1)}, \;\;\;x_4^{(1)}=x_5^{(1)} \leq x_6^{(1)}\;\;\;\mbox{and}\;\;\; x_7^{(1)}=x_8^{(1)} \leq x_9^{(1)},\end{equation}

together with the requirement that all $x_i$ -coordinates have the same projection under $p_1$ . The additional defining inequalities in $c_1$ are

(2·3) \begin{equation}\underbrace{x_4^{(1)} \leq x_7^{(1)}}_2 \mbox{ and } \underbrace{x_7^{(1)}\leq x_1^{(1)}}_{1},\end{equation}

while the additional defining inequalities in $c_2$ are

(2·4) \begin{equation}\underbrace{x_4^{(1)} \leq x_1^{(1)}}_{1} \mbox{ and } \underbrace{x_1^{(1)}\leq x_7^{(1)}}_{2}.\end{equation}

The union of conditions (2·3) and (2·4) can then be stated as

$$\underbrace{x_4^{(1)}\leq x_1^{(1)}}_{1} \mbox{ and } \underbrace{x_4^{(1)}\leq x_7^{(1)}}_2$$

which, together with (2·2), is encoded by

3. The cohomology of ${M_d^{(k)}(n)}$ : products

The cup product $T{{}\smile{}}T'$ of oriented k-forests T and T’ is assessed geometrically in [ Reference Dobrinskaya and Turchin5 ] as the Poincaré dual of the fundamental class of the intersection $c_T\cap c_{T'}$ . We start by setting notation and basic ingredients, which can be found in standard references such as [ Reference Bredon3 , Section V.11], [ Reference Iversen12 , sections II.9, IX.3, IX.4 and IX.5], [ Reference Fulton7 , section 19.1] and [ Reference Spanier21 , theorem 10·4].

For a locally compact space Z, there is a (sheaf theoretic supported) cap product $\frown\colon H^{\text{BM}}_a(Z)\otimes H^b(Z)\to H^{\text{BM}}_{a-b}(Z)$ . This has several properties, including:

1. (1) $f_*(a'\frown f^*\xi)=f_*a'\frown\xi$ , for any proper map $f\colon Z'\to Z$ and arbitrary classes $a'\in H^{\text{BM}}_*(Z')$ and $\xi\in H^*(Z)$ ;

2. (2) $(a\frown\xi)\frown\eta=a\frown(\xi\smile\eta)$ , for arbitrary classes $\xi,\eta\in H^*(Z)$ and $a\in H^{\text{BM}}_*(Z)$ ;

3. (3) for an oriented n-dimensional (Hausdorff paracompact) manifold N, cap product with the fundamental class $[N]\in H^{\text{BM}}_n(N)$ yields a duality isomorphism

   $$D\colon H^*(N)\to H_{n-*}^{\text{BM}}(N);$$

4. (4) for an oriented properly embedded submanifold $V\subset N$ of codimension c, the orientation class $\mathfrak{o}^N_V\in H^{{c}}(N)$ of V in N, i.e., the restriction of the (normal) Thom class $\mathfrak{u}^N_V\in H^{{c}}(N,N-V)$ of V in N, yields $D(\mathfrak{o}^N_V)=[V]_N{{}\in H^{\text{BM}}_{n-c}(N),}$ the image of [V] under the inclusion $V\hookrightarrow N$ .

This information suffices to prove, just as in [ Reference Bredon2 , theorem∼VI·11·9], that cup products on a given oriented n-manifold N can be assessed, in geometrical terms, through the intersection pairing at the bottom of the commutative square

(3·1)

Theorem 3·1. Let the manifold N be as in item (3) above. If the oriented submanifolds X and Y are properly embedded in N and have transverse intersection, then

\[[X]_N \bullet [Y]_N = [X\cap Y]_N.\]

The use of Theorem 3·1 in the case of non-k-equal manifolds leads to:

Definition 3·2. Two oriented k-forests $T_1$ , $T_2 \in H^\ast({M_d^{(k)}(n)})$ are said to be superposable when no square vertex of $T_1$ intersects a square vertex of $T_2$ . In such a case we define the bent superposition $T_1\cup T_2$ as the oriented ${\mathbf{n}}$ -bipartitioned graph obtained by superposition of the oriented ${\mathbf{n}}$ -bipartitioned graphs underlying $T_1$ and $T_2$ . This means that square vertices, their contents, and oriented edges between such vertices in $T_1\cup T_2$ are those holding either on $T_1$ or on $T_2$ . Likewise, round vertices with content i, as well as oriented edges involving such vertices in $T_1\cup T_2$ are those holding either on $T_1$ or on $T_2$ , as long as i has not been accounted for by some square vertex. Instead, if some integer $i \in {\mathbf{n}}$ lies in a square vertex A in, say, $T_{1}$ as well as in a round vertex attached to some square vertex B through an oriented edge in, respectively, $T_{2}$ , then not only i appears in $T_1\cup T_2$ inside the corresponding square vertex A, but a corresponding oriented “bent” edge in $T_1\cup T_2$ between A and B must be added. The left- and right-hand sides in Figure 3.1 sketch the relevant situations for $T_2$ and $T_1\cup T_2$ , respectively.

Fig. 3.

Bending an edge.

Note that $T_1\cup T_2$ might end up having multiple oriented edges between square vertices, as well as round vertices having two square vertices as immediate neighbouring vertices.

Theorem 3·3 ([ Reference Dobrinskaya and Turchin5 , theorem 7·1]). The cup product $T_1\smile T_2$ of two oriented k-forests $T_1$ , $T_2 \in H^\ast({M_d^{(k)}(n)})$ vanishes in either of the following three conditions:

1. (A) $T_1$ and $T_2$ are not superposable;

2. (B) $T_1$ and $T_2$ are superposable and $T_1 \cup T_2$ has unoriented cycles (for instance if two square vertices of $T_1 \cup T_2$ are joined by multiple edges);

3. (C) $T_1$ and $T_2$ are superposable and $T_1 \cup T_2$ has a square vertex with no round vertex attached.

In any other case the intersection $c_{T_1}\cap c_{T_2}$ is transverse and

$$T_1 {{}\smile{}} T_2{{}={}}T_1\cup T_2$$

with orientation set given by the concatenation of the orientation sets of the factors, and with the convention that, if $T_1\cup T_2$ happens not to be a k-forest (in the sense of Definition 2·1), so that $T_1 \cup T_2$ has one or several round vertices of valency 2, then we transform $T_1 \cup T_2$ into a sum of oriented k-forests through repeated use of orientation relations and the following form of the three-term relation:

(3·2)

As above, pictures are local.

Items (B) and (C) in Theorem 3·3 might have to be used in the iterative process of applying relation (3·2) in order to write a non-k-forest $T_1\cup T_2$ as a sum of oriented k-forests. For instance, if the pictures in (3·2) are in fact global (omitting possible isolated round vertices), then each of the two summands on the right of (3·2) would vanish in view of item (C) in Theorem 3·3.

Relevant for us is the fact that $H^\ast({M_d^{(k)}(n)})$ is multiplicatively generated by elementary k-forests, i.e., basic k-forests having a single square vertex. Explicitly, a basic k-forest is, up to sign, the product of its connected components. In turn, each such connected component is, up to sign, a product of elementary k-forests. For example, the basic 3-forest

can be factorised as

Note that factorizations are not unique.

The following additional piece of information regarding items (A)–(C) in Theorem 3·3 will be useful in the next section.

Lemma 3·4. Consider cohomology classes $u,v\in H^*({M_d^{(k)}(n)})$ represented, respectively, by elementary k-forests:

1. (a) If $T_u$ and $T_v$ are not superposable (so that $u\smile v=0$ ), then representing k-forests and linear cells can be chosen so that $c_{T_u} \cap c_{T_v}$ is empty in $M^{(k)}_d(n)$ .

2. (b) Assume $T_u$ and $T_v$ are superposable, still with $u\smile v=0$ , and set $\omega\,:\!=\text{card}\,(\{a_1, \ldots,a_{k-1},b_1,\ldots, b_r \} \cap \{c_1,\ldots,c_{k-1},d_1,\ldots,d_s\})$ —so that $\omega \gt 0$ . Then one of the following options must hold:

   1. (b.1) $\,\,\,{\omega} \gt 1$ ;

   2. (b.2) $\,\,\,{\omega}={r}=1$ and ${b}_1\in {\{c_1,\ldots,c_{k-1}\}}$ ;

   3. (b.3) $\,\,\,{\omega}={s}=1$ and ${d}_1\in {\{a_1,\ldots,a_{k-1}\}}$ ;

   4. (b.4) $\,\,\,{\omega}=r=s=1$ and $b_1=d_1$ .

Furthermore, the conclusion in (a) also holds true in case (b.1), whereas the intersection $c_U\cap c_V$ is a Borel-Moore boundary in cases (b.2)–(b.4).

4. Massey products and duality

For the rest of the paper we deal exclusively with non-3-equal manifolds $M^{(3)}_d(n)$ . Orientation issues will be neglected by working with mod-2 coefficients to simplify arguments. In particular, edges of a 3-forests T will no longer be oriented. Yet, we will keep the convention that linear cells $c_T$ , and the corresponding submanifolds $\text{Int}(c_T)$ of $M^{(3)}_d(n)$ encoded by T, are taken (in terms of (2·1)) as if edge orientations were the canonical ones in Figure 2, i.e., assuming that edges point right or upwards. See for instance (4·3) and (4·4) below.

We assume familiarity with Massey’s triple products as originally defined in [ Reference Massey15 , Reference Uehara and Massey22 ]. We briefly review essential parts, mainly to set notation. Let X be a topological space, and let $u\in H^p(X)$ , $v\in H^q(X)$ and $w\in H^r(X)$ be cohomology classes (recall that we only care for mod 2 coefficients) with trivial cup products

(4·1) \begin{equation}u\cdot v=0 \quad\text{and}\quad v\cdot w=0.\end{equation}

Then Massey’s triple product $\langle u, v, w\rangle$ is defined and is an element of the factor group

$$H^{p+q+r-1}(X)/ \big( u\cdot H^{q+r-1}(X)+H^{p+q-1}(X)\cdot w \big).$$

When we say that a Massey triple product $\langle u,v,w\rangle$ is well defined, we simply mean that (4·1) holds. Additionally, we say that a well defined triple product is trivial when it vanishes, i.e., when it is represented by some (any) element of the indeterminacy

$$u\cdot H^{q+r-1}(X)+H^{p+q-1}(X)\cdot w.$$

In his seminal work [ Reference Massey16 ], Massey introduced a geometric method to assess these products when X is a manifold. The idea is to use Poincaré duality in order to replace cup products at the cochain level by intersections of dual submanifolds at the (locally compact) chain level. Variants of the technique have been used in knot theory to compute higher order linking numbers [ Reference Hain9 , Reference Hsieh, Kauffman and Tsau11 , Reference Porter20 ]. Intersection theory has also been used to evaluate Massey products on classical configuration spaces [ Reference Longoni and Salvatore14 , Reference Miller17 ]. With mod-2 coefficients, the basic (folklore) observation is summarized in Remark 4·1 below, where we use the symbol $\pitchfork$ to indicate that an intersection of submanifolds is transverse.

In the computations below, $M^{(3)}_d(n)$ will play the role of N, while the submanifolds L, K and M will be encoded by suitably chosen 3-forests. On the other hand, the submanifolds X and Y will fail to correspond to honest 3-forests, but will be encoded through a similar terminology (cf. Remark 2·4). For example, the first forest-like graph in

(4·2)

encodes the $(4d-3)$ -codimensional linear submanifold of $M^{(3)}_d(n)$ determined by the conditions

(4·3) \begin{align}&p_1(x_i)=p_1(x_j), \text{ for } i,j\in\{1,2,3,4,5\}, \nonumber\\&x_3=x_4, \\&x^{(1)}_1\leq x^{(1)}_2,\,\,x^{(1)}_1\leq x^{(1)}_3,\,\,x^{(1)} _3\leq x^{(1)}_5.\nonumber\end{align}

Likewise, the forest-like graph on the right hand-side of (4·2) encodes the $(4d-3)$ -codimensional linear submanifold of $M^{(3)}_d(n)$ determined by the conditions

(4·4) \begin{align}&{p_1(x_i)=p_1(x_j), \text{ for } i,j\in\{3,4,5,6,7\},} \nonumber\\&{x_5=x_6,} \\&{x^{(1)}_3\leq x^{(1)}_4,\,\,x^{(1)}_3\leq x^{(1)}_5,\,\,x^{(1)} _5\leq x^{(1)}_7.}\nonumber\end{align}

agree, in view of Theorem 2·2(3). The common cohomology class will simply be denoted by

Part of the subtleties in the search of non-trivial Massey products in the first meaningful case outside the range in (1·1) comes from:

Proof. Consider a well defined triple Massey product $\langle \kappa, \lambda, \mu\rangle$ of classes represented by elementary 3-forests $\kappa$ , $\lambda$ , and $\mu$ (so that $\kappa\smile\lambda=0=\lambda\smile\mu$ ). It suffices to argue that, after perhaps a slight adjustment of representing elementary 3-forests or linear cells,

(4·5) \begin{equation}c_\kappa\cap c_\lambda\text{ and }c_\lambda\cap c_\mu\text{ are empty in} M^{(3)}_d(n),\end{equation}

for then $0\in \langle \kappa, \lambda, \mu\rangle$ , by Remark 4·1. In view of Lemma 3·4, the only way in which (4·5) could fail is if some of the representing elementary 3-forests have a single round vertex attached to its single square vertex. But in those cases, Remark 4·2 can be used to assure the required conditions forcing (4·5).

As a warmup for the proof that $M^{(3)}_d(n)$ supports non-trivial triple Massey products for $n \gt 6$ , we illustrate the arguments in a simpler situation.

(4·6)

lies in the indeterminacy of

(4·7)

Proposition 4·3 implies that (4·6) lies in (4·7). Our aim is to recover the latter assertion through a direct geometric argument. Consider the submanifolds K, L and M of $N:\!=\, M^{(3)}_d(n)$ encoded, respectively, by

The submanifolds X and Y encoded, respectively, by the forest-like diagrams in (4·2) satisfy $K\pitchfork L=\partial X$ and $L\pitchfork M=\partial Y$ in N. Indeed, no boundary condition (i.e., an equality that replaces an inequality) can be taken with respect to the $\leq$ inequalities coming from the edges of the square vertices in (4·2) containing $\{3,4\}$ or $\{5,6\}$ , for otherwise we fall outside N. Furthermore $X\pitchfork M$ and $K\pitchfork Y$ are respectively encoded by

The aim is then achieved in view of Remark 4·1, as the fundamental class $\varphi\,:\!=[(X\cap M) \cup (K\cap Y)]_N\in H^{\text{BM}}_*(M^{(3)}_d(n))$ is Poincaré dual of (4·6). Indeed, the union $(X\cap M) \cup (K\cap Y)$ sits inside the boundary of the submanifold Z of N encoded by

while the rest of the boundary of Z is a manifold representing $\varphi$ and clearly encoded by (4·6).

The verification of the following auxiliary fact is an elementary exercise using the cup-product descriptions in Section 3.

Lemma 4·5 The product of two elementary terms

in $H^*(M^{(3)}_d(n))$ is either zero or a sum of basic elements of one of the two forms

and, in either case, the set of numbers inside the vertices of each of these summands is precisely $\{a,b,c,d,e,x,y,z\}$ .

Theorem 4·6. For $n \gt 6$ , the triple Massey product in $M_d^{(3)}(n)$

(4·8)

is well defined and non-trivial.

Note that $L\cap M_i=\varnothing$ for $5\leq i\leq7$ , so that $L\pitchfork\widetilde{M}=L\pitchfork M=\partial Y$ , where $\widetilde{M}=M\cup M_5\cup M_6\cup M_7$ . Since $X\cap M_i=\varnothing$ for $5\leq i\leq7$ , the conclusion in Example 4·4 extends to yield that

also lies in (4·8). The proof will then be complete once we rule out any possible solution $\alpha,\beta\in H^*(M^{(3)}_d(n))$ to the equation

(4·9)

Here, for $\ell \in \{4,5,6,7\}$ , $T_\ell$ stands for the ordered triple of elements in the set ${\{T_\ell\}}\,:\!=\{4,5,6,7\}\setminus \{\ell\}$ .

In what follows, the expression of a cohomology class $\gamma\in H^*({M_d^{(k)}(n)})$ as a $\mathbb{Z}_2$ -linear combination of basic elements (Definition 2·3) will be referred to as the expansion of $\gamma$ . For instance, by dimensional reasons, the expansions of both classes $\alpha$ and $\beta$ in a potential equation (4·9) would have to involve exclusively elementary basis elements of the form

(4·10)

with $\{a,b,c,d,e\}\subset {\mathbf{n}}$ . Then, looking at the expansions of the two products on the right-hand side of any such expression (4·9), and using Lemma 4·5, we see that no basis element $\varepsilon$ in the expansion of the first product of (4·9) can also appear in the expansion of the second product, nor $\varepsilon$ can be the basic element on the left-hand side of (4·9). Consequently, the first product on the right-hand side of any potential expression (4·9) would have to vanish, and we are left to rule out solutions to the simpler equation

(4·11)

Let $\mathfrak{S}$ denote the sum inside the parenthesis of (4·11), and suppose for a contradiction that (4·11) holds for some $\beta\in H^*(M^{(3)}_d(n))$ . We can assume without loss of generality that no basis element (4·10) in the expansion of $\beta$ has zero product with $\mathfrak{S}$ . In particular, a new application of Lemma 4·5 shows that all basis elements (4·10) in the expansion of $\beta$ must satisfy

(4·12) \begin{equation}\{1,2,3\}\subset\{a,b,c,d,e\}\subset\{1,2,3,4,5,6,7\}.\end{equation}

Let us analyse the product with $\mathfrak{S}$ of any such basis element $\tau$ in the expansion of $\beta$ . To begin with, there must be a (not necessarily unique) $\ell\in\{4,5,6,7\}$ with

(4·13)

In particular, $\{a,b\}\cap {\{T_\ell\}}=\varnothing$ and $|\{c,d,e\}\cap {\{T_\ell\}}|=1$ , in view of (4·12). Say $e\in {\{T_\ell\}}$ , so that $\{c,d\}\cap {\{T_\ell\}}=\varnothing$ . Then (4·13) takes the (perhaps non-basic) form

where $\{1,2,3\}\subset\{a,b,c,d\}\subset\{1,2,3,4,5,6,7\}\setminus {\{T_\ell\}}$ . Altogether, we have

(4·14) \begin{equation}\{a,b,c,d\}=\{1,2,3,\ell\} \quad \text{and} \quad {e\in\{T_\ell\},}\end{equation}

which allows us to evaluate in full the product $\tau\cdot{\mathfrak{S}}$ :

1. (i) if $\ell\in \{a,b\}$ , say $b=\ell$ , then

   

   which is a sum of two basis elements, in view of (4·14);

2. (ii) if $\ell\in\{c,d\}$ , say $d=\ell$ , then for $j\in\{4,5,6,7\}\setminus\{d,e\}$

   

   so that

   

   which is again a sum of two basis elements, in view of (4·14).

This shows that the product $\beta\cdot {\mathfrak{S}}$ is a sum of an even number of basis elements, which is incompatible with (4·11), since the left-hand side term is its own expansion.

Corollary 4·7. For seven pairwise distinct numbers a, b, c, d, e, f, g in ${\mathbf{n}}$ , the triple Massey product in $M_d^{(3)}(n)$

is well defined, non-trivial, and represented by