The Cohomology of Canonical Quotients

of Free Groups and Lyndon Words

Ido Efrat

Received: April 26, 2016 Revised: April 11, 2017

Communicated by Ulf Rehmann

Abstract. For a prime number p and a free profinite group S, let S^{(n,p)} be the nth term of its lower p-central filtration, and S^{[n,p]} the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group H^{2}(S^{[n,p]}, ℤ/p), which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of S^{(n,p)}/S^{(n+1,p)}. We prove that the cohomology group satisfies shuffle relations, which for small values of n fully describe it.

2010 Mathematics Subject Classification: Primary 12G05, Secondary 20J06, 68R15 Keywords and Phrases: Profinite cohomology, lower p-central filtration, Lyndon words, Shuffle relations, Massey products

1

1. Introduction

Let p be a fixed prime number. For a profinite group G one defines the lower p-central filtration G^{(n,p)}, n = 1, 2, …, inductively by

G^{(1,p)} = G, G^{(n+1,p)} = (G^{(n,p)})^{p}[G, G^{(n,p)}].

Thus G^{(n+1,p)} is the closed subgroup of G generated by the powers h^{p} and commutators [g, h] = g^{−1}h^{−1}gh, where g ∈ G and h ∈ G^{(n,p)}. We also set G^{[n,p]} = G/G^{(n,p)}. Now let S be a free profinite group on the basis X, and let n ≥ 2. Then S^{[n,p]} is a free object in the category of pro-p groups G with G^{(n,p)} trivial. As with any pro-p group, the cohomology groups H^{l}(S^{[n,p]}) = H^{l}(S^{[n,p]}, ℤ/p), l = 1, 2, capture the main information on generators and relations, respectively, in a minimal presentation of S^{[n,p]}. The group H^{1}(S^{[n,p]}) is just the dual (S^{[2,p]})^{∨} ≅ ⨁_{x∈X} ℤ/p, and it remains to understand H^{2}(S^{[n,p]}). When n = 2 the quotient S^{[2,p]} is an elementary abelian p-group, and the structure of H^{2}(S^{[2,p]}) is well-known. Namely, for p > 2 one has an isomorphism

1This work was supported by the Israel Science Foundation (grant No. 152/13).

H^{1}(S^{[2,p]}) ⊕^{⋀2}H^{1}(S^{[2,p]}) ⟶^{∼} H^{2}(S^{[2,p]}),

which is the Bockstein map on the first component, and the cup product on the second component. Furthermore, taking a basis χ_{x}, x ∈ X, of H^{1}(S^{[2,p]}) dual to X, there is a fundamental duality between pth powers and commutators in the presentation of S and Bockstein elements and cup products, respectively, of the χ_{x} (see [NSW08, Ch. III, §9] for details). These facts have numerous applications in Galois theory, ranging from class field theory ([Koc02], [NSW08]), the works by Serre and Labute on the pro-p Galois theory of p-adic fields ([Ser63], [Lab67]), the structure theory of general absolute Galois groups ([MS96], [EM11]), the birational anabelian phenomena ([Bog91], [Top16]), Galois groups with restricted ramification ([Vog05], [Sch10]), and mild groups ([Lab06], [For11], [LM11]), to mention only a few of the pioneering works in these areas. In this paper we generalize the above results from the case n = 2 to arbitrary n ≥ 2. Namely, we give a complete description of H^{2}(S^{[n,p]}) in terms of a canonical linear basis of this cohomology group. This basis is constructed using tools from the combinatorics of words – in particular, the Lyndon words in the alphabet X, i.e., words which are lexicographically smaller than all their proper suffixes (for a fixed total order on X). We call it the Lyndon basis, and use it to prove several structural results on H^{2}(S^{[n,p]}), and in particular to compute its size (see below). The Lyndon basis constructed here can be most naturally described in terms of central extensions, as follows: For 1 ≤ s ≤ n let 𝕌 denote the group of all unipotent upper-triangular (s + 1) × (s + 1)-matrices over the ring ℤ/p^{n−s+1}. There is a central extension

0 → ℤ/p → 𝕌 → 𝕌^{[n,p]} → 1

(Proposition 6.3). It corresponds to a cohomology element γ_{n,s} ∈ H^{2}(𝕌^{[n,p]}). For a profinite group G and a continuous homomorphism ρ: G → 𝕌 we write ρ̄: G^{[n,p]} → 𝕌^{[n,p]} for the induced homomorphism, and ρ̄^{∗}γ_{n,s} for the pullback to H^{2}(G^{[n,p]}). Now for any word w = (x_{1} ⋯ x_{s}) in the alphabet X we define a homomorphism ρ^{w}: S → 𝕌 by setting the entry (ρ^{w}(σ))_{ij} to be the coefficient of the subword (x_{i} ⋯ x_{j−1}) in the power series Λ(σ), where Λ: S → (ℤ/p^{n−s+1})⟨⟨X⟩⟩^{×} is the Magnus homomorphism, defined on the generators x ∈ X by Λ(x) = 1+x (see §3 and §6 for more details). The Lyndon basis is now given by:

Main Theorem. The pullbacks α_{w,n} = (ρ̄^{w})^{∗}γ_{n,s}, where w ranges over all Lyndon words of length 1 ≤ s ≤ n in the alphabet X, form a linear basis of H^{2}(S^{[n,p]}) over ℤ/p.

We further show a duality between the Lyndon basis and certain canonical elements σ_{w} ∈ S^{(n,p)}, with w a Lyndon word of length ≤ n, generalizing the above mentioned duality in the case n = 2 (see Corollary 8.3). The cohomology elements ρ̄^{∗}γ_{n,s} include the Bockstein elements (for s = 1), the cup products (for n = s = 2), and more generally, the elements of n-fold Massey products in H^{2}(G^{[n,p]}) (for n = s ≥ 2); see Examples 7.4. The full spectrum ρ̄^{∗}γ_{n,s}, 1 ≤ s ≤ n, appears to give new significant “external” objects in profinite cohomology, which to our knowledge have not been investigated so far in general. Lyndon words are known to have tight connections with the shuffle algebra, and indeed, the α_{w,n} for arbitrary words w of length ≤ n in X satisfy natural shuffle relations (Theorem 9.4). In §10-§11 we show that for n = 2, 3 these shuffle relations fully describe H^{2}(S^{[n,p]}), provided that p > 2, p > 3, respectively (for n = 2 this was essentially known). Interestingly, related considerations arise also in the context of multiple zeta values, see e.g. [MP00], although we are not aware of a direct connection. The Lyndon words on X form a special instance of Hall sets, which are wellknown to have fundamental role in the structure theory of free groups and free Lie algebras (see [Reu93], [Ser92]). In addition, the Lyndon words have a triangularity property (see Proposition 4.4(b)). This property allows us to construct certain upper-triangular unipotent matrices that express a (semi- )duality between the σ_{w} and the cohomology elements α_{w,n}. We now outline the proof that the α_{w,n} form a linear basis of H^{2}(S^{[n,p]}). For simplicity we assume for the moment that X is finite. To each Lyndon word w of length 1 ≤ s ≤ n one associates in a canonical way an element τ_{w} of the s-th term of the lower central series of S (see §4). The cosets of the powers σ_{w} = τ_{w}^{p^{n−s}} generate S^{(n,p)}/S^{(n+1,p)} (Theorem 5.3). Using the special structure of the lower p-central filtration of 𝕌 we define for any two Lyndon words w, w′ of length ≤ n a value ⟨w, w′⟩_{n} ∈ ℤ/p (see §6). The triangularity property of Lyndon words implies that the matrix (⟨w, w′⟩_{n}) is unipotent upper-triangular, whence invertible. Turning now to cohomology, we define a natural perfect pairing

(·, ·)_{n}: S^{(n,p)}/S^{(n+1,p)} × H^{2}(S^{[n,p]}) → ℤ/p.

Cohomological computations show that, for Lyndon words w, w′ of length ≤ n, one has ⟨w, w′⟩_{n} = (σ_{w}, α_{w′,n})_{n}. Hence the matrix ((σ_{w}, α_{w′,n})_{n}) is also invertible. We then conclude that the α_{w′,n} form a basis of H^{2}(S^{[n,p]}) (Theorem 8.5). This immediately determines the dimension of the latter cohomology group, in terms of Witt’s necklace function, which counts the number of Lyndon words over X of a given length (Corollary 8.6). In the special case n = 2, the theory developed here generalizes the above description of H^{2}(S^{[2,p]}) in terms of the Bockstein map and cup products (see §10 for details). Namely, the matrix (⟨w, w′⟩_{2}) is the identity matrix, which gives the above duality between pth powers/commutators and Bockstein elements/cup products. Likewise, the shuffle relations just recover the basic fact that the cup product factors via the exterior product. In §11 we describe our theory explicitly also for the (new) case n = 3. While here we focus primarily on free profinite groups, it may be interesting to study the canonical elements ρ̄_{∗}γ_{n,s} for more general profinite groups G, in particular, when G = G_{F} is the absolute Galois group of a field F. For instance, when n = 2, they were used in [EM11] (following [MS96] and [AKM99]) and [CEM12] to describe the quotient G_{F}^{[3,p]}. Triple Massey products for G_{F} (which correspond to the case n = s = 3) were also extensively studied in recent years – see [EM15], [MT15b], [MT16], [MT17], and [Wic12] and the references therein. I thank Claudio Quadrelli and the anonymous referee for their careful reading of this paper and for their very valuable comments and suggestions on improving the exposition.

2. Words

Let X be a nonempty set, considered as an alphabet. Let X^{∗} be the free monoid on X. We view its elements as associative words on X. The length of a word w is denoted by |w|. We write ∅ for the empty word, and ww′ for the concatenation of words w and w′. Recall that a magma is a set M with a binary operation (·, ·): M × M → M. A morphism of magmas is a map which commutes with the associated binary operations. There is a free magma M_{X} on X, unique up to an isomorphism; that is, X ⊆ M_{X}, and for every magma (·, ·): N ×N → N and a map f_{0} : X → N there is a magma morphism f: M_{X} → N extending f_{0}. See [Ser92, Ch. IV, §1] for an explicit construction of M_{X}. The elements of M_{X} may be viewed as non-associative words on X. The monoid X^{∗} is a magma with respect to concatenation, so the universal property of M_{X} gives rise to a unique magma morphism f: M_{X} → X^{∗}, called the foliage (or brackets dropping) map, such that f(x) = x for x ∈ X. Let H be a subset of M_{X} and ≤ a total order on H. We say that (H, ≤) is a Hall set in M_{X}, if the following conditions hold [Reu93, §4.1]:

(i) X ⊆ H; (ii) If h = (h′, h′′) ∈ H \ X, then h < h′′; (iii) For h = (h′, h′′) ∈ M_{X} \ X, one has h ∈ H if and only if

• h′, h′′ ∈ H and h′ < h′′; and

• either h′ ∈ X, or h′ = (v, u) where u ≥ h′′.

Given a Hall set (H, ≤) in M_{X} we call H = f(H) a Hall set in X^{∗}. Every w ∈ H can be written as w = f(h) for a unique h ∈ H [Reu93, Cor. 4.5]. If w ∈ H \ X, then we can uniquely write h = (h′, h′′) with h′, h′′ ∈ H, and call w = w′w′′, where w′ = f(h′) and w′′ = f(h′′), the standard factorization of w [Reu93, p. 89].

Next we fix a total order ≤ on X, and define a total order ≤_{alp} (the alphabetical order) on X^{∗} as follows: Let w_{1}, w_{2} ∈ X^{∗}. Then w_{1} ≤_{alp} w_{2} if and only if w_{2} = w_{1}v for some v ∈ X^{∗}, or w_{1} = vx_{1}u_{1}, w_{2} = vx_{2}u_{2} for some words v, u_{1}, u_{2} and some letters x_{1}, x_{2} ∈ X with x_{1} < x_{2}. Note that the restriction of ≤_{alp} to X^{n} is the lexicographic order. In addition, we order ℤ_{≥0} × X^{∗} lexicographically with respect to the usual order on ℤ_{≥0} and the order ≤_{alp} on X^{∗}. We then define a second total order ⪯ on X^{∗} by setting

(2.1)  w_{1} ⪯ w_{2} ⟺ (|w_{1}|, w_{1}) ≤ (|w_{2}|, w_{2})

with respect to the latter order on ℤ_{≥0} × X^{∗}. A nonempty word w ∈ X^{∗} is called a Lyndon word if it is smaller in ≤_{alp} than all its non-trivial proper right factors. Equivalently, no non-trivial rotation leaves w invariant, and w is lexicographically strictly smaller than all its rotations ≠ w ([CFL58, Th. 1.4], [Reu93, Cor. 7.7]). We denote the set of all Lyndon words on X by Lyn(X), and the set of all such words of length n (resp., ≤ n) by Lyn_{n}(X) (resp., Lyn_{≤n}(X)). The set Lyn(X), totally ordered with respect to ≤_{alp}, is a Hall set [Reu93, Th. 5.1].

Example 2.1. The Lyndon words of length ≤ 4 on X are

(x) for x ∈ X,

(xy), (xxy), (xyy), (xxxy), (xxyy), (xyyy) for x, y ∈ X, x < y,

(xyz), (xzy), (xxyz), (xxzy), (xyxz), (xyyz), (xyzy), (xyzz),

(xzyy), (xzyz), (xzzy) for x, y, z ∈ X, x < y < z,

(xyzt), (xytz), (xzyt), (xzty), (xtyz), (xtzy) for x, y, z, t ∈ X, x < y < z < t

The necklace map (also called the Witt map) is defined for integers n, m ≥ 1 by

φ_{n}(m) = \frac{1}{n} ∑_{d|n} μ(d)m^{n/d}.

Here μ is the Möbius function, defined by μ(d) = (−1)^{k}, if d is a product of k distinct prime numbers, and μ(d) = 0 otherwise; Alternatively, 1/ζ(s) = ∑_{n=1}^{∞} μ(n)/n^{s}, where ζ(s) is the Riemann zeta function and s is a complex number with real part > 1. When m = q is a prime power, φ_{n}(q) also counts the number of irreducible monic polynomials of degree n over a field of q elements [Reu93, §7.6.2]. We also define φ_{n}(∞) = ∞. One has [Reu93, Cor. 4.14]

(2.2)  | Lyn_{n}(X)| = φ_{n}(|X|).

3. Power series

We fix a commutative unital ring R. Recall that a bilinear map M × N → R of R-modules is non-degenerate if its left and right kernels are trivial, i.e., the induced maps M → Hom(N, R) and N → Hom(M, R) are injective. The bilinear map is perfect if these two maps are isomorphisms.

Let R⟨X⟩ be the free associative R-algebra on X. We may view its elements as polynomials over R in the non-commuting variables x ∈ X. The additive group of R⟨X⟩ is the free R-module on the basis X^{∗}. We grade R⟨X⟩ by total degree. Let R⟨⟨X⟩⟩ be the ring of all formal power series in the noncommuting variables x ∈ X and coefficients in R. For f ∈ R⟨⟨X⟩⟩ we write f_{w} for the coefficient of f at w ∈ X^{∗}. Thus f = ∑_{w∈X^{∗}} f_{w}w. There are natural embedings X^{∗} ⊆ R⟨X⟩ ⊆ R⟨⟨X⟩⟩, where we identify w ∈ X^{∗} with the series f such that f_{w} = 1 and f_{w′} = 0 for w′ ≠ w. There is a well-defined non-degenerate bilinear map of R-modules

(3.1)  R⟨⟨X⟩⟩ × R⟨X⟩ → R, (f, g) ↦ ∑_{w∈X^{∗}} f_{w}g_{w};

See [Reu93, p. 17]. For every integer n ≥ 0, it restricts to a non-degenerate bilinear map on the homogenous components of degree n. We may identify the additive group of R⟨⟨X⟩⟩ with R^{X^{∗}} via the map f ↦ (f_{w})_{w}. When R is equipped with a profinite ring topology (e.g. when R is finite), the product topology on R^{X^{∗}} induces a profinite group topology on R⟨⟨X⟩⟩. Moreover, the multiplication map of R⟨⟨X⟩⟩ is continuous, making it a profinite ring. The group R⟨⟨X⟩⟩^{×} of all invertible elements in R⟨⟨X⟩⟩ is then a profinite group. Next we recall from [FJ08, §17.4] the following terminology and facts on free profinite groups. Let G be a profinite group and X a set. A map ψ: X → G converges to 1, if for every open normal subgroup N of G, the set X\ψ^{−1}(N) is finite. We say that a profinite group S is a free profinite group on basis X with respect to a map ι: X → S if

(i) ι: X → S converges to 1 and ι(X) generates S as a profinite group; (ii) For every profinite group G and a map ψ: X → G converging to 1, there is a unique continuous homomorphism ψ̂: S → G such that ψ = ψ̂ ∘ ι on X.

A free profinite group on X exists, and is unique up to a continuous isomorphism. We denote it by S_{X}. The map ι is then injective, and we identify X with its image in S_{X}. We define the (profinite) Magnus homomorphism Λ_{X,R}: S_{X} → R⟨⟨X⟩⟩^{×} as follows (compare [Efr14, §5]): Assume first that X is finite. For x ∈ X one has 1 = (1 + x) ∑_{i=0}^{∞}(−1)^{i}x^{i} in R⟨⟨X⟩⟩, so 1 + x ∈ R⟨⟨X⟩⟩^{×}. Hence, by (ii), the map ψ: X → R⟨⟨X⟩⟩^{×}, x ↦ 1+x, uniquely extends to a continuous homomorphism Λ_{X,R}: S_{X} → R⟨⟨X⟩⟩^{×}. Now suppose that X is arbitrary. Let Y range over all finite subsets of X. The map ψ: X → S_{Y}, which is the identity on Y and 1 on X \ Y, converges to 1. Hence it extends to a unique continuous group homomorphism S_{X} → S_{Y}. Also, there is a unique continuous R-algebra homomorphism R⟨⟨X⟩⟩ → R⟨⟨Y ⟩⟩, which is the identity on Y and 0 on X \ Y. Then

S_{X} = \underleftarrow{lim} S_{Y} , R⟨⟨X⟩⟩ = \underleftarrow{lim} R⟨⟨Y ⟩⟩, R⟨⟨X⟩⟩^{×} = \underleftarrow{lim} R⟨⟨Y ⟩⟩^{×}.

We define Λ_{X,R} = \underleftarrow{lim} Λ_{Y,R}. It is functorial in both X and R in the natural way. Note that Λ_{X,R}(x) = 1 + x for x ∈ X. In the sequel, X will be fixed, so we abbreviate S = S_{X} and Λ_{R} = Λ_{X,R}.

For σ ∈ S and a word w ∈ X^{∗} we denote the coefficient of w in Λ_{R}(σ) by ϵ_{w,R}(σ). Thus

Λ_{R}(σ) = ∑_{w∈X^{∗}} ϵ_{w,R}(σ)w.

By the construction of Λ_{R}, we have ϵ_{∅,R}(σ) = 1. Since Λ_{R} is a homomorphism, for every σ, τ ∈ S and w ∈ X^{∗} one has

(3.2)  ϵ_{w,R}(στ) = ∑_{w=u_{1}u_{2}} ϵ_{u_{1},R}(σ)ϵ_{u_{2},R}(τ).

We will also need the classical discrete version of the Magnus homomorphism. To define it, assume that X is finite, and let F_{X} be the free group on basis X. There is a natural homomorphism F_{X} → S_{X}. The discrete Magnus homomorphism Λ_{ℤ}^{disc} : F_{X} → ℤ⟨⟨X⟩⟩^{×} is defined again by x ↦ 1 + x. There is a commutative square

(3.3)  \xymatrix{F_{X} \ar[r] \ar[d]_{Λ_{ℤ}^{disc}} & S_{X} \ar[d]^{Λ_{ℤp}} \\ ℤ⟨⟨X⟩⟩^{×} \ar@{^{(}->}[r] & ℤ_{p}⟨⟨X⟩⟩^{×}.}

4. Lie algebra constructions

Recall that the lower central filtration G^{(n,0)}, n = 1, 2, …, of a profinite group G is defined inductively by

G^{(1,0)} = G, G^{(n+1,0)} = [G^{(n,0)}, G].

Thus G^{(n+1,0)} is generated as a profinite group by all elements of the form [σ, τ] with σ ∈ G^{(n,0)} and τ ∈ G. One has [G^{(n,0)}, G^{(m,0)}] ≤ G^{(n+m,0)} for every n, m ≥ 1 (compare [Ser92, Part I, Ch. II, §3]).

Proposition 4.1. Let S = S_{X} and let σ ∈ S. Then:

(a) σ ∈ S^{(n,0)} if and only if ϵ_{w,ℤ_{p}}(σ) = 0 for every w ∈ X^{∗} with 1 ≤ |w| <

n;

(b) σ ∈ S^{(n,p)} if and only if ϵ_{w,ℤ_{p}}(σ) ∈ p^{n−|w|}ℤ_{p} for every w ∈ X^{∗} with

1 ≤ |w| < n.

Proof. In the discrete case (a) and (b) are due to Grün and Magnus (see [Ser92, Part I, Ch. IV, th. 6.3]) and Koch [Koc60], respectively. The results in the profinite case follow by continuity. For other approaches see [Mor12, Prop. 8.15], [CE16, Example 4.5], and [MT15a, Lemma 2.2(d)]. □

We will need the following profinite analog of [Fen83, Lemma 4.4.1(iii)].

Lemma 4.2. Let σ ∈ S^{(n,0)} and τ ∈ S^{(m,0)}, and let w ∈ X^{∗} have length n + m. Write w = u_{1}u_{2} = u′_{2}u′_{1} with |u_{1}| = |u′_{1}| = n and |u_{2}| = |u′_{2}| = m. Then

ϵ_{w,ℤ_{p}}([σ, τ]) = ϵ_{u_{1},ℤ_{p}}(σ)ϵ_{u_{2},ℤ_{p}}(τ) − ϵ_{u′_{2},ℤ_{p}}(τ)ϵ_{u′_{1},ℤ_{p}}(σ).

Proof. By Proposition 4.1(a), we may write Λ_{ℤ_{p}}(σ) = 1 + P + O(n + 1) and Λ_{ℤ_{p}}(τ) = 1 + Q + O(m + 1), where P, Q ∈ ℤ_{p}⟨⟨X⟩⟩ are homogenous of degrees n, m, respectively, and where O(r) denotes a power series containing only terms of degree ≥ r. Then

Λ_{ℤ_{p}}([σ, τ]) = 1 + P Q − QP + O(n + m + 1).

(compare e.g., [Mor12, Proof of Prop. 8.5]). By (3.2), it follows that

\begin{aligned}ϵ_{w,ℤ_{p}}([σ, τ]) & = (P Q − QP)_{w} = P_{u_{1}}Q_{u_{2}} − Q_{u′_{2}}P_{u′_{1}} \\ & = ϵ_{u_{1},ℤ_{p}}(σ)ϵ_{u_{2},ℤ_{p}}(τ) − ϵ_{u′_{2},ℤ_{p}}(τ)ϵ_{u′_{1},ℤ_{p}}(σ). \end{aligned}

□

The commutator map induces on the graded ring ⨁_{n=1}^{∞} S^{(n,0)}/S^{(n+1,0)} a graded Lie algebra structure [Ser92, Part I, Ch. II, Prop. 2.3]. Let 𝔡 be the ideal in the ℤ_{p}-algebra ℤ_{p}⟨⟨X⟩⟩ generated by X. Then ⨁_{n=1}^{∞} 𝔡^{n}/𝔡^{n+1} is a Lie algebra with the Lie brackets defined on homogenous components by [f̄, ḡ] = \overline{fg − gf} for f ∈ 𝔡^{n}, g ∈ 𝔡^{m} [Ser92, p. 25]. By Proposition 4.1(a), Λ_{ℤ_{p}} induces a graded ℤ_{p}-algebra homomorphism

gr Λ_{ℤ_{p}}: ⨁_{n=1}^{∞} S^{(n,0)}/S^{(n+1,0)} → ⨁_{n=1}^{∞} 𝔡^{n}/𝔡^{n+1}, σS^{(n+1,0)} ↦ ∑_{|w|=n} ϵ_{w,ℤ_{p}}(σ)+𝔡^{n+1}.

Then Lemma 4.2 means that gr Λ_{ℤ_{p}} is a Lie algebra homomorphism. For w ∈ Lyn(X) we inductively define an element τ_{w} of S and a noncommutative polynomial P_{w} ∈ ℤ⟨X⟩ ⊆ ℤ_{p}⟨X⟩ as follows:

• If w = (x) has length 1, then τ_{w} = x and P_{w} = x;

• If |w| > 1, then we take the standard factorization w = w′w′′ of w with respect to the Hall set Lyn(X) (see §2), and set

τ_{w} = [τ_{w′}, τ_{w′′}], P_{w} = P_{w′}P_{w′′} − P_{w′′}P_{w′}.

For w ∈ Lyn_{n}(X) one has τ_{w} ∈ S^{(n,0)}. Moreover:

Proposition 4.3. Let n ≥ 1. The cosets of τ_{w}, with w ∈ Lyn_{n}(X), generate S^{(n,0)}/S^{(n+1,0)}.

Proof. See [Reu93, Cor. 6.16] for the discrete version. The profinite version follows by taking closure. □

Let ≤_{alp} and ⪯ be the total orders on X^{∗} defined in §2. The importance of the Lyndon words in our context, beside forming a Hall set, is part (b) of the following Proposition, called the triangularity property.

Proposition 4.4. Let w ∈ Lyn(X). Then (a) Λ_{ℤ_{p}}(τ_{w}) − 1 − P_{w} is a combination of words of length > |w|.

(b) P_{w} − w is a combination of words of length |w| which are strictly larger than w with respect to ≤_{alp}.

(c) Λ_{ℤ_{p}}(τ_{w})−1−w is a combination of words which are strictly larger than w in ⪯.

Proof. (a) Since gr Λ_{ℤ_{p}} is a Lie algebra homomorphism, for w ∈ Lyn_{n}(X) we have by induction

(gr Λ_{ℤ_{p}})(τ_{w}S^{(n+1,0)}) = P_{w} + 𝔡^{n+1},

and the assertion follows. See also [Reu93, Lemma 6.10(ii)].

(b) See [Reu93, Th. 5.1] and its proof.

(c) This follows from (a) and (b). □

5. Generators for S^{(n,p)}/S^{(n+1,p)}

Let π be an indeterminate over the ring ℤ/p and let ℤ/p[π] be the polynomial ring. Let A_{•} = ⨁_{n=1}^{∞} A_{n} be a graded Lie ℤ/p-algebra with Lie bracket [·, ·]. Suppose that there is a map ℤ/p[π] × A_{•} → A_{•}, (α, ξ) ↦ αξ, which is ℤ/plinear in ℤ/p[π], such that πξ ∈ A_{s+1} for ξ ∈ A_{s}, and such that for every ξ_{1}, ξ_{2} ∈ A_{s},

(5.1)  π(ξ_{1} + ξ_{2}) = \begin{cases}πξ_{1} + πξ_{2}, & if p > 2 or s > 1, \\ πξ_{1} + πξ_{2} + [ξ_{1}, ξ_{2}], & if p = 2, s = 1. \end{cases}

By induction, this extends to:

Lemma 5.1. Let r, k, s ≥ 1 and let ξ_{1}, …, ξ_{k} ∈ A_{s}. Then

π^{r}(∑_{i=1}^{k} ξ_{i}) = \begin{cases}∑_{i=1}^{k} π^{r}ξ_{i}, & if p > 2 or s > 1, \\ ∑_{i=1}^{k} π^{r}ξ_{i} + ∑_{i<j} π^{r−1}[ξ_{i}, ξ_{j}], & if p = 2, s = 1. \end{cases}

We write ⟨T ⟩ for the submodule of A_{•} generated by a subset T.

Lemma 5.2. Let n ≥ 2 and for each 1 ≤ s ≤ n let T_{s} be a subset of A_{s}. When p = 2 assume also that [τ_{1}, τ_{2}] ∈ T_{2} ∪ {0} for every τ_{1}, τ_{2} ∈ T_{1}. If the sets π^{n−s}⟨T_{s}⟩, s = 1, 2, …, n, generate A_{n}, then the sets π^{n−s}T_{s}, s = 1, 2, …, n, also generate A_{n}.

Proof. When p > 2 or s > 1 Lemma 5.1 shows that π^{n−s}⟨T_{s}⟩ = ⟨π^{n−s}T_{s}⟩. When p = 2 and s = 1, it shows that

π^{n−1}⟨T_{1}⟩ ⊆ ⟨π^{n−1}T_{1}⟩ + ⟨π^{n−2}T_{2}⟩ ⊆ A_{n}.

Therefore the subgroup of A_{n} generated by the sets π^{n−s}T_{s}, s = 1, 2, …, n, contains the sets π^{n−s}⟨T_{s}⟩, s = 1, 2, …, n, and hence equals A_{n}. □

Motivated by e.g., [Laz54], [Ser63], [Lab67], we now specialize to a graded Lie algebra defined using the lower p-central filtration. We refer to [NSW08, Ch. III, §8] for the following facts. For the free profinite group S on the basis X and for n ≥ 1 we set gr_{n}(S) = S^{(n,p)}/S^{(n+1,p)}. It is an elementary abelian p-group, which we write additively. The commutator map induces a map [·, ·]: gr_{n}(S) × gr_{m}(S) → gr_{n+m}(S), which endows a graded Lie algebra structure on gr_{•}(S) = ⨁_{n=1}^{∞} gr_{n}(S). The map τ ↦ τ^{p} maps S^{(r,p)} into S^{(r+1,p)}, and induces a map π_{r}: gr_{r}(S) → gr_{r+1}(S). The map (π^{r}, ξ) ↦ π_{r}(ξ) for ξ ∈ gr_{r}(S) extends to a map ℤ/p[π] × gr_{•}(S) → gr_{•}(S) which is ℤ/p-linear in the first component and which satisfies (5.1).

Theorem 5.3. Let n ≥ 1. The cosets of τ_{w}^{p^{n−s}}, with 1 ≤ s ≤ n and w ∈ Lyn_{s}(X), generate S^{(n,p)}/S^{(n+1,p)}.

Proof. The case n = 1 is immediate, so we assume that n ≥ 2. For every 1 ≤ s ≤ n let T_{s} be the set of cosets of τ_{w}, w ∈ Lyn_{s}(X), in gr_{s}(S). By Proposition 4.3, ⟨T_{s}⟩ is the image of S^{(s,0)} in gr_{s}(S). Hence π^{n−s}⟨T_{s}⟩ consists of the cosets in gr_{n}(S) of the p^{n−s}-powers of S^{(s,0)}. One has

S^{(n,p)} = ∏_{s=1}^{n}(S^{(s,0)})^{p^{n−s}}

[NSW08, Prop. 3.8.6]. Thus the sets π^{n−s}⟨T_{s}⟩, s = 1, 2, …, n, generate gr_{n}(S). Further, T_{1} consists of the cosets of x, with x ∈ X, and T_{2} consists of the cosets of commutators [x, y] with x < y in X. Moreover, when p = 2 the cosets of [x, y] and [y, x] = [x, y]^{−1} in gr_{2}(S) coincide. Lemma 5.2 therefore implies that even the sets π^{n−s}T_{s}, s = 1, 2, …, n, generate gr_{n}(S), as required. □

6. The pairing ⟨w, w′⟩_{n}

For a commutative unitary ring R and a positive integer m, let 𝕌_{m}(R) be the group of all m × m upper-triangular unipotent matrices over R. We write I_{m} for the identity matrix in 𝕌_{m}(R), and E_{ij} for the matrix with 1 at entry (i, j) and 0 elsewhere. As above, X will be a totally ordered set. For the following fact we refer, e.g., to [Efr14, Lemma 7.5]. We recall from §3 that ϵ_{u,R}(σ) is the coefficient of the word u ∈ X^{∗} in the formal power series Λ_{R}(σ) ∈ R⟨⟨X⟩⟩^{×}.

Proposition 6.1. Given a profinite ring R and a word w = (x_{1} ⋯ x_{s}) in X^{∗} there is a well defined homomorphism of profinite groups

ρ_{R}^{w}: S → 𝕌_{s+1}(R), σ ↦ (ϵ_{(x_{i}···x_{j−1}),R}(σ))_{1≤i<j≤s+1}.

Remark 6.2. In particular, for each x ∈ X the map χ_{x,R} = ϵ_{(x),R}: S → R is a group homomorphism, where R is considered as an additive group. The homomorphisms χ_{x,R}, x ∈ X, are dual to the basis X, in the sense that χ_{x,R}(x) = 1, and χ_{x,R}(y) = 0 for x ≠ y in X.

Proposition 6.3. Let 1 ≤ s ≤ n. For R = ℤ/p^{n−s+1} one has:

(a) 𝕌_{s+1}(R)^{(n,p)} = I_{s+1} + ℤp^{n−s}E_{1,s+1}.

(b) 𝕌_{s+1}(R)^{(n,p)} is central in 𝕌_{s+1}(R).

Proof. (a) We follow the argument of [MT15a, Lemma 2.4]. Take X = {x_{1}, …, x_{s}} be a set of s elements, let S = S_{X}, and let w = (x_{1} ⋯ x_{s}). The matrices ρ_{R}^{w}(x_{i}) = I_{s+1} + E_{i,i+1}, i = 1, 2, …, s, generate 𝕌_{s+1}(R) [Wei55, p. 55], so ρ_{R}^{w} is surjective. Therefore it maps S^{(n,p)} onto 𝕌_{s+1}(R)^{(n,p)}. By Proposition 4.1(b), for σ ∈ S^{(n,p)} and u ∈ X^{∗} of length 1 ≤ |u| ≤ s one has ϵ_{u,ℤ_{p}}(σ) ∈ p^{n−|u|}ℤ_{p}. If |u| < s, then ϵ_{u,ℤ_{p}}(σ) ∈ p^{n−|u|}ℤ_{p} ⊆ p^{n−s+1}ℤ_{p}. Hence ϵ_{u,R}(σ) = 0 in this case. If |u| = s, then ϵ_{u,ℤ_{p}}(σ) ∈ p^{n−s}ℤ_{p}, so ϵ_{u,R}(σ) ∈ p^{n−s}R. Moreover, τ_{w}^{p^{n−s}} ∈ (S^{(s,0)})^{p^{n−s}} ≤ S^{(n,p)}. By Proposition 4.4(c), Λ_{ℤ_{p}}(τ_{w}) = 1+w +f, where f is a combination of words strictly larger than w in ⪯. Hence Λ_{ℤ_{p}}(τ_{w}^{p^{n−s}}) = 1 + p^{n−s}w + g, where g is also a combination of words strictly larger than w in ⪯, which implies that ϵ_{w,R}(τ_{w}^{p^{n−s}}) = p^{n−s} · 1_{R}. Consequently, 𝕌_{s+1}(R)^{(n,p)} = ρ_{R}^{w}(S^{(n,p)}) = I_{s+1} + ℤp^{n−s}E_{1,s+1}.

(b) It is straightforward to see that I_{s+1} + ℤE_{1,s+1} is central in 𝕌_{s+1}(R), so the assertion follows from (a). □

See [Bor04, §2] for a related analysis of the lower p-central filtration of 𝕌_{s+1}(ℤ/p^{n−s+1}). Consider the obvious isomorphism

ι_{n,s}: p^{n−s}ℤ/p^{n−s+1}ℤ ⟶^{∼} ℤ/p, ap^{n−s} (mod p^{n−s+1}) ↦ a (mod p).

In view of Proposition 6.3(a), we may define a group isomorphism

ι_{n,s}^{𝕌}: 𝕌_{s+1}(ℤ/p^{n−s+1})^{(n,p)} ⟶^{∼} ℤ/p, (a_{ij}) ↦ ι_{n,s}(a_{1,s+1}).

Next let w, w′ ∈ X^{∗} be words of lengths 1 ≤ s, s′ ≤ n, respectively, where w is Lyndon. We have τ_{w}^{p^{n−s}} ∈ (S^{(s,0)})^{p^{n−s}} ≤ S^{(n,p)}. By Proposition 4.1(b), ϵ_{w′,ℤ_{p}}(τ_{w}^{p^{n−s}}) ∈ p^{n−s′}ℤ_{p}, and therefore ϵ_{w′,ℤ/p^{n−s′+1}}(τ_{w}^{p^{n−s}}) ∈ p^{n−s′}ℤ/p^{n−s′+1}ℤ. We set

(6.1)  ⟨w, w′⟩_{n} = ι_{n,s′}(ϵ_{w′,ℤ/p^{n−s′+1}}(τ_{w}^{p^{n−s}})) ∈ ℤ/p.

Alternatively,

(6.2)  ⟨w, w′⟩_{n} = ι_{n,s′}^{𝕌}(ρ_{ℤ/p^{n−s′+1}}^{w′}(τ_{w}^{p^{n−s}})).

Let ⪯ be as in (2.1).

Proposition 6.4. Let w, w′ be words in X^{∗} of lengths 1 ≤ s, s′ ≤ n, respectively, with w Lyndon.

(a) If w′ ≺ w, then ⟨w, w′⟩_{n} = 0;

(b) ⟨w, w⟩_{n} = 1;

(c) If w′ contains letters which do not appear in w, then ⟨w, w′⟩_{n} = 0;

(d) If s < s′ < 2s, then ⟨w, w′⟩_{n} = 0.

Proof. (a), (b) Proposition 4.4(c) implies that Λ_{ℤ_{p}}(τ_{w}^{p^{n−s}}) − 1 − p^{n−s}w is a combination of words strictly larger than w with respect to ⪯, and the same therefore holds over the coefficient ring ℤ/p^{n−s′+1}. Hence, if w′ ≺ w, then ϵ_{w′,ℤ/p^{n−s′+1}}(τ_{w}^{p^{n−s}}) = 0, so ⟨w, w′⟩_{n} = 0. If w = w′, then ϵ_{w,ℤ/p^{n−s+1}}(τ_{w}^{p^{n−s}}) = p^{n−s} · 1_{ℤ/p^{n−s+1}}, whence ⟨w, w⟩_{n} = 1.

(c) Here we clearly have ϵ_{w′,ℤ/p^{n−s′+1}}(τ_{w}^{p^{n−s}}) = 0.

(d) Since τ_{w} ∈ S^{(s,0)}, one may write Λ_{ℤ_{p}}(τ_{w}) = 1 + P + O(s′ + 1), where P is a combination of words w′′ of length s ≤ |w′′| ≤ s′, and O(s′ + 1) denotes a combination of words of length ≥ s′+1 (Proposition 4.1(a)). Since s′ < 2s, this implies that Λ_{ℤ_{p}}(τ_{w}^{p^{n−s}}) = 1+p^{n−s}P +O(s′ +1). In particular, ϵ_{w′,ℤ_{p}}(τ_{w}^{p^{n−s}}) ∈ p^{n−s}ℤ_{p}, and therefore

ϵ_{w′,ℤ/p^{n−s′+1}}(τ_{w}^{p^{n−s}}) ∈ p^{n−s}(ℤ/p^{n−s′+1}) = {0},

since s < s′. Hence ⟨w, w′⟩_{n} = 0. □

7. Transgressions

Given a profinite group G and a discrete G-module A, we write as usual C^{i}(G, A), Z^{i}(G, A), and H^{i}(G, A) for the corresponding group of continuous i-cochains, group of continuous i-cocycles, and the ith profinite cohomology group, respectively. For x ∈ Z^{i}(G, A) let [x] be its cohomology class in H^{i}(G, A). For a normal closed subgroup N of G, let trg: H^{1}(N, A)^{G} → H^{2}(G/N, A^{N}) be the transgression homomorphism. It is the map d_{2}^{0,1} of the Lyndon–Hochschild– Serre spectral sequence associated with G and N [NSW08, Th. 2.4.3]. We recall the explicit description of trg, assuming for simplicity that the G-action on A is trivial [NSW08, Prop. 1.6.6]: If x ∈ Z^{1}(N, A), then there exists y ∈ C^{1}(G, A) such that y|_{N} = x and (∂y)(σ_{1}, σ_{2}) depends only on the cosets of σ_{1}, σ_{2} modulo N, so that ∂y may be viewed as an element of Z^{2}(G/N, A). For any such y one has trg([x]) = [∂y]. We fix for the rest of this section a finite group 𝕌 and a normal subgroup N of 𝕌 satisfying:

(i) N ≅ ℤ/p; and (ii) N lies in the center of 𝕌.

We set 𝕌̄ = 𝕌/N, and let it act trivially on 𝕌. We denote the image of u ∈ 𝕌 in 𝕌̄ by ū. We may choose a section λ of the projection 𝕌 → 𝕌̄ such that λ(¯1) = 1. We define a map δ ∈ C^{2}(𝕌̄, N) by

δ(ū, ū′) = λ(ū) · λ(ū′) · λ(ūū′)^{−1}.

It is normalized, i.e., δ(ū, 1) = δ(1, ū) = 1 for every ū ∈ 𝕌̄. We also define y ∈ C^{1}(𝕌, N) by y(u) = uλ(ū)^{−1}. Note that y|_{N} = id_{N}.

Lemma 7.1. For every u, u′ ∈ 𝕌 one has

δ(ū, ū′) · y(u) · y(u′) = y(uu′).

Proof. Since y(u) and y(u′) are in N, they are central in 𝕌, so

δ(ū, ū′) · y(u) · y(u′) = λ(ū) · λ(ū′) · λ(ūū′)^{−1} · y(u) · y(u′)
= y(u) · λ(ū) · y(u′) · λ(ū′) · λ(ūū′)^{−1}
= uu′λ(ūū′)^{−1} = y(uu′).
□

For the correspondence between elements of H^{2} and central extensions see e.g., [NSW08, Th. 1.2.4].

Proposition 7.2. Using the notation above, the following holds.

(a) δ ∈ Z^{2}(𝕌̄, N);

(b) One has trg(id_{N}) = −[δ] for the transgression map trg: H^{1}(N, N)^{𝕌} → H^{2}(𝕌̄, N).

(c) The cohomology class [δ] ∈ H^{2}(𝕌̄, N) corresponds to the equivalence class of the central extension

(7.1)  1 → N → 𝕌 → 𝕌̄ → 1.

Proof. (a), (b): For u, u′ ∈ 𝕌 Lemma 7.1 gives

(∂y)(u, u′) = y(u) · y(u′) · y(uu′)^{−1} = δ(ū, ū′)^{−1}.

This shows that δ is a 2-cocycle, and that (∂y)(u, u′) depends only on the cosets ū, ū′. Further, id_{N} ∈ Z^{1}(N, N). By the explicit description of the transgression above, trg(id_{N}) = −[δ].

(c) Consider the set B = N × 𝕌̄ with the binary operation

(u, v̄) ∗ (u′, v̄′) = (δ(v̄, v̄′)uu′, v̄v̄′).

The proof of [NSW08, Th. 1.2.4] shows that this makes B a group, and [δ] corresponds to the equivalence class of the central extension

(7.2)  1 → N → B → 𝕌̄ → 1.

Moreover, the map h: 𝕌 → B, u ↦ (y(u), ū) is clearly bijective. We claim that it is a homomorphism, whence an isomorphism. Indeed, for u, u′ ∈ 𝕌 Lemma 7.1 gives:

\begin{aligned}h(u) ∗ h(u′) & = (y(u), ū) ∗ (y(u′), ū′) = (δ(ū, ū′)y(u)y(u′), ūū′) \\ & = (y(uu′), ūū′) = h(uu′). \end{aligned}

We obtain that the central extension (7.2) is equivalent to the central extension (7.1). □

Next let Ḡ be a profinite group, and let ρ̄: Ḡ → 𝕌̄ be a continuous homomorphism. Let ι: N ⟶^{∼} ℤ/p be a fixed isomorphism (see (i)). Set

(7.3)  α = (ρ̄^{∗} ∘ ι_{∗})([δ]) = −(ρ̄^{∗} ∘ ι_{∗} ∘ trg)(id_{N}) ∈ H^{2}(Ḡ, ℤ/p),

where the second equality is by Proposition 7.2(b). Then α corresponds to the equivalence class of the central extension

(7.4)  0 → ℤ/p ⟶^{ι^{−1}×1} 𝕌 ×𝕌̄ Ḡ → Ḡ → 1,

where 𝕌×𝕌̄ Ḡ is the fiber product with respect to the natural projection 𝕌 → 𝕌̄ and to ρ̄; See [Hoe68, Proof of 1.1]. Suppose further that there is a profinite group G, a closed normal subgroup M of G, and a continuous homomorphism ρ: G → 𝕌 such that Ḡ = G/M, ρ(M) ≤ N, and ρ̄: Ḡ → 𝕌̄ is induced from ρ. The functoriality of transgression yields a commutative diagram

\xymatrix{H^{1}(N, N)^{𝕌} \ar[r]^{ι_{∗}}_{∼} \ar[d]^{trg} & H^{1}(N, ℤ/p)^{𝕌} \ar[r]^{ρ^{∗}} \ar[d]^{trg} & H^{1}(M, ℤ/p)^{G} \ar[d]^{trg} \\ H^{2}(𝕌̄, N) \ar[r]^{ι_{∗}}_{∼} & H^{2}(𝕌̄, ℤ/p) \ar[r]^{ρ̄^{∗}} & H^{2}(Ḡ, ℤ/p).}

The image of id_{N} ∈ H^{1}(N, N) in H^{1}(M, ℤ/p)^{G} is

(7.5)  θ = ι ∘ (ρ|_{M}) ∈ H^{1}(M, ℤ/p).

By (7.3) and the commutativity of the diagram,

(7.6)  α = − trg(θ) ∈ H^{2}(Ḡ, ℤ/p)

Remark 7.3. Suppose that 𝕌̄ is abelian and that Ḡ acts trivially on 𝕌̄. A 2-cocycle representing α is

(σ̄, σ̄′) ↦ ι(λ(ρ̄(σ̄)) · λ(ρ̄(σ̄′)) · λ(ρ̄(σ̄σ̄′))^{−1}).

But λ(ρ̄(σ̄)) · λ(ρ̄(σ̄′)) · λ(ρ̄(σ̄σ̄′))^{−1} is a 2-cocycle representing the image of ρ̄ under the connecting homomorphism H^{1}(Ḡ, 𝕌̄) → H^{2}(Ḡ, N) arising from (7.1). Thus α is the image of ρ̄ under the composition

H^{1}(Ḡ, 𝕌̄) → H^{2}(Ḡ, N) ⟶^{ι_{∗}} H^{2}(Ḡ, ℤ/p).

Example 7.4. We give several examples of the above construction with the group 𝕌 = 𝕌_{s+1}(ℤ/p^{n−s+1}) (where 1 ≤ s ≤ n), a continuous homomorphism ρ: G → 𝕌, where G is a profinite group, and the induced homomorphism ρ̄: G^{[n,p]} → 𝕌^{[n,p]}. Note that assumptions (i) and (ii) then hold for N = 𝕌^{(n,p)}, by Proposition 6.3. We will be especially interested in the case where G = S = S_{X} is a free profinite group, M = S^{(n,p)}, ρ = ρ_{ℤ/p^{n−s+1}}^{w} for a word w ∈ X^{∗} of length 1 ≤ s ≤ n, and ρ̄ = ρ̄_{ℤ/p^{n−s+1}}^{w}: S^{[n,p]} → 𝕌^{[n,p]} is the induced homomorphism. In this setup we write α_{w,n} for α.

(1) Bocksteins. For a positive integer m and a profinite group Ḡ, the connecting homomorphism arising from the short exact sequence of trivial Ḡ-modules

0 → ℤ/p → ℤ/pm → ℤ/m → 0

is the Bockstein homomorphism

Bock_{m,Ḡ}: H^{1}(Ḡ, ℤ/m) → H^{2}(Ḡ, ℤ/p).

Let 𝕌 = 𝕌_{2}(ℤ/p^{n}) (i.e., s = 1). There is a commutative diagram of central extensions

\xymatrix{1 \ar[r] & 𝕌^{(n,p)} \ar[r] \ar[d]_{≀} & 𝕌 \ar[r] \ar[d]_{≀} & 𝕌^{[n,p]} \ar[r] \ar[d]_{≀} & 1 \\ 0 \ar[r] & p^{n−1}ℤ/p^{n}ℤ \ar[r] & ℤ/p^{n} \ar[r] & ℤ/p^{n−1} \ar[r] & 0.}

For a profinite group Ḡ and a homomorphism ρ̄: Ḡ → ℤ/p^{n−1}, Remark 7.3 therefore implies that α = Bock_{p^{n−1},Ḡ}(ρ̄). In particular, for x ∈ X take

ρ̄ = ρ̄_{ℤ/p^{n}}^{(x)} : Ḡ = S^{[n,p]} → 𝕌_{2}(ℤ/p^{n})^{[n,p]}.

Identifying 𝕌_{2}(ℤ/p^{n})^{[n,p]} = ℤ/p^{n−1}, we obtain

α_{(x),n} = Bock_{p^{n−1},S^{[n,p]}}(ϵ_{(x),ℤ/p^{n−1}}).

(2) Massey products. Let n = s ≥ 2, so 𝕌 = 𝕌_{n+1}(ℤ/p). Let Ḡ be a profinite group, let ρ̄: Ḡ → 𝕌^{[n,p]} be a continuous homomorphism, and let ρ_{i,i+1} : Ḡ → ℤ/p denote its projection on the (i, i + 1)-entry, i = 1, 2, …, n. By a result of Dwyer [Dwy75, Th. 2.6], the extension (7.4) then corresponds to a defining system for the n-fold Massey product

⟨ρ_{12}, ρ_{23}, …, ρ_{n,n+1}⟩ ⊆ H^{2}(Ḡ, ℤ/p)

(see [Efr14, Prop. 8.3] for the profinite analog of this fact). Thus, for a fixed Ḡ and for homomorphisms ρ̄_{1}, …, ρ̄_{n}: Ḡ → ℤ/p, with ρ̄ varying over all homomorphisms such that ρ̄_{i,i+1} = ρ̄_{i}, i = 1, 2, …, n, the cohomology element α ranges over the elements of the Massey product ⟨ρ̄_{1}, …, ρ̄_{n}⟩. In particular, for Ḡ = S^{[n,p]} and for a word w = (x_{1} ⋯ x_{n}) ∈ X^{∗} of length n, the cohomology elements α_{w,n} range over the Massey product ⟨ϵ_{(x_{1}),ℤ/p}, …, ϵ_{(x_{n}),ℤ/p}⟩ ⊆ H^{2}(S^{[n,p]}, ℤ/p), where the ϵ_{(x_{i}),ℤ/p} are viewed as elements of H^{1}(S^{[n,p]}, ℤ/p).

(3) Cup products. In the special case n = s = 2, the Massey product contains only the cup product. Hence for every profinite group Ḡ and a homomorphism ρ̄: Ḡ → 𝕌_{3}(ℤ/p) the cohomology element α ∈ H^{2}(Ḡ, ℤ/p) is the cup product ρ̄_{12} ∪ ρ̄_{23}. In particular, for w = (xy) we have

α_{(xy),ℤ/p} = ϵ_{(x),ℤ/p} ∪ ϵ_{(y),ℤ/p} ∈ H^{2}(S^{[2,p]}, ℤ/p).

8. Cohomological duality

Let S = S_{X} be again a free profinite group on the totally ordered set X, and let it act trivially on ℤ/p. Let n ≥ 2, so S^{(n,p)} ≤ S^{p}[S, S]. Then the inflation map H^{1}(S^{[n,p]}, ℤ/p) → H^{1}(S, ℤ/p) is an isomorphism. Further, H^{2}(S, ℤ/p) = 0. By the five-term sequence of cohomology groups [NSW08, Prop. 1.6.7], trg: H^{1}(S^{(n,p)}, ℤ/p)^{S} → H^{2}(S^{[n,p]}, ℤ/p) is an isomorphism.

There is a natural non-degenerate bilinear map

S^{(n,p)}/S^{(n+1,p)} × H^{1}(S^{(n,p)}, ℤ/p)^{S} → ℤ/p, (σ̄, φ) ↦ φ(σ)

(see [EM11, Cor. 2.2]). It induces a bilinear map

(·, ·)_{n}: S^{(n,p)} × H^{2}(S^{[n,p]}, ℤ/p) → ℤ/p, (σ, α)_{n} = −(trg^{−1}(α))(σ),

with left kernel S^{(n+1,p)} and trivial right kernel. Now let w ∈ X^{∗} be a word of length 1 ≤ s ≤ n. As in Examples 7.4, we apply the computations in Section 7 to the group 𝕌 = 𝕌_{s+1}(ℤ/p^{n−s+1}), the open normal subgroup N = 𝕌^{(n,p)}, the homomorphism ρ = ρ_{ℤ/p^{n−s+1}}^{w}: S → 𝕌, the induced homomorphism ρ̄ = ρ̄_{ℤ/p^{n−s+1}}^{w}: S^{[n,p]} → 𝕌^{[n,p]}, and the closed normal subgroup M = S^{(n,p)} of S. We write θ_{w,n}, α_{w,n} for θ, α, respectively.

Lemma 8.1. For σ ∈ S^{(n,p)} and a word w ∈ X^{∗} of length 1 ≤ s ≤ n one has

(σ, α_{w,n})_{n} = ι_{n,s}(ϵ_{w,ℤ/p^{n−s+1}}(σ)).

Proof. By (7.6) and (7.5),

(σ, α_{w,n})_{n} = θ_{w,n}(σ) = ι_{n,s}^{𝕌}(ρ_{ℤ/p^{n−s+1}}^{w}(σ)) = ι_{n,s}(ϵ_{w,ℤ/p^{n−s+1}}(σ)).  □

This and (6.1) give:

Corollary 8.2. Let w, w′ be words in X^{∗} of lengths 1 ≤ s, s′ ≤ n, respectively, with w Lyndon. Then

(τ_{w}^{p^{n−s}}, α_{w′,n})_{n} = ⟨w, w′⟩_{n}.

Proposition 6.4(a)(b) now gives:

Corollary 8.3. Let Lyn_{≤n}(X) be totally ordered by ⪯. The matrix

((τ_{w}^{p^{n−|w|}}, α_{w′,n})_{n}),

where w, w′ ∈ Lyn_{≤n}(X), is upper-triangular unipotent.

In general the above matrix need not be the identity matrix – see e.g., Proposition 11.2 below. Next we observe the following general fact:

Lemma 8.4. Let R be a commutative ring and let (·, ·): A × B → R be a nondegenerate bilinear map of R-modules. Let (L, ≤) be a finite totally ordered set, and for every w ∈ L let a_{w} ∈ A, b_{w} ∈ B. Suppose that the matrix ((a_{w}, b_{w′}))_{w,w′∈L} is invertible, and that a_{w}, w ∈ L, generate A. Then a_{w}, w ∈ L, is an R-linear basis of A, and b_{w}, w ∈ L, is an R-linear basis of B.

Proof. Let b ∈ B, and consider r_{w′} ∈ R, with w′ ∈ L. The assumptions imply that b = ∑_{w′} r_{w′}b_{w′} if and only if (a_{w}, b − ∑_{w′} r_{w′}b_{w′}) = 0 for every w. Equivalently, the r_{w′} solve the linear system ∑_{w′}(a_{w}, b_{w′})X_{w′} = (a_{w}, b), for w ∈ L. By the invertibility, the latter system has a unique solution. This shows that b_{w}, w ∈ L, is an R-linear basis of B. By reversing the roles of a_{w}, b_{w}, we conclude that the a_{w}, w ∈ L, form an R-linear basis of A. □

Theorem 8.5. (a) The cohomology elements α_{w,n}, where w ∈ Lyn_{≤n}(X),

form a ℤ/p-linear basis of H^{2}(S^{[n,p]}, ℤ/p).

(b) When X is finite, the cosets of the powers τ_{w}^{p^{n−s}}, w ∈ Lyn_{≤n}(X), form

a basis of the ℤ/p-module S^{(n,p)}/S^{(n+1,p)}.

Proof. When X is finite, the set Lyn_{≤n}(X) is also finite. By Theorem 5.3, the cosets a_{w} of τ_{w}^{p^{n−|w|}}, where w ∈ Lyn_{≤n}(X), generate the ℤ/pmodule S^{(n,p)}/S^{(n+1,p)}. We apply Lemma 8.4 with the ℤ/p-modules A = S^{(n,p)}/S^{(n+1,p)} and B = H^{2}(S^{[n,p]}, ℤ/p), the non-degenerate bilinear map A × B → ℤ/p induced by (·, ·)_{n}, the generators a_{w} of A, and the elements b_{w} = α_{w,n} of B. Corollary 8.3 implies that the matrix (a_{w}, b_{w′}) is invertible. Therefore Lemma 8.4 gives both assertions in the finite case. The general case of (a) follows from the finite case by a standard limit argument.

□

We call α_{w,n}, w ∈ Lyn_{≤n}(X), the Lyndon basis of H^{2}(S^{[n,p]}, ℤ/p). Recall that the number of relations in a minimal presentation of a pro-p group G is given by dim H^{2}(G, ℤ/p) [NSW08, Cor. 3.9.5]. In view of (2.2), Theorem 8.5 gives this number for G = S^{[n,p]}:

Corollary 8.6. One has

dim_{𝔽_{p}} H^{2}(S^{[n,p]}, ℤ/p) = dim_{𝔽_{p}}(S^{(n,p)}/S^{(n+1,p)}) = ∑_{s=1}^{n} φ_{s}(|X|),

where φ_{s} is the necklace map.

9. The shuffle relations

We recall the following constructions from [CFL58], [Reu93, pp. 134–135]. Let u_{1}, …, u_{t} ∈ X^{∗} be words of lengths s_{1}, …, s_{t}, respectively. We say that a word w ∈ X^{∗} of length 1 ≤ n ≤ s_{1} +⋯+s_{t} is an infiltration of u_{1}, …, u_{t}, if there exist sets I_{1}, …, I_{t} of respective cardinalities s_{1}, …, s_{t} such that {1, 2, …, n} = I_{1} ∪ ⋯ ∪ I_{t} and the restriction of w to the index set I_{j} is u_{j}, j = 1, 2, …, t. We then write w = w(I_{1}, …, I_{t}, u_{1}, …, u_{t}). We write Infil(u_{1}, …, u_{t}) for the set of all infiltrations of u_{1}, …, u_{t}. The infiltration product u_{1} ↓ ⋯ ↓ u_{t} of u_{1}, …, u_{t} is the polynomial ∑ w in ℤ⟨X⟩, where the sum is over all such infiltrations, taken with multiplicity. If in the above setting, the sets I_{1}, …, I_{t} are pairwise disjoint, then w(I_{1}, …, I_{t}, u_{1}, …, u_{t}) is called a shuffle of u_{1}, …, u_{t}. We write Sh(u_{1}, …, u_{t}) for the set of all shuffles of u_{1}, …, u_{t}. It consists of the words in Infil(u_{1}, …, u_{t}) of length s_{1} + ⋯ + s_{t}. The shuffle product u_{1}� ⋯ �u_{t} is the polynomial ∑ w(I_{1}, …, I_{t}, u_{1}, …, u_{t}) in ℤ⟨X⟩, where the sum is over all shuffles of u_{1}, …, u_{t}, taken with multiplicity. Thus u_{1}� ⋯ �u_{t} is the homogenous part of u_{1} ↓ ⋯ ↓ u_{t} of (maximal) degree s_{1} + ⋯ + s_{t}. For instance

\begin{aligned}(xy) ↓ (xz) & = (xyxz) + 2(xxyz) + 2(xxzy) + (xzxy) + (xyz) + (xzy), \\ (xy)�(xz) & = (xyxz) + 2(xxyz) + 2(xxzy) + (xzxy) \\ (x) ↓ (x) & = 2(xx) + (x), (x)�(x) = 2(xx). \end{aligned}

We may view infiltration and shuffle products also as elements of ℤ_{p}⟨X⟩. Let Shuffles(X) be the ℤ-submodule of ℤ⟨X⟩ generated by all shuffle products u�v, with ∅ ≠ u, v ∈ X^{∗}. Let Shuffles_{n}(X) be its homogenous component of degree n.

Examples 9.1. Shuffles_{1}(X) = {0},

Shuffles_{2}(X) = ⟨(xy) + (yx) | x, y ∈ X⟩,
Shuffles_{3}(X) = ⟨(xyz) + (xzy) + (zxy) | x, y, z ∈ X⟩.

Let (·, ·) be the pairing of (3.1) for the ring R = ℤ_{p}. As before, S = S_{X} is the free profinite group on the set X. The following fact is due to Chen, Fox, and Lyndon in discrete case [CFL58, Th. 3.6] (see also [Mor12, Prop. 8.6], [Reu93, Lemma 6.7]), as well as [Vog05, Prop. 2.25] in the profinite case.

Proposition 9.2. For every ∅ ≠ u, v ∈ X^{∗} and every σ ∈ S one has

ϵ_{u,ℤ_{p}}(σ)ϵ_{v,ℤ_{p}}(σ) = (Λ_{ℤ_{p}}(σ), u ↓ v).

Corollary 9.3. Let u, v be nonempty words in X^{∗} with s = |u| + |v| ≤ n. For every σ ∈ S^{(n,p)} one has (Λ_{ℤ_{p}}(σ), u�v) ∈ p^{n−s+1}ℤ_{p}.

Proof. If w is a nonempty word of length |w| < s, then by Proposition 4.1(b), ϵ_{w,ℤ_{p}}(σ) ∈ p^{n−|w|}ℤ_{p} ⊆ p^{n−s+1}ℤ_{p}. In particular, this is the case for w = u, w = v, and when w ∈ Infil(u, v) \ Sh(u, v). It follows from Proposition 9.2 that (Λ_{ℤ_{p}}(σ), u�v) ∈ p^{n−s+1}ℤ_{p}. □

We obtain the following shuffle relations (see also [Vog04, Cor. 1.2.10] and [FS84, Th. 6.8]). We write X^{s} for the set of words in X^{∗} of length s.

Theorem 9.4. For every ∅ ≠ u, v ∈ X^{∗} with s = |u| + |v| ≤ n one has

∑_{w∈X^{s}}(u�v)_{w}α_{w,n} = 0.

Proof. For σ ∈ S^{(n,p)}, Corollary 9.3 gives

∑_{w∈X^{s}}(u�v)_{w}ϵ_{w,ℤ_{p}}(σ) = ∑_{w∈X^{∗}}(u�v)_{w}ϵ_{w,ℤ_{p}}(σ) = (Λ_{ℤ_{p}}(σ), u�v) ∈ p^{n−s+1}ℤ_{p}.

Therefore, by Lemma 6.2,

(σ, ∑_{w∈X^{s}}(u�v)_{w}α_{w,n})_{n} = ∑_{w∈X^{s}}(u�v)_{w}(σ, α_{w,n})_{n}

= ∑_{w∈X^{s}}(u�v)_{w} ι_{n,s}(ϵ_{w,ℤ/p^{n−s+1}}(σ))

= ι_{n,s}( ∑_{w∈X^{s}}(u�v)_{w}ϵ_{w,ℤ/p^{n−s+1}}(σ)) = 0.

Now use the fact that (·, ·)_{n}: S^{(n,p)} × H^{2}(S^{[n,p]}, ℤ/p) → ℤ/p has a trivial right kernel. □

Corollary 9.5. There is a canonical epimorphism

\begin{matrix}⨁_{s=1}^{n}(( ⨁_{w∈X^{s}} ℤ)/ & Shuffles_{s}(X)) ⊗ (ℤ/p) → H^{2}(S^{[n,p]}, ℤ/p) \\ (r̄_{w})_{w} ↦ ∑_{w} r_{w}α_{w,n}. \end{matrix}

Proof. By Theorem 9.4 this homomorphism is well defined. By Theorem 8.5(a), it is surjective. □

Remark 9.6. In view of Lemma 6.2, the epimorphism of Corollary 9.5 and the canonical pairing (·, ·)_{n} induce a bilinear map

\begin{aligned}S^{(n,p)} & × ⨁_{s=1}^{n}(( ⨁_{w∈X^{s}} ℤ)/ Shuffles_{s}(X)) → ℤ/p, \\ & (σ, \overline{(r_{w})}_{w}) = ∑_{w} r_{w}ι_{n,s}(ϵ_{w,ℤ/p^{n−s+1}}(σ)) \end{aligned}

with left kernel S^{(n+1,p)}.

Example 9.7. We show that for every x_{1}, x_{2}, …, x_{k} ∈ X one has

(x_{1}x_{2} ⋯ x_{k}) + (−1)^{k−1}(x_{k} ⋯ x_{2}x_{1}) ∈ Shuffles_{n}(X).

We may assume that x_{1}, x_{2}, …, x_{k} are distinct. For 1 ≤ l ≤ k − 1 let u_{l} = (x_{l} ⋯ x_{2}x_{1}) and v_{l} = (x_{l+1} ⋯ x_{k}). We consider the polynomial ∑_{l=1}^{k−1}(−1)^{l−1}u_{l}�v_{l} in ℤ⟨X⟩. It is homogenous of degree k. If w ∈ Sh(u_{l}, v_{l}), then either:

(1) x_{l} appears before x_{l+1} in w, and then w appears with an opposite sign also in Sh(u_{l−1}, v_{l−1}); or

(2) x_{l+1} appears before x_{l} in w, and then w appears with opposite sign

also in Sh(u_{l+1}, v_{l+1}). The only exceptions are w = (x_{1}x_{2} ⋯ x_{k}) ∈ Sh(u_{1}, v_{1}) and w = (x_{k} ⋯ x_{2}x_{1}) ∈ Sh(u_{k−1}, v_{k−1}). This shows that

(x_{1}x_{2} ⋯ x_{k}) + (−1)^{k}(x_{k} ⋯ x_{2}x_{1}) = ∑_{l=1}^{k−1}(−1)^{l−1}u_{l}�v_{l}.

For 1 ≤ k ≤ n Corollary 9.5 therefore implies that

α_{(x_{1}x_{2}···x_{k}),n} = (−1)^{k−1}α_{(x_{k}···x_{2}x_{1}),n}.

10. Example: The case n = 2

Our results in this case are fairly well known, and are brought here in order to illustrate the general theory. As before let S = S_{X} with X totally ordered. Here S^{(2,p)} = S^{p}[S, S] and S̄ = S^{[2,p]} is the maximal elementary p-abelian quotient of S. We may identify H^{1}(S, ℤ/p) = H^{1}(S̄, ℤ/p) ≅ ⨁_{x∈X} ℤ/p. Let χ_{x,ℤ/p} = ϵ_{(x),ℤ/p}, x ∈ X, be the basis of H^{1}(S, ℤ/p) dual to X (see Remark 6.2). The Lyndon words w of length ≤ 2 are (x), where x ∈ X, and (xy), where x, y ∈ X and x < y. For these words we have τ_{(x)} = x and τ_{(xy)} = [x, y]. By Examples 7.4(1)(3),

α_{(x),2} = Bock_{p,S̄}(χ_{x,ℤ/p}), α_{(xy),2} = χ_{x,ℤ/p} ∪ χ_{y,ℤ/p}.

Hence, by Theorem 8.5, Bock_{p,S̄}(χ_{x,ℤ/p}) and χ_{x,ℤ/p} ∪ χ_{y,ℤ/p}, where x < y, form a ℤ/p-linear basis of H^{2}(S̄, ℤ/p). Furthermore, when X is finite, the elements of the form x^{p} and [x, y] with x, y ∈ X, x < y, form a ℤ/p-linear basis of S^{(2,p)}/S^{(3,p)}. In view of Examples 9.1 and Corollary 9.5, the map (r̄_{w}) ↦ ∑_{w} r_{w}α_{w,2} induces an epimorphism

⨁_{x∈X} ℤ/p ⊕ (( ⨁_{x,y∈X} ℤ)/⟨(xy) + (yx) | x, y ∈ X⟩) ⊗ (ℤ/p) → H^{2}(S̄, ℤ/p).

It coincides with the map

H^{1}(S̄, ℤ/p) ⊕ ⋀^{2} H^{1}(S̄, ℤ/p) → H^{2}(S̄, ℤ/p)

which is Bock_{p,S̄} on the first component and ∪ on the second component. When p ≠ 2 the direct sum is a free ℤ/p-module on Lyn_{≤2}(X), and by comparing dimensions we see that the epimorphism is in fact an isomorphism (compare [EM11, Cor. 2.9(a)]). However when p = 2 one has Bock_{2,S̄}(χ) = χ ∪ χ [EM11, Lemma 2.4], so the above epimorphism is not injective. Next, Proposition 6.4 shows that the matrix (⟨w, w′⟩_{2}), where w, w′ ∈ Lyn_{≤2}(X), is the identity matrix. In view of Corollary 8.2, it coincides with the matrix ((τ_{w}^{p^{2−|w|}}, α_{w′,2})_{2}). Thus

(x^{p}, Bock_{p,S̄}(χ_{x,ℤ/p}))_{2} = 1 for every x ∈ X,
(x^{p}, Bock_{p,S̄}(χ_{y,ℤ/p}))_{2} = 0 for every x, y ∈ X, x ≠ y,
(x^{p}, χ_{y,ℤ/p} ∪ χ_{z,ℤ/p})_{2} = 0 for every x, y, z ∈ X,
([x, y], Bock_{p,S̄}(χ_{z,ℤ/p}))_{2} = 0 for every x, y, z ∈ X,
([x, y], χ_{z,ℤ/p} ∪ χ_{t,ℤ/p})_{2} = 0 for every x, y, z, t ∈ X, (xy) ≠ (zt), (tz),
([x, y], χ_{x,ℤ/p} ∪ χ_{y,ℤ/p})_{2} = 1 for every x, y ∈ X with x < y.

This recovers well known facts from [Lab66, §2.3], [Koc02, §7.8] and [NSW08, Th. 3.9.13 and Prop. 3.9.14]

11. Example: The case n = 3.

Here S^{(3,p)} = S^{p^{2}}[S, S]^{p}[S, [S, S]]. We abbreviate S̄ = S^{[3,p]}. Recall that Lyn_{≤3}(X) consists of the words

(x) for x ∈ X,
(xy), (xxy), (xyy) for x, y ∈ X with x < y,
(xyz), (xzy) for x, y, z ∈ X with x < y < z.

For these words

\begin{aligned}τ_{(x)} = x, & τ_{(xy)} = [x, y], τ_{(xxy)} = [x, [x, y]], τ_{(xyy)} = [[x, y], y], \\ & τ_{(xyz)} = [x, [y, z]], τ_{(xzy)} = [[x, z], y]. \end{aligned}

By Theorem 5.3, the cosets of

x^{p^{3}}, [x, y]^{p}, [x, [x, y]], [[x, y], y], [x, [y, z]], [[x, z], y],

with x, y, z as above, generate S^{(3,p)}/S^{(4,p)}. When X is finite, they form a linear basis of S^{(3,p)}/S^{(4,p)} over ℤ/p (Theorem 8.5(b)). Furthermore, Theorem 8.5(a) gives:

Theorem 11.1. The following cohomology elements form a ℤ/p-linear basis of H^{2}(S̄, ℤ/p):

α_{(x),3}, α_{(xy),3}, α_{(xxy),3}, α_{(xyy),3}, α_{(xyz),3}, α_{(xzy),3},

where x, y, z ∈ X and we assume that x < y < z.

By Examples 7.4, α_{(x),3} = Bock_{p^{2},S̄}(χ_{x,ℤ/p^{2}}), and for every x, y, z ∈ X, α_{(xyz),3} belongs to the triple Massey product ⟨χ_{x,ℤ/p}, χ_{y,ℤ/p}, χ_{z,ℤ/p}⟩ ⊆ H^{2}(S̄, ℤ/p). We further recall that α_{(xy),3} is the pullback to H^{2}(S̄, ℤ/p) under ρ̄_{ℤ/p^{2}}^{(xy)} : S̄ → 𝕌_{3}(ℤ/p^{2})^{[3,p]} of the cohomology element in H^{2}(𝕌_{3}(ℤ/p^{2})^{[3,p]}, ℤ/p) corresponding to the central extension

0 → ℤ/p → 𝕌_{3}(ℤ/p^{2}) → 𝕌_{3}(ℤ/p^{2})^{[3,p]} → 1.

Alternatively, it has the following explicit description: By Proposition 6.3(a),

𝕌_{3}(ℤ/p^{2})^{(3,p)} = I_{3} + ℤpE_{13}, and let ι = ι_{3,2}^{𝕌}: 𝕌_{3}(ℤ/p^{2})^{(3,p)} ⟶^{∼} ℤ/p be

the natural isomorphism. By (7.5) and (7.6), α_{(xy),3} = − trg(θ), where

θ = ι ∘ (ρ_{ℤ/p^{2}}^{(xy)}|_{S^{(3,p)}}) and trg: H^{1}(S^{(3,p)}, ℤ/p)^{S} ⟶^{∼} H^{2}(S̄, ℤ/p) is the trans-

gression isomorphism. Next we compute the matrix ((τ_{w}^{p^{3−|w|}}, α_{w′,3})_{3}) = (⟨w, w′⟩_{3}), where w, w′ ∈ Lyn_{≤3}(X):

Proposition 11.2. For w, w′ ∈ Lyn_{≤3}(X) one has

⟨w, w′⟩_{3} = \begin{cases}1, & if w = w′; \\ −1, & if w = (xyz), w′ = (xzy) for some x, y, z ∈ X, x < y < z; \\ 0, & otherwise. \end{cases}

Proof. In view of Proposition 6.4, it is enough to show the assertion when w ≠ w′, either |w| = |w′| or 2|w| ≤ |w′|, and the letters of w′ appear in w. Furthermore, when |w| = |w′| we may assume that w ≤_{alp} w′. Thus when w has one of the forms (x), (xy), (xyy), (xzy) (where x < y < z) there is nothing more to show. When w = (xxy) with x < y we need to check only the word w′ = (xyy). Then Lemma 4.2 gives

⟨(xxy), (xyy)⟩_{3} = ϵ_{(xyy),ℤ/p}([x, [x, y]])

= ϵ_{(x),ℤ/p}(x) · ϵ_{(yy),ℤ/p}([x, y]) − ϵ_{(y),ℤ/p}(x) · ϵ_{(xy),ℤ/p}([x, y])

= 1 · 0 − 0 · 1 = 0.

When w = (xyz) with x < y < z we need to check only the word w′ = (xzy). Then Lemma 4.2 gives

⟨(xyz), (xzy)⟩_{3} = ϵ_{(xzy),ℤ/p}([x, [y, z]])

= ϵ_{(x),ℤ/p}(x) · ϵ_{(zy),ℤ/p}([y, z]) − ϵ_{(y),ℤ/p}(x) · ϵ_{(xz),ℤ/p}([y, z])

= 1 · (−1) − 0 · 0 = −1.

This completes the verification in all cases. □ In view of Examples 9.1, Corollary 9.5 gives rise to an epimorphism

\begin{aligned}⨁_{x∈X} ℤ/p & ⊕ (( ⨁_{x,y∈X} ℤ(xy))/⟨(xy) + (yx) | x, y ∈ X⟩) ⊗ (ℤ/p) \\ & ⊕ (( ⨁_{x,y,z∈X} ℤ(xyz))/⟨(xyz) + (xzy) + (zxy) | x, y, z ∈ X⟩) ⊗ (ℤ/p) \\ & → H^{2}(S̄, ℤ/p). \end{aligned} (11.1)

Moreover, for x, y, z ∈ X, x < y < z, we have (yx) = (x)�(y) − (xy) 2(xx) = (x)�(x) (xyx) = (x)�(xy) − 2(xxy) (yxx) = (x)�(yx) − (xx)�(y) + (xxy) (yxy) = (xy)�(y) − 2(xyy) (yyx) = (yy)�(x) − (y)�(xy) + (xyy) (yxz) = (y)�(xz) − (xyz) − (xzy) (zxy) = (z)�(xy) − (xzy) − (xyz) (yzx) = (zx)�(y) − (x)�(zy) + (xzy) (zyx) = (yx)�(z) − (x)�(yz) + (xyz) 3(xxx) = (x)�(xx).

These congruences and Example 2.1 imply that

∑_{w∈X^{2}} ℤw ≡ ∑_{w∈Lyn_{2}(X)} ℤw + 2-torsion (mod Shuffles_{2}(X)),
∑_{w∈X^{3}} ℤw ≡ ∑_{w∈Lyn_{3}(X)} ℤw + 3-torsion (mod Shuffles_{3}(X)).

Therefore, for p > 3, the direct sum in (11.1) is the free ℤ/p-module on the basis Lyn_{≤3}(X). Thus the epimorphism (11.1) maps the ℤ/p-linear basis 1w, w ∈ Lyn_{≤3}(X), bijectively onto the ℤ/p-linear basis α_{w,3}, w ∈ Lyn_{≤3}(X) (see Theorem 8.5). Consequently we have:

Theorem 11.3. For n = 3 and p > 3, (11.1) is an isomorphism. Thus all relations in H^{2}(S^{[3,p]}, ℤ/p) are consequences of the shuffle relations of Theorem 9.4.

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Ido Efrat

Department of Mathematics

Ben-Gurion University of the Negev P.O. Box 653 Be’er-Sheva 84105 Israel efrat@math.bgu.ac.il 998

Documenta Mathematica 22 (2017)