A differential in the Lyndon-Hochschild-Serre spectral sequence by I. J. Leary* School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS

Abstract. We consider the Lyndon-Hochschild-Serre spectral sequence with F_{p} coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence d_{2} and d_{3} are known, and we give a description for d_{4}. Using this result we deduce a similar formula for the Serre spectral sequence for a fibration with fibre BC_{p^{n}}. The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in the cohomology of a finite abelian group. As an example we determine the Poincaré series for the mod-3 cohomology of various 3-groups.

Introduction. Let G be a discrete group, let C be a cyclic group of order p^{m} for some prime p and m ≥ 1, and let E be a central extension with kernel C and quotient G. Such an extension is described by its extension class, which may be any element of H^{2}(G; C). Since C is central, the E_{2}-page of the Lyndon-Hochschild-Serre spectral sequence with F_{p} coefficients for this extension is isomorphic to the graded tensor product of H^{∗}(G) and H^{∗}(C). (Note that throughout this paper H^{∗}(.) will denote mod-p cohomology.) There is an isomorphism

H^{∗}(C) ≅ F_{p}[u, t]/I,

where u has degree one, t has degree two, and I is generated by u^{2} except when p^{m} = 2 in which case I is generated by u^{2} − t. In particular this implies that each row of the E_{2}-page of the spectral sequence is isomorphic to a single copy of H^{∗}(G).

The first two differentials in the spectral sequence have been known since the calculations of the cohomology of Eilenberg-MacLane spaces by Cartan and Serre [2], [9]. Modulo choice of various isomorphisms, d_{2}(u) is the mod-p reduction of the extension class, d_{2}(t) is zero, and d_{3}(t) is the image of the extension class under the Bockstein for the following coefficient sequence.

0 → F_{p} → Z/p^{m+1} → Z/p^{m} ≅ C → 0

We shall describe the next differential d_{4}, in Theorem 3, but first we require two lemmata concerning the spectral sequence. In Corollary 4 we deduce a similar result for the Serre spectral sequence for a fibration with fibre BC_{p^{m}}.

* supported by SERC post-doctoral fellowship number B90 RFH 8960 The spectral sequence. We set up the Lyndon-Hochschild-Serre spectral sequence as described in Evens’ book [5]. Thus we let P_{∗} be the standard (or bar) resolution for F_{p} over F_{p}G, and Q_{∗} be the standard resolution for F_{p} over F_{p}E. Now we define

E_{0}^{i,j} = Hom_{E}(P_{i} ⊗ Q_{j}, F_{p}),

where E acts diagonally on P_{∗} ⊗ Q_{∗}, d_{1} is the map induced by the differential on P_{∗}, and d_{0} is the map induced by the differential on Q_{∗} except that a sign must be introduced (in fact on E_{0}^{i,j} d_{0} is the adjoint of (−1)^{i}1 ⊗ d^{Q}). Using the Kunneth theorem it may be checked that the singly graded total complex (P_{∗} ⊗ Q_{∗})_{∗} with differential d_{0} + d_{1} is an E-free resolution for F_{p}.

To introduce products into the spectral sequence we need a diagonal approximation for (P_{∗} ⊗ Q_{∗})_{∗}, that is a chain map from (P_{∗} ⊗ Q_{∗})_{∗} to (P_{∗} ⊗ Q_{∗} ⊗ P_{∗} ⊗ Q_{∗})_{∗} inducing the identity map on homology. For any two chain complexes A_{∗} and B_{∗}, let τ denote the chain isomorphism from A_{∗} ⊗ B_{∗} to B_{∗} ⊗ A_{∗} given by τ(a ⊗ b) = (−1)^{|a||b|}. If Δ_{0}^{P} and Δ_{0}^{Q} are diagonal approximations for P_{∗} and Q_{∗} respectively, then (1 ⊗ τ ⊗ 1)(Δ_{0}^{P} ⊗ Δ_{0}^{Q}) is a diagonal approximation for P_{∗} ⊗ Q_{∗}, which is coassociative provided that Δ_{0}^{P} and Δ_{0}^{Q} are. Since it is possible to choose Δ_{0}^{P} and Δ_{0}^{Q} to be coassociative using the Alexander-Whitney formula (given on the usual F_{p} basis of P_{n} by

Δ_{0}^{P} (g_{0}, …, g_{n}) = ∑_{i=0}^{n}(g_{0}, …, g_{i}) ⊗ (g_{i}, …, g_{n})

and similarly for Δ_{0}^{Q}) we do so. We thus obtain an associative product (which we shall call the cup product) on the E_{0}-page of the spectral sequence given by

ϕ ⌣ θ(x) = μ(ϕ ⊗ θ)(1 ⊗ τ ⊗ 1)(Δ_{0}^{P} ⊗ Δ_{0}^{Q})(x)

for all ϕ and θ in E_{0} and all x ∈ P_{∗} ⊗ Q_{∗}, where μ is the product map from F_{p} ⊗ F_{p} to F_{p}. The only reason why we insisted that P_{∗} and Q_{∗} should be the standard resolution was to ensure that E_{0} could be endowed with an associative product. The differentials d_{0} and d_{1} are derivations for this product.

We also need to introduce analogues within the spectral sequence of the cup-1 product [10]. The map τΔ_{0}^{P} is also a diagonal approximation for P_{∗}, so is chain homotopic to Δ_{0}^{P}. Let Δ_{1}^{P} be a chain homotopy from Δ_{0}^{P} to τΔ_{0}^{P}. For example, Δ_{1}^{P} could be the map given by Steenrod [10]

Δ_{1}^{P} (g_{0}, …, g_{n}) = ∑_{0≤i<j≤n}(−1)^{f(i,j,n)}(g_{0}, …, g_{i}, g_{j}, …, g_{n}) ⊗ (g_{i}, …, g_{j}),

where f(i, j, n) = n + (n − i − 1)(j − i − 1). We define Δ_{1}^{Q} similarly. We now consider three different diagonal approximations for P_{∗} ⊗ Q_{∗}, which are defined as follows.

(1 ⊗ τ ⊗ 1)(Δ_{0}^{P} ⊗ Δ_{0}^{Q})  (1 ⊗ τ ⊗ 1)(τ ⊗ 1 ⊗ 1)(Δ_{0}^{P} ⊗ Δ_{0}^{Q})  (1 ⊗ τ ⊗ 1)(τ ⊗ τ)(Δ_{0}^{P} ⊗ Δ_{0}^{Q})

As above, each of these induces a product on the E_{0}-page of the spectral sequence. By definition the first induces the cup product, and it is easily checked that the third induces the product

(ϕ, θ) ↦ (−1)^{|ϕ||θ|}θ ⌣ ϕ,

where |.| refers to total degree. The second induces another product which we shall refer to as

(ϕ, θ) ↦ ϕ∧θ.

It is easily checked that (1 ⊗ τ ⊗ 1)(Δ_{1}^{P} ⊗ Δ_{0}^{Q}) is a chain homotopy between the first two diagonal approximations, and that the map defined to be (−1)^{i}(1 ⊗ τ ⊗ 1)(τ ⊗ 1 ⊗ 1)(Δ_{0}^{P} ⊗ Δ_{1}^{Q}) on P_{i} ⊗ Q_{j} is a chain homotopy between the second two. We may use these chain homotopies to define cup-1 products of bidegrees (−1, 0) and (0, −1) respectively, whose properties are summarised in the following lemma.

Lemma 1. For ϕ, θ ∈ E_{0}, and x ∈ P_{i} ⊗ Q_{j}, define products of ϕ and θ as below.

\begin{aligned}ϕ⌣_{1,0}θ(x) & = μ(ϕ ⊗ θ)(1 ⊗ τ ⊗ 1)(Δ_{1}^{P} ⊗ Δ_{0}^{Q})(x) \\ ϕ⌣_{0,1}θ(x) & = (−1)^{i}μ(ϕ ⊗ θ)(1 ⊗ τ ⊗ 1)(τ ⊗ 1 ⊗ 1)(Δ_{0}^{P} ⊗ Δ_{1}^{Q})(x) \end{aligned}

Then ⌣_{1,0} and ⌣_{0,1} satisfy the following coboundary formulae.

\begin{aligned}d_{0}(ϕ⌣_{1,0}θ) & = −(d_{0}ϕ)⌣_{1,0}θ − (−1)^{|ϕ|}ϕ⌣_{1,0}(d_{0}θ) \\ d_{1}(ϕ⌣_{1,0}θ) & = −(d_{1}ϕ)⌣_{1,0}θ − (−1)^{|ϕ|}ϕ⌣_{1,0}(d_{1}θ) + ϕ ⌣ θ − ϕ∧θ \\ d_{0}(ϕ⌣_{0,1}θ) & = −(d_{0}ϕ)⌣_{0,1}θ − (−1)^{|ϕ|}ϕ⌣_{0,1}(d_{0}θ) + ϕ∧θ − (−1)^{|ϕ||θ|}θ ⌣ ϕ \\ d_{1}(ϕ⌣_{0,1}θ) & = −(d_{1}ϕ)⌣_{0,1}θ − (−1)^{|ϕ|}ϕ⌣_{0,1}(d_{1}θ). \end{aligned}

Proof. Left as an exercise for the reader. QED Differentials.

In any spectral sequence, if ϕ ∈ E_{0}^{i,j} is to survive until the E_{2}-page, d_{0}(ϕ) must be zero, and also d_{1}(ϕ) must represent zero on the E_{1}-page, or equivalently there must be ϕ_{1} ∈ E_{0}^{i+1,j−1} such that d_{0}(ϕ_{1}) = d_{1}(ϕ). In this case d_{2}(ϕ) is represented by d_{1}(ϕ_{1}). Similarly, ϕ can only survive until the E_{3}-page if there is ϕ_{2} ∈ E_{0}^{i+2,j−2} such that d_{1}(ϕ_{1}) = d_{0}(ϕ_{2}), in which case d_{3}(ϕ) is represented by d_{1}(ϕ_{2}). In this way the higher differentials in the spectral sequence may be computed.

Returning to the spectral sequence we are studying, let u and t be cochains in E_{0}^{0,1} and E_{0}^{0,2} respectively, representing the elements of E_{2}^{0,∗} with the same names (as used in the introduction). The known description of d_{2} implies that d_{0}(u) = 0, and there is an element θ in E_{0}^{1,0} such that d_{1}(u) = d_{0}(θ), and d_{1}(θ) = ξ is a cocycle representing d_{2}(u). Similarly, there are cochains η_{1} ∈ E_{0}^{1,1} and η_{2} ∈ E_{0}^{2,0} such that d_{0}(t) = 0, d_{1}(t) = d_{0}(η_{1}), d_{1}(η_{1}) = d_{0}(η_{2}), and d_{1}(η_{2}) = ξ′, where ξ′ is a cocycle representing d_{3}(t). Note that ξ and ξ′ are universal cycles. The situation may be summarised in the following diagram.

0  0
↑  ↑
u → •  t → •
↑  ↑
θ → ξ  η_{1} → •
↑
η_{2} → ξ′

From now on we fix this notation, so that u, t, θ, ξ, η_{1}, η_{2} and ξ′ will always refer to these cochains. Recall from the introduction that the classes represented by ξ and ξ′ in H^{∗}(G) may be easily described in terms of the extension class of the group E in H^{2}(G; C). We are now able to state our second lemma, which describes a sequence of cochains related to t^{n}.

Lemma 2. For all n ≥ 1 there exist cochains η_{1}(n), …, η_{4}(n), with η_{i}(n) ∈ E_{0}^{i,2n−i} such that

\begin{aligned}d_{1}(t^{n}) & = d_{0}(η_{1}(n)) \\ d_{1}(η_{2}(n)) & = nt^{n−1}ξ′ + d_{0}(η_{3}(n)) \end{aligned}  \begin{aligned}d_{1}(η_{1}(n)) & = d_{0}(η_{2}(n)) \\ d_{1}(η_{3}(n)) & = nη_{1}(n − 1)ξ′ + d_{0}(η_{4}(n)). \end{aligned}

Remark. The existence of η_{1}(n), η_{2}(n) and η_{3}(n) and the relations between them follow immediately from the fact that d_{3}(t^{n}) is represented by nt^{n−1}ξ′. Thus the content of this lemma is the existence of η_{4}(n) and the equation for d_{1}(η_{3}(n)).

Proof. Define η_{i}(0) = η_{3}(1) = η_{4}(1) = 0, η_{1}(1) = η_{1}, and η_{2}(1) = η_{2}, where η_{1} and η_{2} are as above. Then define η_{i}(n) inductively by the following formulae.

η_{1}(n) =η_{1}(n − 1)t + t^{n−1}η_{1}
η_{2}(n) =η_{2}(n − 1)t + η_{1}(n − 1)η_{1} + t^{n−1}η_{2} − (n − 1)t^{n−2}(ξ′⌣_{1,0}t)
η_{3}(n) =η_{3}(n − 1)t + η_{2}(n − 1)η_{1} + η_{1}(n − 1)η_{2}
− (n − 1)(η_{1}(n − 2)(ξ′⌣_{1,0}t) + t^{n−2}(ξ′⌣_{1,0}η_{1})) + (n − 1)t^{n−2}(η′⌣_{0,1}t)
η_{4}(n) =η_{4}(n − 1)t + η_{3}(n − 1)η_{1} + η_{2}(n − 1)η_{2}
− (n − 1)(η_{2}(n − 2)(ξ′⌣_{1,0}t) + η_{1}(n − 2)(ξ′⌣_{1,0}η_{1}) + t^{n−2}(ξ′⌣_{1,0}η_{2}))
+ (n − 1)(η_{1}(n − 2)(ξ′⌣_{0,1}t) + t^{n−2}(ξ′⌣_{0,1}η_{1}))

The equations given in the statement hold for n = 1, and may be verified for all n by induction. QED

We are now ready to consider d_{4}. First we decide which elements of E_{2} will survive until E_{4}. Let χ be an element of H^{∗}(G). Since d_{2}(t^{n}uχ) = t^{n}ξχ we deduce that t^{n}uχ survives until E_{3} iff ξχ = 0 in H^{∗}(G). In this case d_{3}(t^{n}uχ) = −nt^{n−1}uξ′χ, and since d_{2} had trivial image in E_{2}^{∗,2n−1} it follows that t^{n}uχ survives until E_{4} iff ξχ = 0 in H^{∗}(G) and either p divides n or ξ′χ = 0 in H^{∗}(G). If p divides n then since uχ is a universal cycle and t^{p} survives until E_{2p+1} (by the Serre transgression theorem) then d_{4}(t^{n}uχ) = 0. Thus we only need determine d_{4}(t^{n}uχ) for χ such that χξ and χξ′ are zero. Similarly, for t^{n}χ to survive, we must have that d_{3}(t^{n}χ) represents zero in E_{3} , which will happen if either p divides n (in which case d_{4}(t^{n}χ) = 0) or if ξ′χ = ξχ′ in H^{∗}(G) for some χ′.

Theorem 3. In the Lyndon-Hochschild-Serre spectral sequence with F_{p} coefficients for a central extension of a cyclic p-group C by a group G, let u generate E_{2}^{0,1}, let t generate E_{2}^{0,2}, and let d_{2}(u) = ξ, d_{3}(t) = ξ′.

a) If χ ∈ H^{∗}(G) is such that ξχ = 0 and ξ′χ = 0 in H^{∗}(G), then for all n, t^{n}uχ survives until E_{4}, and

d_{4}(t^{n}uχ) = nt^{n−1}⟨ξ′, χ, ξ⟩,

where ⟨ξ′, χ, ξ⟩ is the Massey product (see next section for a description of the Massey product).

b) If χ ∈ H^{∗}(G) is such that ξ′χ = ξχ′ for some ξ′, then for all n t^{n}χ survives until E_{4}, and

d_{4}(t^{n}χ) = n(n − 1)t^{n−2}uξ′χ′.

Remark. The ‘generic’ case of statement b) above for C of order p was conjectured by P. H. Kropholler. He made this conjecture by looking for a natural map between the relevant subquotients of H^{∗}(G). The author thanks P. H. Kropholler for showing him this conjecture, which inspired the work contained in this paper. Proof. In either case let χ be an element of E_{0}^{∗,0} yielding the element of the same name in E_{2}^{∗,0}, and in case b) define χ′ similarly. The proof now splits into two cases. The conditions in case a) are equivalent to the existence of ψ and ψ′ in E_{0}^{∗,0} such that d_{0}(ψ) = 0, d_{0}(ψ′) = 0, d_{1}(ψ) = χξ, and d_{1}(ψ′) = ξ′χ. Define cochains ϕ_{0}, …, ϕ_{4} by the equations

\begin{aligned}ϕ_{0} & = t^{n}χu, \\ ϕ_{1} & = η_{1}(n)χu + t^{n}(χθ − (−1)^{|χ|}ψ), \\ ϕ_{2} & = η_{2}(n)χu + η_{1}(n)(χθ − (−1)^{|χ|}ψ) − nt^{n−1}ψ′u, \\ ϕ_{3} & = η_{3}(n)χu + η_{2}(n)(χθ − (−1)^{|χ|}ψ) − nη_{1}(n − 1)ψ′u − nt^{n−1}ψ′θ, \\ ϕ_{4} & = η_{4}(n)χu + η_{3}(n)(χθ − (−1)^{|χ|}ψ) − nη_{2}(n − 1)ψ′u − nη_{1}(n − 1)ψ′θ, \end{aligned}

where η_{i}(j) and θ are as described during and just before Lemma 2. The following equations may be seen to hold by applying Lemma 2.

d_{0}(ϕ_{0}) = 0, d_{1}(ϕ_{i}) = d_{0}(ϕ_{i−1}) for i = 0, 1, 2,
d_{1}(ϕ_{3}) = d_{0}(ϕ_{4}) − (−1)^{|χ|}nt^{n−1}(ξ′ψ + ψ′ξ).

Now note that in E_{2}^{∗,∗} ϕ_{0} represents (−1)^{|χ|}t^{n}uχ, and that −(ξ′ψ + ψ′ξ) represents the Massey product ⟨ξ′, χ, ξ⟩.

The proof for case b) is similar. Here the hypotheses give ψ ∈ E_{0}^{∗,0} such that d_{0}(ψ) = 0 and d_{1}(ψ) = ξ′χ − ξχ′. Now define cochains ϕ_{0}, …, ϕ_{4} by the following equations.

\begin{aligned}ϕ_{0} & = t^{n}χ, \\ ϕ_{1} & = η_{1}(n)χ − nt^{n−1}uχ′, \\ ϕ_{2} & = η_{2}(n)χ − nη_{1}(n − 1)uχ′ − nt^{n−1}(θχ′ + ψ), \\ ϕ_{3} & = η_{3}(n)χ − nη_{2}(n − 1)uχ′ − nη_{1}(n − 1)(θχ′ + ψ), \\ ϕ_{4} & = η_{4}(n)χ − nη_{3}(n − 1)uχ′ − nη_{2}(n − 1)(θχ′ + ψ). \end{aligned}

Usng Lemma 2 the following equations may be verified.

d_{0}(ϕ_{0}) = 0, d_{1}(ϕ_{i}) = d_{0}(ϕ_{i+1}) for i = 0, 1, 2,
d_{1}(ϕ_{3}) = d_{0}(ϕ_{4}) − n(n − 1)t^{n−2}ξ′uχ′.
QED

Corollary 4. Let C be a cyclic p-group. Then in the Serre spectral sequence with F_{p} coefficients for a fibration with fibre BC and path connected base space d_{4} may be described as in Theorem 3.

Proof. For any path connected X, Kan-Thurston [7] exhibit an aspherical space T X (in other words a space having π_{i}(X) = 0 unless i = 1) and a map from T X to X inducing an isomorphism on cohomology for any local coefficients on X. If we let X be the base space of the BC bundle, it follows that the map from T X to X induces an isomorphism between the Serre spectral sequence for the original bundle and that for the induced bundle over T X. It follows from the homotopy long exact sequence that the total space of the induced bundle is also aspherical, and so the Serre spectral sequence for this bundle is isomorphic to the Lyndon-Hochschild-Serre spectral sequence for the extension given by taking fundamental groups. QED Remark. The author has not tried to find a direct proof of Corollary 4.

Massey Products. We recall first the definition of the Massey (triple) product as in [1]. Let P_{∗} be any projective F_{p}G resolution for F_{p}, let Δ be a diagonal approximation for P_{∗}, and let H be a chain homotopy between (Δ⊗1)Δ and (1⊗Δ)Δ, that is a map from P_{∗} to (P_{∗}⊗P_{∗}⊗P_{∗})_{∗} satisfying

dH + Hd = (Δ ⊗ 1)Δ − (1 ⊗ Δ)Δ.

(Of course such a homotopy exists because (Δ⊗1)Δ and (1⊗Δ)Δ are chain maps between projective resolutions inducing the same map on homology.) If ϕ_{1}, ϕ_{2} and ϕ_{3} are cochains from P_{∗} to F_{p}, we may define a triple product h(ϕ_{1}, ϕ_{2}, ϕ_{3}) by the formula

h(ϕ_{1}, ϕ_{2}, ϕ_{3})(x) = μ′(ϕ_{1} ⊗ ϕ_{2} ⊗ ϕ_{3})H(x),

where μ′ is the usual map from F_{p} ⊗ F_{p} ⊗ F_{p} to F_{p}. If ϕ_{1}, ϕ_{2} and ϕ_{3} are cocycles then it is easily checked that dh(ϕ_{1}, ϕ_{2}, ϕ_{3}) is (ϕ_{1}ϕ_{2})ϕ_{3} − ϕ_{1}(ϕ_{2}ϕ_{3}). If also ϕ_{1}ϕ_{2} and ϕ_{2}ϕ_{3} are coboundaries, with dψ_{1} = ϕ_{1}ϕ_{2} and dψ_{2} = ϕ_{2}ϕ_{3}, then it may be verified that the following cochain is a cocycle.

⟨ϕ_{1}, ϕ_{2}, ϕ_{3}⟩ = (−1)^{|ϕ_{1}|}ϕ_{1}ψ_{2} − ψ_{1}ϕ_{3} + h(ϕ_{1}, ϕ_{2}, ϕ_{3})

For any such ϕ_{1}, ϕ_{2} and ϕ_{3} we define the Massey product of the cohomology classes that they represent to be the cohomology class of the above cocycle. If Δ is strictly coassociative then H and hence h(, ,) may be chosen to be zero. This is the case in which Massey products are usually discussed, especially in topology, and is the case used in the previous two sections of this paper. From this case it is easy to see that varying the ϕ_{i} by a coboundary or the ψ_{i} by a cocycle changes the Massey product by an element of the ideal of H^{∗}(G) generated by ϕ_{1} and ϕ_{3}, so that the Massey product is well-defined modulo this ideal. Benson and Evens suggest another extreme case of interest [1], the case when G is finite and P_{∗} is a minimal resolution. In this case the differential on the mod-p cochains on P_{∗} is trivial [5], so ψ_{1} and ψ_{2} may be chosen to be zero, and the Massey product is determined by h. We shall use this case below to determine Massey products in the mod-p cohomology of finite abelian groups. First we discuss cyclic groups.

For G a cyclic group of order n with generator g, Cartan-Eilenberg describe a projective F_{p}G resolution P_{∗} for F_{p} [3] in which P_{i} is free on one generator e_{i} with boundary map

d(e_{i}) = \{ \begin{matrix}(g − 1)e_{i−1} & for i odd \\ ∑_{j=0}^{n−1} g^{j}e_{i−1} & for i even. \end{matrix}

Cartan-Eilenberg also give a diagonal approximation for this resolution whose composite with the projection from P_{∗} ⊗ P_{∗} to P_{a} ⊗ P_{b} is given by

Δ(e_{a+b}) = \{ \begin{matrix}e_{a} ⊗ e_{b} & for a even, \\ e_{a} ⊗ ge_{b} & for a odd, b even, \\ ∑_{0≤i<j<n} g^{i}e_{a} ⊗ g^{j}e_{b} & for a and b odd. \end{matrix}

If n is a power of p then this resolution is minimal. We now describe a map H as above for this resolution.

Proposition 5. Let P_{∗} with diagonal approximation Δ be the Cartan-Eilenberg resolution for a cyclic group G of order n generated by g as above. Define a map H of degree one from P_{∗} to (P_{∗} ⊗ P_{∗} ⊗ P_{∗})_{∗} by defining its composite with projection to P_{a} ⊗ P_{b} ⊗ P_{c} to be

H(e_{a+b+c−1}) = \{ \begin{matrix}∑_{0≤i<j<k<n} g^{i}e_{a} ⊗ g^{j}e_{b} ⊗ g^{k}e_{c} & if a, b and c are all odd, \\ 0 & otherwise. \end{matrix}

Then H satisfies the formula dH + Hd = (Δ ⊗ 1)Δ − (1 ⊗ Δ)Δ.

Proof. Check that the maps on each side of the above equation when composed with projection to P_{a} ⊗ P_{b} ⊗ P_{c} give the following map.

e_{a+b+c} ↦ \{ \begin{matrix}∑_{0≤i<j<n}(g^{i}e_{a} ⊗ g^{j}e_{b} ⊗ e_{c} − e_{a} ⊗ g^{i+1}e_{b} ⊗ g^{j+1}e_{c}) & for a, b and c all odd, \\ ∑_{0≤i<j<n} g^{i}e_{a} ⊗ g^{j}e_{b} ⊗ (1 − g^{j+1})e_{c} & for a and b odd, \\ ∑_{0≤i<j<n} g^{i}e_{a} ⊗ (g^{i+1} − g^{j})e_{b} ⊗ g^{j}e_{c} & for a and c odd, \\ ∑_{0≤i<j<n}(g^{i} − 1)e_{a} ⊗ g^{i}e_{b} ⊗ g^{j}e_{c} & for b and c odd, \\ 0 & otherwise. \end{matrix}

QED

Remark. If the group G has order two then Δ is strictly coassociative, and the empty sum occuring in the definition of H may be regarded as zero.

Corollary 6. Let G be a cyclic group of order p^{m}, where p^{m} > 2, and let u and t be generators for H^{∗}(G) as in the introduction, where if m = 1 then t is the Bockstein of u. The only Massey products that are defined in H^{∗}(G) are ⟨t^{i}u, t^{j}u, t^{k}u⟩, which are defined modulo zero and take the following value.

⟨t^{i}u, t^{j}u, t^{k}u⟩ = \{ \begin{matrix}t^{i+j+k+1} & for p^{m} = 3, \\ 0 & otherwise. \end{matrix}

Proof. If p^{m} is greater than 3 then \binom{p^{m}}{3} is divisible by p, and the result follows easily from Proposition 5. If p^{m} = 3 the argument requires slightly more care (and an explicit definition of the Bockstein map). Since this Corollary is well known we omit the remainder of the proof. QED

Proposition 7. Let G be a finite group expressible as G = G′ × G′′, let P′_{∗}, P′′_{∗} be minimal resolutions for G′ and G′′, and let h′, h′′ be functions as used above to define Massey products in H^{∗}(G′) and H^{∗}(G′′) respectively. Then (P′_{∗} ⊗ P′′_{∗})_{∗} is a minimal resolution for G, and the function h defined (for cochains ϕ_{i} from P′_{∗} to F_{p} and θ_{i} from P′′_{∗} to F_{p}) by

\begin{matrix}h(ϕ_{1} ⊗ θ_{1}, ϕ_{2} ⊗ θ_{2}, ϕ_{3} ⊗ θ_{3}) = & (−1)^{|ϕ_{2}||θ_{1}|+|ϕ_{3}||θ_{1}|+|ϕ_{3}||θ_{2}|}(h′(ϕ_{1}, ϕ_{2}, ϕ_{3}) \\ ⊗ (θ_{1}θ_{2})θ_{3} + (−1)^{|ϕ_{1}|+|ϕ_{2}|+|ϕ_{3}|}ϕ_{1}(ϕ_{2}ϕ_{3}) ⊗ h′′(θ_{1}, θ_{2}, θ_{3})) \end{matrix}

may be used to define Massey products in H^{∗}(G).

Proof. Let Δ′ be a diagonal approximation on P′_{∗} and H′ a chain homotopy between (Δ′ ⊗ 1)Δ′ and (1 ⊗ Δ′)Δ′ inducing h′, and define Δ′′ and H′′ similarly. Now the map which is defined to be H′ ⊗ (Δ′′ ⊗ 1)Δ′′ + (−1)^{i}(1 ⊗ Δ′)Δ′ ⊗ H′′ on P′_{i} ⊗ P′′_{j} is a chain homotopy between (Δ′ ⊗ 1 ⊗ Δ′′ ⊗ 1)(Δ′ ⊗ Δ′′) and (1 ⊗ Δ′ ⊗ 1 ⊗ Δ′′)(Δ′ ⊗ Δ′′). The composite of this map with the obvious map from P_{∗}^{′⊗3} ⊗ P_{∗}^{′′⊗3} to (P′_{∗} ⊗ P′′_{∗})^{⊗3} which interlaces the factors is therefore a chain homotopy between (Δ ⊗ 1)Δ and (1 ⊗ Δ)Δ (where Δ = (1 ⊗ τ ⊗ 1)(Δ′ ⊗ Δ′′)), and this is the map which induces h. QED Remark. Using Propositions 5 and 7 and Corollary 6 it is easy to calculate any Massey product in the mod-p cohomology of a finite abelian group. We do not include the general formula because it is long and unilluminating. Corollary 8 and Theorem 9 contain examples of such calculations.

Corollary 8. Let G be a finite abelian group. Then all the Massey products that are defined in H^{∗}(G) contain zero unless p = 3 and G has a direct summand of order three.

Examples. In this short section we consider the 3-groups expressible as an extension with kernel cyclic of order three and quotient abelian of rank two. For these groups we determine all of the differentials in the Lyndon-Hochschild-Serre spectral sequence with F_{p} coefficients, and hence the Poincaré series of their cohomology rings. The best examples among these (from the point of view of using our description of d_{4}) are the non-metacyclic examples, but we shall include the others for completeness. The spectral sequence for the nonabelian group of order 27 and exponent 3, which we shall temporarily refer to as E, was first calculated by Huynh-Mui [6]. Our description of d_{4} greatly simplifies this calculation. The cohomology rings of the metacyclic examples are contained in work of Diethelm [4], who used a different spectral sequence. The other nontrivial cases of Theorem 9 are new.

Let H^{∗}(C_{3^{m}} ⊕C_{3^{n}}) be generated by y_{1}, y_{2}, x_{1} and x_{2} of degrees one, one, two and two respectively, where y_{i} and x_{i} are in the image of the inflation from projection onto the ith factor. The Bockstein maps y_{1} to x_{1} (resp. to zero) if m = 1 (resp. m > 1), and similarly for y_{2}. Modulo choice of generators, we need only consider four distinct extension classes in H^{2}(C_{3^{m}} ⊕ C_{3^{n}}), 0, x_{1}, x_{1} + y_{1}y_{2}, and y_{1}y_{2}. The first two of these give abelian groups and the third gives a metacyclic group.

Theorem 9. Let G be an extension with kernel cyclic of order three and quotient C_{3^{m}} ⊕ C_{3^{n}}, and let ξ be its extension class. Then the Poincaré series for H^{∗}(G) is one of the following power series.

a) 1/(1 − s)^{3},  b) 1/(1 − s)^{2},
c) (1 + s)/(1 − s)(1 − s^{6}),  d) (1 + s + s^{2})/(1 − s^{2})^{2}(1 − s),
e) (1 + s^{2})/(1 − s^{6})(1 − s)^{2},  f) (1 + s + 2s^{2} + 2s^{3} + s^{4} + s^{5})/(1 − s^{6})(1 − s).

If ξ = 0 then case a) occurs. If ξ = x_{1} then case b) occurs. If ξ = x_{1} + y_{1}y_{2} then case b) occurs if n is greater than one and case c) occurs if n = 1. If ξ = y_{1}y_{2} then d) occurs if both m and n are greater than one, e) if one of m and n is one, and f) if m and n are both one.

Proof. As stated above, we examine the Lyndon-Hochschild-Serre spectral sequences for these extensions. It is convenient to introduce the bigraded Poincaré series for the pages of the spectral sequence, defined by the following formula.

P_{r}(s, s′) = ∑_{i,j} s^{i}s^{′j} dim E_{r}^{i,j}

The Poincaré series that we wish to determine is equal to P_{∞}(s, s). For any central extension of p-groups the image of d_{r} is a finitely generated graded module for the kernel of d_{r} which in turn is a finitely generated graded F_{p}-algebra. It follows that P_{r} and P_{r+1} satisfy a relation of the following form,

P_{r}(s, s′) − P_{r+1}(s, s′) = (s^{r} + s^{′r−1})f(s, s′)/g(s, s′),

where f(s, s′) is an integer polynomial and g(s, s′) is a product of terms of the forms 1−s^{k} and 1 − s^{′l}. This relation provides a useful check when calculating P_{r}(s, s′).

The E_{2}-page of the spectral sequence we are considering is Λ[u, y_{1}, y_{2}] ⊗ F_{p}[t, x_{1}, x_{2}], and P_{2} = 1/(1 − s)^{2}(1 − s′). All of the groups have an abelian subgroup of index three and an element in the cohomology of this subgroup that maps to t in the cohomology of the kernel. A simple Evens’ norm map argument [5] now implies that in the spectral sequence t^{3} is a permanent cycle, and so E_{7} = E_{∞}. The first two differentials are given by d_{2}(u) = ξ, d_{2}(t) = 0, d_{3}(t) = β(ξ).

If ξ is zero then the spectral sequence collapses. If ξ is either x_{1} or x_{1} + y_{1}y_{2} then d_{2} is injective from odd rows to even rows, and P_{3} = (1 + s)/(1 − s^{′2})(1 − s). Now d_{3}(t) is zero if either ξ is x_{1} or ξ is x_{1} + y_{1}y_{2} and m, n are both greater than one. If m = 1 and n > 1 then β(x_{1} + y_{1}y_{2}) = (x_{1} + y_{1}y_{2})y_{2}, so that d_{3}(t) is zero in this case too. In the other cases when ξ = x_{1} + y_{1}y_{2}, d_{3}(t) is equal to −x_{2}y_{1} modulo the image of d_{2}, and in these cases P_{4} = (1 + (s − s^{3})(1 + s^{′2}) + ss^{′4})/(1 − s^{′6})(1 − s). The E_{4}-page is concentrated in even vertical degree, so d_{4} must be zero. The only possible non-zero differential now is d_{5}(t^{2}y_{1}), but d_{5}(t^{2}x_{2}y_{1}) is zero by Kudo’s transgression theorem [8], and hence d_{5}(t^{2}y_{1}) is zero since multiplication by x_{2} is injective from E_{5}^{6,0} to E_{5}^{8,0}. Hence E_{4} = E_{∞}.

We now move on to the cases when ξ = y_{1}y_{2}. Here P_{3} is equal to (1 + 2s + s′(2s + s^{2}))/(1 − s^{2})(1 − s^{2})^{2}. If m and n are both greater than one, then d_{3}(t) is zero and so E_{3} = E_{∞}. If exactly one of m and n is equal to one, we may assume that n = 1. Now d_{3}(t) = −x_{2}y_{1}, and

\begin{aligned}P_{4} = (1 + 2s − s^{3} & + s′(2s + s^{2} − s^{4}) + s^{′2}(2s − s^{3}) \\ & + s^{′3}(s + s^{2} − s^{4}) + 2s^{′4}s + s^{′5}(s + s^{2}))/(1 − s^{′6})(1 − s^{2})^{2}. \end{aligned}

We now need to determine the images of tuy_{1}, t^{2}y_{1}, t^{2}y_{2} and t^{2}uy_{1} under d_{4}. From Proposition 7 and Corollary 6 it follows that

⟨x_{2}y_{1}, y_{1}, y_{1}y_{2}⟩ = ⟨y_{1}, y_{1}, y_{1}⟩x_{2}y_{2} = 0,

and so by Theorem 3, d_{4}(t^{i}uy_{1}) is zero. Also (x_{2}y_{1})y_{1} = 0, whereas (−x_{2}y_{1})y_{2} = −(y_{1}y_{2})x_{2}, and so Theorem 3 implies that d_{4}(t^{2}y_{1}) = 0 and d_{4}(t^{2}y_{2}) = −ux_{2}^{2}y_{1}. The Poincaré series for the E_{5}-page is given by the following formula.

\begin{aligned}P_{5} = (1 + 2s − s^{3} & + s′(2s + s^{2} − s^{4} − s^{5}) + s^{′2}(2s − s^{3}) \\ & + s^{′3}(s + s^{2} − s^{4}) + s^{′4}s + s^{′5}(s + s^{2}))/(1 − s^{′6})(1 − s^{2})^{2}. \end{aligned}

The remaining possibly non-zero differentials are d_{5}(t^{2}y_{1}) and d_{5}(t^{2}uy_{1}), which we claim are zero, and then d_{6}(t^{2}uy_{1}), which we also claim to be zero. It may be shown that d_{5}(t^{2}y_{1}) = 0 either using a Kudo transgression theorem argument or by noting that no polynomial in x and x′ can be hit, since E_{5}^{∗,j} = E_{6}^{∗,j} for j equal to 2 and 3, and each of these rows contains a free submodule for the subring of E_{5}^{∗,0} generated by x and x′. Showing that d_{5}(t^{2}uy_{1}) is zero is more difficult. Since E_{5}^{6,1} is generated by uy_{1}y_{2}x_{2}^{2}, it suffices to show that a cohomology element may be chosen which yields uy_{1}y_{2}x_{2}^{2} and whose Bockstein yields a non-zero element in E_{∞}^{7,1} (which is clearly isomorphic to E_{5}^{7,1}). Let Ê_{r}^{i,j} stand for the corresponding spectral sequence with integer coefficients. There is a Bockstein map from Ê_{r}^{i,j} to Ê_{r}^{i,j+1} if j > 0 (resp. Ê_{r}^{i+1,j} if j = 0) converging to the usual Bockstein from mod-p cohomology to integral cohomology. Under this Bockstein uy_{1}y_{2} maps to a generator for Ê_{∞}^{2,2} (one need only compute d_{3} to check this), which is clearly not divisible by three as a cohomology element because Ê_{∞}^{1,3} and Ê_{∞}^{0,4} are trivial. It follows that the Bockstein of an element yielding uy_{1}y_{2} must yield a nonzero multiple of ux_{2}y_{1} in E_{∗}, and hence that d_{5}(t^{2}uy_{1}) cannot be uy_{1}y_{2}x_{1}^{2}, so must be zero. Similarly, d_{6}(t^{2}uy_{1}) must be in the kernel of the Bockstein from E_{6}^{7,0} to E_{6}^{8,0}, so can only be a multiple of x_{1}^{3}y_{1}. Comparison with the spectral sequence for the subextension with quotient C_{3^{m}} ⊕{0} shows now that d_{6}(t^{2}uy_{1}) = 0.

Now we consider the case when ξ = y_{1}y_{2} and m = n = 1. Here d_{3}(t) is equal to x_{1}y_{2} − x_{2}y_{1}, and P_{4} is given by the following formula.

\begin{aligned}P_{4} = (1 + 2s − s^{3} & + s′(2s + s^{2} − 2s^{4} + s^{6}) + s^{′2}(2s − s^{3}) \\ & + s^{′3}(s^{2} + s^{3} − 2s^{4} + s^{6}) + 2s^{′4}s + s^{′5}(s^{2} + s^{3}))/(1 − s^{2})^{2}(1 − s^{′6}) \end{aligned}

We need to determine the images under d_{4} of tu(x_{1}y_{2} − x_{2}y_{1}), t^{2}y_{1}, t^{2}y_{2} and t^{2}u(x_{1}y_{2} − x_{2}y_{1}). From Propostion 7 and Corollary 6 we see that

⟨x_{1}y_{2} − y_{1}x_{2}, x_{1}y_{2} − y_{1}x_{2}, y_{1}y_{2}⟩ = ⟨y_{1}, y_{1}, y_{1}⟩x_{2}^{2}y_{2} − x_{1}^{2}y_{1}⟨y_{2}, y_{2}, y_{2}⟩ = x_{1}x_{2}^{2}y_{2} − x_{1}^{2}x_{2}y_{1},

and hence that d_{4}(t^{i}u(x_{1}y_{2} −x_{2}y_{1})) = it^{i−1}(x_{1}x_{2}^{2}y_{2} −x_{1}^{2}x_{2}y_{1}). Note that modulo multiples of x_{1}y_{2} − x_{2}y_{1}, x_{1}x_{2}^{2}y_{2} − x_{1}^{2}x_{2}y_{1} is equal to the first reduced power of x_{1}y_{2} − x_{2}y_{1}, which we know must be hit by some differential. Also note that

(x_{1}y_{2} − x_{2}y_{1})y_{1} = −(y_{1}y_{2})x_{1} (x_{1}y_{2} − x_{2}y_{1})y_{2} = −(y_{1}y_{2})x_{2},

from which it follows that d_{4}(t^{2}y_{i}) = u(x_{1}y_{2} − x_{2}y_{1})x_{i}. The Poincaré series for E_{5} page is the following expression.

\begin{aligned}P_{5} = (1 + 2s − s^{3} − s^{7} & + s′(2s + s^{2} − 2s^{4} − 2s^{5} + s^{6}) + s^{′2}(2s − s^{3} − s^{7}) \\ & + s^{′3}(s^{2} − 2s^{4} + s^{6}) + s^{′4}s + s^{′5}s^{2})/(1 − s^{2})^{2}(1 − s^{′6}) \end{aligned}

It remains now to calculate d_{5}(t^{2}(x_{1}y_{2} − x_{2}y_{1})), which is x_{1}^{3}x_{2} − x_{2}^{3}x_{1} by Kudo’s transgression theorem, and d_{5}(t^{2}uy_{1}y_{2}), which must be a non-zero multiple of u(x_{1}^{3}y_{2} − x_{2}^{3}y_{1}) to ensure that E_{6}^{∗,1} has a well defined module structure for E_{6}^{∗,0}. The Poincaré series for E_{6}, which is equal to E_{∞}, is the following.

\begin{aligned}P_{6} = (1 + 2s + s^{2} & + s^{3} + s^{4} + s^{5} + s^{6} + s′(2s + s^{2} + 2s^{3} − s^{4}) \\ & + s^{′2}(2s + s^{3} + s^{5}) + s^{′3}(s^{2} − s^{4}))/(1 − s^{2})(1 − s^{′6}) \end{aligned}

QED

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