The mod-p cohomology rings of some p-groups. I.J. Leary* School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London. E1 4NS

Introduction. Throughout this paper p denotes an odd prime. The groups we shall consider are central extensions of a cyclic subgroup by C_{p} ⊕ C_{p}, and may be presented as

P (n) = ⟨A, B, C | A^{p} = B^{p} = C^{p^{n−2}} = [A, C] = [B, C] = 1, [A, B] = C^{p^{n−3}}⟩.

The group P (n) is defined for each n ⩾ 3, and has order p^{n}. The group P (3) is the nonabelian group of order p^{3} and exponent p. The mod-p cohomology rings of the other groups of order p^{3} have been known for some time (see [14] or [5] for that of the nonabelian metacyclic group), and so this paper completes the calculation of the mod-p cohomology rings of the groups of order p^{3}. The corresponding calculations for integral cohomology were completed by Lewis [11]. In the case when p = 3, Milgram and Tezuka have determined the mod-p cohomology of P (3) by showing that it is detected by the restrictions to proper subgroups [13]. For p ⩾ 5 the cohomology of P (3) is not detected by restrictions to proper subgroups, so the method of [13] cannot be used. There is some overlap between this paper and the paper of Benson and Carlson on the cohomology of extraspecial groups [2].

We calculate H^{∗}(BP (n); 𝔽_{p}) using a similar method to our calculation of the integral cohomology of the same groups [10]. First, we let P̃ be the unique nonabelian Lie group whose underlying topological space consists of p^{2} circles. P̃ may be ‘presented’ as follows, where we consider S^{1} as a subgroup of ℂ.

P̃ = ⟨S^{1}, A, B | A^{p} = B^{p} = 1, S^{1} central, [A, B] = exp(2πi/p)⟩.

The subgroup of P̃ generated by A, B, and C = exp(2πi/p^{n−2}) is a normal subgroup isomorphic to P (n), with quotient isomorphic to S^{1}, and from now on we shall regard P (n) as a subgroup of P̃, using this fixed embedding. It follows that BP (n) is a principal S^{1}-bundle over BP̃. We determine H^{∗}(BP̃; 𝔽_{p}), and then use the two row spectral sequence (or Gysin sequence) for the fibration

S^{1} ⟶ BP (n) ⟶ BP̃

to find H^{∗}(BP (n); 𝔽_{p}). This method, which may be applied to any group with nontrivial centre, was suggested independently by P.H. Kropholler and J. Huebschmann [6], [7].

* Supported by SERC post-doctoral fellowship B90 RFH 8960.

Calculations. We begin by calculating H^{∗}(BP̃; 𝔽_{p}). This may be achieved by using the spectral sequence for P̃ as an extension of S^{1} by C_{p} ⊕ C_{p}, which has only two non-zero differentials, d_{3} and d_{2p−1}. This approach was taken in the author’s thesis [9], and in work of B. Moselle [15], who found H^{∗}(BP̃; 𝔽_{p}) while investigating extraspecial p-groups. It is also possible to calculate H^{∗}(BP̃; 𝔽_{p}) directly from H^{∗}(BP̃; ℤ), and this is the method we shall employ. The following statement is taken from theorem 2 of [10].

Theorem 1. Let P̃ be the group defined above. Then H^{∗}(P̃; ℤ) is generated by elements α,β,χ_{1}, …, χ_{p−1}, ζ, with

deg(α) = deg(β) = 2 deg(χ_{i}) = 2i deg(ζ) = 2p,

subject to the following relations:

pα = pβ = 0
α^{p}β = β^{p}α

αχ_{i} = \{ \begin{matrix}0 \\ −α^{p} \end{matrix} βχ_{i} = \{ \begin{matrix}0 & for i < p − 1 \\ −β^{p} & for i = p − 1 \end{matrix}

χ_{i}χ_{j} = \{ \begin{matrix}pχ_{i+j} & i + j < p \\ p^{2}ζ & i + j = p \\ pζχ_{i+j−p} & p < i + j < 2p − 2 \\ pζχ_{p−2} + α^{2p−2} + β^{2p−2} − α^{p−1}β^{p−1} & i = j = p − 1 \end{matrix}

An endomorphism of P̃ that restricts to S^{1} as z ↦ z^{j} sends χ_{i} to j^{i}χ_{i} and ζ to j^{p}ζ The effect of an automorphism on α, β may be determined from their definition. Considered as elements of Hom(P̃, S^{1}), α and β may be defined by α(A^{a}B^{b}z) = η^{a}, β(A^{a}B^{b}z) = η^{b}, where η = exp 2πi/p. If we let H be the subgroup generated by B and elements of S^{1} we may define

χ_{i} = \{ \begin{matrix}Cor_{P} ; H(τ^{′i}) & for i < p − 1 \\ Cor_{P} ; H(τ^{′p−1}) − α^{p−1} & for i = p − 1 \end{matrix}

where τ′ is any element of H^{2}(BH; ℤ) restricting to S^{1} as the standard generator τ. Similarly, ζ may be defined to be the pth Chern class of a p-dimensional irreducible representation of P̃ that restricts to S^{1} as p copies of the identity.

■

Theorem 2. H^{∗}(BP̃; 𝔽_{p}) is generated by elements y, y′, x, x′, c_{2}, …, c_{p−1}, z with

deg(y) = deg(y′) = 1, deg(x) = deg(x′) = 2, deg(c_{i}) = 2i, deg(z) = 2p,

subject to the following relations:

xy′ = x′y,

\begin{aligned}x^{p}y′ & = x^{′p}y, \\ β(y) & = x, \end{aligned}  \begin{aligned}x^{p}x′ & = x^{′p}x, \\ β(y′) & = x′, \end{aligned}

c_{i}y = \{ \begin{matrix}0 \\ −x^{p−1}y \end{matrix} c_{i}y′ = \{ \begin{matrix}0 & for i < p − 1 \\ −x^{′p−1}y′ & for i = p − 1, \end{matrix}

c_{i}x = \{ \begin{matrix}0 \\ −x^{p} \end{matrix} c_{i}x′ = \{ \begin{matrix}0 & for i < p − 1 \\ −x^{′p} & for i = p − 1, \end{matrix}

c_{i}c_{j} = \{ \begin{matrix}0 & for i + j < 2p − 2 \\ x^{2p−2} + x^{′2p−2} − x^{p−1}x^{′p−1} & for i = j = p − 1. \end{matrix}

If we let π_{∗} stand for the projection map from H^{n}(BG; ℤ) to H^{n}(BG; 𝔽_{p}), and δ_{p} for the Bockstein from H^{n}(BG; 𝔽_{p}) to H^{n+1}(BG; ℤ), then we may define the generators by the following equations.

π_{∗}(χ_{i}) = c_{i}, π_{∗}(ζ) = z, π_{∗}(α) = x, π_{∗}(β) = x′,
δ_{p}(y) = α, δ_{p}(y′) = β.

This determines the effect of automorphisms of P̃ on the cohomology ring. The element yy′ is a non-zero multiple of π_{∗}(χ_{1}).

Proof. From Theorem 1 we see that H^{n}(BP̃; ℤ) is trivial for n odd, and has torsion of exponent at most p for n even. It follows that π_{∗} maps H^{2n}(BP̃; ℤ) onto H^{2n}(BP̃; 𝔽_{p}), with kernel the elements divisible by p, and that δ_{p} maps H^{2n−1}(BP̃; 𝔽_{p}) isomorphically to the torsion subgroup of H^{2n}(BP̃; ℤ). This determines the ring structure of H^{even}(BP̃; 𝔽_{p}), and the module structure for this ring of H^{odd}(BP̃; 𝔽_{p}). More explicitly, the relations of odd degree given in the statement must hold, because they map under the injective map δ_{p} to valid relations, for example δ_{p}(xy′) = αβ = δ_{p}(x′y). No more generators of odd degree are required, because the torsion in H^{even}(BP̃; ℤ) is spanned by multiples of α and β. The given relations of odd degree suffice to show that monomials of the forms z^{i}x^{j}y and z^{i}x^{j}x^{′k}y′, where either k ⩽ p − 2 or j = 0, span H^{odd}(BP̃; 𝔽_{p}), and δ_{p} maps these to a basis for the torsion in H^{∗}(BP̃; ℤ), so no more relations of odd degree are needed. Since yy′x = yy′x′ = 0, it follows that yy′ must be some multiple of π_{∗}(χ_{1}), but it remains to show that yy′ is non-zero. The easiest way to check this is by examining the spectral sequence for P̃ expressed as an extension of S^{1} by C_{p} ⊕ C_{p}. In this spectral sequence y and y′ yield generators for E_{2}^{1,0}, their product is non-zero in E_{2}^{2,0}, and E_{2}^{0,1} is trivial, so E_{2}^{2,0} = E_{∞}^{2,0}. Alternatively, it may be verified using explicit cocycles that π_{∗}(χ_{1}) = y′y, which must therefore be non-zero. ■

Since H^{∗}(BP̃; 𝔽_{p}) is generated by elements of degree at most 2p, and we already know the action of the Bockstein, the following proposition suffices to determine the action of the Steenrod algebra on H^{∗}(BP̃; 𝔽_{p}).

Proposition 3. With notation as in Theorem 2, and writing c_{1} for y′y, the following relations hold.

P^{1}(z) = zc_{p−1}, P^{1}(c_{i}) = \{ \begin{matrix}izc_{i−1} & for i < p − 1, \\ −zc_{p−2} + x^{2p−2} + x^{′2p−2} − x^{p−1}x^{′p−1} & for i = p − 1. \end{matrix}

Proof. As in the statement of Theorem 1, let H be the subgroup of P̃ generated by B and S^{1}. Let t be the image under π_{∗} of the element τ′ of H^{2}(BH; ℤ), write Cor and Res for the corestriction and restriction maps between P̃ and H, and let x̄′ = Res(x′). Then the definition of c_{i} given above is equivalent to c_{i} = Cor(t^{i}) for i < p − 1, and c_{p−1} = Cor(t^{p−1}) − x^{p−1}. The p-dimensional representation of P̃ with pth Chern class z restricts to H as a sum of one copy of each of the representations of H that restrict to S^{1} as the identity. The first Chern classes of these representations are t + jx̄′ for each choice of j, and so

Res(z) = ∏_{j=0}^{p−1}(t + jx̄′) = t^{p} − x̄^{′p−1}t.

The action of conjugation by A on H^{1}(BH; 𝔽_{p}) sends t to t+x̄′ and fixes x̄′. The restrictioncorestriction formula [3] therefore implies that

ResCor(t^{i}) = ∑_{j=0}^{p−1}(t + jx̄′)^{i} = ∑_{k=0}^{i} \binom{i}{k}t^{i−k}x̄^{′k} (∑_{j=0}^{p−1} j^{k}) .

Newton’s formula states that ∑_{j=1}^{p−1} j^{k} is divisible by p except when k is divisible by p − 1, in which case it is congruent to −1 mod p. Applying this formula we see that

ResCor(t^{i}) = \{ \begin{matrix}0 & for i < p − 1 \\ −x̄^{′p−1} & for i = p − 1. \end{matrix}

Now we may determine P^{1}(c_{i}) by repeated application of Frobenius reciprocity, as below.

P^{1}Cor(t^{i}) = Cor(P^{1}(t^{i})) = iCor(t^{p+i−1})
= iCor(t^{i−1}Res(z) + t^{i}x̄^{′p−1})
= izCor(t^{i−1}) + ix^{p−1}Cor(t^{i}).

To check that P^{1}(z) is as claimed, note that a general element of H^{4p−2}(BP̃; 𝔽_{p}) has the form λzc_{p−1} + zP (x, x′) + Q(x, x′), so we may assume that P^{1}(z) has that form. First we show that Q(x, x′) is zero. Let H′ be any subgroup of P̃ of index p, and let t̂, x̂ ∈ H^{2}(BH; 𝔽_{p})′ be such that t̂ restricts to a generator for H^{2}(BS^{1}; 𝔽_{p}) and x̂ restricts

trivially. As in the case H′ = H considered above, it may be shown that Res_{P} ; H′(z), and hence also Res_{P} ; H′(P^{1}z), are divisible by t̂, whereas Q(x, x′) restricts to H′ as a scalar multiple of x̂^{2p−1}. This multiple is zero for each choice of H′ if and only if Q(x, x′) is a multiple of each x + ix′ and of x′. In this case Q is a multiple of x^{p}x′ − x^{′p}x so represents zero in H^{4p−2}(BP̃; 𝔽_{p}).

To show that P (x, x′) is zero, note that z and P^{1}(z) are invariant under the action of SL_{2}(𝔽_{p}) considered as a subgroup of Aut(P̃), but no non-zero P (x, x′) of degree p − 1 is invariant. Thus we have shown that P^{1}(z) is a multiple of zc_{p−1}. This multiple may be determined by restricting to the subgroup H. ■

We can now state our result for H^{∗}(BP (n); 𝔽_{p}) for n ⩾ 4.

Theorem 4. Let P (n) be the group presented in the introduction, and view it as a subgroup of P̃. Then for n ⩾ 4 the restriction map from H^{∗}(BP̃; 𝔽_{p}) to H^{∗}(BP (n); 𝔽_{p}) is injective, and if we write Λ[u] for an exterior algebra over 𝔽_{p} on a generator u of degree one, then H^{∗}(BP (n); 𝔽_{p}) is isomorphic to the graded tensor product Λ[u]⊗H^{∗}(BP (n); 𝔽_{p}). We may take for u the element given by the cocycle u([A^{i}B^{j}C^{k}]) = k. Since any automorphism of P (n) extends to an endomorphism of P̃, Theorem 2 determines the action of Aut(P (n)) on H^{∗}(BP (n); 𝔽_{p}). The action of the Steenrod algebra is determined by Proposition 3, except for β(u), which is zero for n > 4 and y′y for n = 4.

Proof. The E_{2} page of the spectral sequence for BP (n) as an S^{1} bundle over BP̃ is isomorphic to Λ[u] ⊗ H^{∗}(BP (n); 𝔽_{p}), so E_{2}^{1,0} ≅ C_{p} ⊕ C_{p} and E_{2}^{0,1} ≅ C_{p}. We know that H^{1}(BP (n); 𝔽_{p}) ≅ Hom(P (n), C_{p}) ≅ (C_{p})^{3}, so u must survive, and the spectral sequence collapses. It remains to check that β(u) is as claimed. The image of δ_{p} from H^{1}(BP (n); 𝔽_{p}) to H^{2}(BP (n); ℤ) is the subgroup of elements of order p, and the kernel of π_{∗} from H^{2}(BP (n); ℤ) to H^{2}(BP (n); 𝔽_{p}) is the subgroup of elements divisible by p. Since H^{2}(BP (n); ℤ) ≅ Hom(P (n), S^{1}) ≅ C_{p^{n−3}} ⊕ C_{p} ⊕ C_{p}, it follows that u may be chosen such that β(u) is as claimed. It may be checked using cochains that u as described in the statement is such an element. ■

In the case when n = 3 the spectral sequence considered above does not collapse, and many generators not in the image of the restriction from P̃ (or equivalently, yielding elements on the E_{∞}^{∗,1} line of the spectral sequence) are required. Some of these may be described as corestrictions from subgroups of P (3), but others require us to consider Massey products. Many of the relations between these elements will be implied by relations that Massey products always satisfy, which we summarise in the following lemma.

Lemma 5. Let C^{∗} stand for the cochain complex for a space X with 𝔽_{p} coefficients, and let H^{∗} be the cohomology of C^{∗}. Given homogeneous cocycles u, v, w in C^{∗} such that [uv] and [vw] represent 0 in H^{∗}, choose cochains a and b such that δa = uv, δb = vw, and define the Massey product of [u], [v] and [w] by the formula

⟨[u], [v], [w]⟩ = [(−1)^{|u|}ub − aw] ∈ H^{|u|+|v|+|w|−1}/(uH^{|v|+|w|−1} + wH^{|u|+|v|−1}).

This product is linear in each of its arguments, and satisfies the following relations whenever all the terms are defined.

⟨u, v, w⟩x + (−1)^{|u|}u⟨v, w, x⟩ ≡ 0 mod uH^{∗}x  (1)
(−1)^{|u|}⟨⟨u, v, w⟩, x, y⟩ + ⟨u, ⟨v, w, x⟩, y⟩+(−1)^{|v|}⟨u, v, ⟨w, x, y⟩⟩ ≡ 0
mod uH^{∗} + H^{|u|+|v|−1}wH^{|x|+|y|−1} + yH^{∗} (2)

(−1)^{|w||u|}⟨u, v, w⟩ + (−1)^{|u||v|}⟨v, w, u⟩ + (−1)^{|v||w|}⟨w,u, v⟩ ≡ 0
mod uH^{∗} + vH^{∗} + wH^{∗} (3)
⟨u, v, w⟩ + (−1)^{|u||v|+|v||w|+|w||u|}⟨w, v, u⟩ ≡ 0 mod uH^{∗} + wH^{∗}  (4)

Proof. See for example [8] or [12]. ■

We are ready now to calculate H^{∗}(BP (3); 𝔽_{p}), which we split into the cases p > 3 and p = 3. It should not be surprising that these cases are quite different, because the groups of order p^{4} behave differently in these cases. For example, there are two generator groups of order p^{4} and exponent p if and only if p > 3 [4]. Any such group is expressible as a nonsplit central extension of C_{p} by P (3), such that every ‘subextension’ of C_{p} by C_{p} is split. Translating these statements into a statement about extension classes in H^{2}(BP (3); 𝔽_{p}), we see that H^{2}(BP (3); 𝔽_{p}) is detected by cyclic subgroups if and only if p = 3.

Theorem 6 Let p be greater than 3. Then H^{∗}(BP (3); 𝔽_{p}) is generated by elements y, y′, x, x′, Y, Y′, X, X′, d_{4}, …, d_{p}, c_{4}, …, c_{p−1}, z, with

deg(y) = deg(y′) = 1, deg(x) = deg(x′) = deg(Y) = deg(Y′) = 2,
deg(X) = deg(X′) = 3, deg(d_{i}) = 2i − 1, deg(c_{i}) = 2i, deg(z) = 2p,

\begin{aligned}β(y) & = x, \\ β(Y) & = X, \end{aligned} \begin{aligned}β(y′) & = x′, \\ β(Y′) & = X′, \end{aligned}

β(d_{i}) = \{ \begin{matrix}c_{i} & for i < p \\ 0 & for i = p, \end{matrix}

subject to the following relations:

yy′ = 0, xy′ = x′y,
yY = y′Y′ = 0, yY′ = y′Y,
Y^{2} = Y^{′2} = Y Y′ = 0,
\begin{aligned}yX & = xY, \\ Xy′ & = 2xY′ + x′Y, \end{aligned}  \begin{aligned}y′X′ & = x′Y′, \\ X′y & = 2x′Y + xY′, \end{aligned}

XY = X′Y′ = 0,  XY′ = −X′Y,  xX′ = −x′X,
x(xY′ + x′Y) = x′(xY′ + x′Y) = 0,

\begin{aligned}x^{p}y′ − x^{′p}y & = 0, \\ x^{p}Y′ + x^{′p}Y & = 0, \end{aligned}  \begin{matrix}x^{p}x′ − x^{′p}x = 0, \\ x^{p}X′ + x^{′p}X = 0, \end{matrix}

c_{i}y = \{ \begin{matrix}0 \\ −x^{p−1}y \end{matrix}_{c_{i}x = \{ \begin{matrix}0 \\ −x^{p} \end{matrix}}_{c_{i}Y = \{ \begin{matrix}0 \\ −x^{p−1}Y \end{matrix}}_{c_{i}X = \{ \begin{matrix}0 \\ −x^{p−1}X \end{matrix}} c_{i}y′ = \{ \begin{matrix}0 & for i < p − 1 \\ −x^{′p−1}y′ & for i = p − 1, \end{matrix}

c_{i}x′ = \{ \begin{matrix}0 & for i < p − 1 \\ −x^{′p} & for i = p − 1, \end{matrix}
c_{i}Y′ = \{ \begin{matrix}0 & for i < p − 1 \\ −x^{′p−1}Y′ & for i = p − 1, \end{matrix}
c_{i}X′ = \{ \begin{matrix}0 & for i < p − 1 \\ −x^{′p−1}X′ & for i = p − 1, \end{matrix}

c_{i}c_{j} = \{ \begin{matrix}0 & for i + j < 2p − 2 \\ x^{2p−2} + x^{′2p−2} − x^{p−1}x^{′p−1} & for i = j = p − 1, \end{matrix}

d_{i}y = \{ \begin{matrix}0 \\ −x^{p−1}Y \end{matrix}_{d_{i}x = \{ \begin{matrix}0 \\ −x^{p−1}y \\ x^{p−1}X \end{matrix}}_{d_{i}Y = 0}_{d_{i}X = \{ \begin{matrix}0 \\ −x^{p−1}Y \end{matrix}} d_{i}y′ = \{ \begin{matrix}0 & for i < p \\ x^{′p−1}Y′ & for i = p, \end{matrix}

d_{i}x′ = \{ \begin{matrix}0 & for i < p − 1 \\ −x^{′p−1}y′ & for i = p − 1 \\ −x^{′p−1}X′ & for i = p, \end{matrix}
d_{i}Y′ = 0,
d_{i}X′ = \{ \begin{matrix}0 & for i ≠ p − 1 \\ −x^{′p−1}Y′ & for i = p − 1, \end{matrix}

d_{i}d_{j} = \{ \begin{matrix}0 & for i < p − 1 or j < p − 1 \\ x^{2p−3}Y − x^{′2p−3}Y′ + x^{p−1}x^{′p−2}Y′ & for i = p and j = p − 1, \end{matrix}

d_{i}c_{j} = \{ \begin{matrix}0 & for i < p − 1 or j < p − 1 \\ x^{2p−3}y + x^{′2p−3}y′ − x^{p−1}x^{′p−2}y′ & for i = j = p − 1 \\ −x^{2p−3}X + x^{′2p−3}X′ − x^{p−1}x^{′p−2}X′ & for i = p, j = p − 1. \end{matrix}

We define y, y′ by the cocycles y(A^{i}B^{j}C^{k}) = i and y′(A^{i}B^{j}C^{k}) = j. The equation yy′ = 0 implies that we may define unique elements Y, Y′ by

Y = ⟨y, y, y′⟩, Y′ = ⟨y′, y′, y⟩.

If we define d′ ∈ H^{1}(B⟨B, C⟩; 𝔽_{p}) by the cocycle d′(B^{r}C^{s}) = s, and c′ = β(d′), then we may define

d_{i} = \{ \begin{matrix}Cor_{⟨B,C⟩}^{P (3)}(c^{′i−1}d′) & for i < p − 1 \\ Cor_{⟨B,C⟩}^{P (3)}(c^{′p−2}d′) − x^{p−2}y & for i = p − 1 \\ Cor_{⟨B,C⟩}^{P (3)}(c^{′p−1}d′) + x^{p−2}X & for i = p. \end{matrix}

We may define the elements x, x′, X, X′, c_{i} from the elements already defined, using the Bockstein, and we may define the elements y, y′, x, x′, c_{i}, z to be the restrictions from P̃ of the generators with the same name. The two definitions given for y, y′, x, x′, c_{i} are equivalent. The effect of automorphisms of P (3) on y, y′, x, x′, Y, Y′, X, X′ is determined by the above definitions. An automorphism of P (3) that restricts to its centre as C ↦ C^{j} sends d_{i} to j^{i}d_{i}, c_{i} to j^{i}c_{i} and z to jz. We note that the restriction from P̃ to P (3) sends c_{2} to a non-zero multiple of xY′ + x′Y, and c_{3} to a non-zero multiple of XX′.

Proof. Throughout this proof we shall adopt the convention that Cor and Res will stand for Cor_{⟨B,C⟩}^{P (3)} and Res_{⟨B,C⟩}^{P (3)} respectively. As in the previous theorem we consider the spectral

sequence for P (3) as an S^{1} bundle over P̃. First we prove that H^{n}(BP (3); ℤ) has exponent p for n < 2p by considering the spectral sequence with integer coefficients. (This is proved in [11] and [10], but we include it for completeness.) Since H^{2k+1}(BP̃; ℤ) is trivial, it follows that H^{2k+1}(BP (3); ℤ) is isomorphic to a subgroup of H^{2k}(BP̃; ℤ), which has torsion only of order p. It also follows that H^{2k}(BP̃; ℤ) maps onto H^{2k}(BP (3); ℤ). Since for k < p H^{2k}(BP̃; ℤ) is generated by elements of order p and corestrictions from proper subgroups the same must be true for H^{2k}(BP (3); ℤ), which therefore has exponent p. It follows that for n < 2p the image and kernel of the Bockstein β in H^{n}(BP (3); 𝔽_{p}) coincide.

Returning now to the spectral sequence with 𝔽_{p} coefficients, the E_{2} page is isomorphic to Λ[u]⊗H^{∗}(BP̃; 𝔽_{p}), where u has bidegree (0, 1). Since H^{1}(BP (3); 𝔽_{p}) has order p^{2}, d_{2}(u) must be non-zero. All the subgroups of P (3) of index p are isomorphic to C_{p} ⊕C_{p}, so d_{2}(u) must restrict trivially to each S^{1} ⊕ C_{p} subgroup of P̃. It follows that d_{2}(u) is a non-zero multiple of yy′, and so

E_{∞}^{∗,0} ≅ H^{∗}(BP̃; 𝔽_{p})/(yy′), E_{∞}^{∗,1} ≅ Ann(yy′) ⊂ H^{∗}(BP̃; 𝔽_{p}).

Hence E_{∞}^{∗,∗} is generated by the elements [uy], [uy′], [ux], [ux′], [uc_{2}], …, [uc_{p−1}], y, y′, x, x′, c_{2}, …, c_{p−1}, z, subject to the relations implied by those that hold in E_{2}^{∗,∗}, together with the relation yy′ = 0. The cohomology relations that involve only generators in the image of the restriction from P̃ follow from the spectral sequence.

Since unique elements Y, Y′ may be defined as in the statement, our next task is to show that these elements together with x, x′ form a basis for H^{2}. We may calculate Res_{⟨A,C⟩}^{P (3)}(Y′) and Res(Y′) = Res_{⟨B,C⟩}^{P (3)}(Y′) by explicit calculation using cochains in the bar resolution for P (3). Define 1-cochains a and b by the equations a([g]) = −1/2(y′([g]))^{2}, b([A^{r}B^{s}C^{t}]) = t. Then

δa([g|h]) = −1/2 (y′([g])^{2} + y′([h])^{2} − (y′([g]) + y′([h]))^{2}) = y′([g])y′([h]), and

\begin{aligned}δb([A^{i}B^{j}C^{k}|A^{r}B^{s}C^{t}]) & = b([A^{i}B^{j}C^{k}]) + b([A^{r}B^{s}C^{t}]) − b([A^{i+r}B^{j+s}C^{k+t−jr}]) \\ & = y′([A^{i}B^{j}C^{k}])y([A^{r}B^{s}C^{t}]), \end{aligned}

so we may choose a cocycle representing Y′ by the formula

Y′([g|h]) = y′([g])(1/2y′([g])y([h]) − b([h])).

The cocycle y vanishes on ⟨B, C⟩, and y′ vanishes on ⟨A, C⟩, so it is now easy to check that even as cochains Res_{⟨A,C⟩}^{P (3)}(Y′) = 0 and Res(Y′) = Res(y′)d′, where d′ is the cocycle defined in the statement. A similar argument may be used to show that Y restricts to zero on ⟨B, C⟩ and to a non-zero product of elements of degree one on ⟨A, C⟩. Since x and x′ restrict to subgroups as elements in the image of the Bockstein, it follows that x, x′, Y, Y′ form a basis for H^{2}. The identities of Lemma 5 suffice to express any Massey product of elements of H^{1} in terms of Y and Y′, so this determines the action of Aut(P (3)) on Y and Y′. For example, the ‘Jacobi identity’ ((3) of Lemma 5) implies that 3⟨y, y, y⟩ = 0.

The relations in H^{3} between Y, Y′ and y, y′ follow from properties of the Massey product, for example

Y y′ = ⟨y, y, y′⟩y′ ≡ y⟨y, y′, y′⟩ ≡ y⟨y′, y′, y⟩ = yY′,

where the congruences are modulo {0}, and come from applications of (1) and (4) of Lemma 5. Before moving on to H^{4} we note that ⟨y, y, Y ⟩ is defined, and is congruent to zero modulo H^{2}H^{1}, which follows from (2) of Lemma 5. Applying each of (1) and (4) it is now easy to show that Y^{2} ≡ 0 modulo H^{2}H^{1}H^{1} = {0}. Similarly Y^{′2} = (Y + Y′)^{2} = 0, because there are automorphisms of P (3) sending Y to Y′ and to Y + Y′. The expressions for yX and y′X′ follow by applying the Bockstein to the relations yY = y′Y′ = 0. For the remaining relations in H^{4} we introduce some matrix Massey products [12]. We consider

⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩, which is defined modulo H^{2}H^{1}, so has a well defined product with any

element of H^{1}. These products are determined by the following calculations.

⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩y ≡ −(x, x′) ( \begin{matrix}⟨y′, y, y⟩ \\ −⟨y, y, y⟩ \end{matrix} ) = −(x, x′) ( \begin{matrix}Y \\ 0 \end{matrix} ) = −xY,

⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩y′ ≡ −(x, x′) ( \begin{matrix}⟨y′, y, y′⟩ \\ −⟨y, y, y′⟩ \end{matrix} ) ≡ −(x, x′) ( \begin{matrix}−2Y′ \\ −Y \end{matrix} ) = 2xY′ + x′Y,

where each congruence is modulo H^{2}H^{1}H^{1} = {0}. Similarly, the following expressions may be verified.

⟨(x′, x), ( \begin{matrix}y \\ −y′ \end{matrix} ) , y′⟩y′ = −x′Y′, ⟨(x′, x), ( \begin{matrix}y \\ −y′ \end{matrix} ) , y′⟩y = 2x′Y + xY′.

Hence we deduce that X ∈ ⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩ and X′ ∈ ⟨(x′, x), ( \begin{matrix}y \\ −y′ \end{matrix} ) , y′⟩, and the

remaining relations in H^{4} follow. Our results allow us to deduce that X, X′, yY′, xy, xy′, x′y′ form a basis for H^{3}. In H^{5} the relations XY = 0, X′Y′ = 0, XY′ = −X′Y and xX′ = −x′X follow by applying the Bockstein to the relations Y^{2} = 0, Y^{′2} = 0, Y Y′ = 0 and X′y = 2x′Y + xY′ respectively. For the relations stated in H^{6}, we note that xyY′ = x′yY′ = 0, and then apply the Bockstein to these relations, noting also that β(yY′) = 2xY′ + 2x′Y.

We see now that in the E_{∞}^{∗,∗} page of the spectral sequence Y yields λ[uy] and Y′ yields −λ[uy′] for some non-zero λ, and deduce that xY′ + x′Y ≡ 0 modulo the image of the restriction from P̃. The relations in H^{6} imply that xY′ + x′Y = λ′Res_{P} ; P (3)(c_{2}) for some λ′. The element yY′ is not in the kernel of the Bockstein, so λ′ must be non-zero, and we see that products of elements in H^{2} generate H^{4}.

The spectral sequence shows that xH^{3} + x′H^{3} has index p in H^{5}, and it may be shown that XY′ ∉ xH^{3} + x′H^{3} by considering another spectral sequence, that for P (3) expressed as a central extension of C_{p} by C_{p} ⊕ C_{p}. Only the first two differentials need be computed for this purpose. This calculation is contained in [10]. Since XY′ is therefore not in the image of the Bockstein it follows that β(XY′) = −XX′ is not zero. XX′ is however annihilated by x, so must be a multiple of c_{3}. Hence H^{6} is generated by XX′ and products of elements of H^{2}.

We already have the relations x^{p}y′ = x^{′p}y, x^{p}x′ = x^{′p}x, and we obtain x^{p}Y′ + x^{′p}Y = 0 by applying P^{1} to the relation xY′ + x′Y = λRes_{P} ; P (3)(c_{2}). (Recall that P^{1}(c_{2}) was determined in Proposition 3.) The relation x^{p}X′ + x^{′p}X = 0 now follows by applying the Bockstein.

We may verify that in degrees greater than 6 all products of the generators y, y′, x, x′, Y, Y′, X, X′ (which we shall call the ‘low dimensional generators’) may be expressed in the form

f_{1} + f_{2}Y + f_{3}Y′  for even total degree
f_{1}y + f_{2}y′ + f_{3}X + f_{4}X′  for odd total degree

where f_{i} is a polynomial in x and x′. With the exception of xY′ + x′Y, such expressions satisfy ‘the same’ relations as elements of H^{∗} as they do as elements of E_{∞}^{∗,∗}. Elements that are expressible as above form a subspace of H^{n} of codimension 1 for 7 ⩽ n ⩽ 2p, so we introduce the elements c_{i} for i > 3 and z to our generating set. The Bockstein β sends elements of the above form of odd total degree to elements of the above form of even total degree, so any element sent by β to c_{i} will suffice to complete a basis for H^{2i−1}. Hence we may add d_{i} for i < p to our generating set. Assuming that the relations given involving d_{p} hold, it follows that d_{p} is not an element of the form described above, so suffices to complete a basis for H^{2p−1}. Using Frobenius reciprocity and the fact that CorRes is the zero map, it may be checked that β(d_{p}) = 0.

The relations involving the c_{i} and d_{i} may be checked using Frobenius reciprocity, and we shall only prove some examples and indicate an economical order in which to prove the rest. Before starting it is helpful to notice that the automorphism θ of P (3) defined by θ(A) = B, θ(B) = A has the effect of exchanging the ‘primed’ and ‘unprimed’ low dimensional generators, and sends c_{i} to (−1)^{i}c_{i}.

The relations between c_{i} and y, y′, x, x′ follow from the spectral sequence. The expressions for c_{i}Y are easily determined since Cor(c^{′i})Y = Cor(c^{′i}Res(Y)) = 0. The expressions for c_{i}Y′ follow by applying θ^{∗}, and the expressions for c_{i}X and c_{i}X′ by applying β. The relations between d_{i} and the low dimensional generators are slightly more involved, and we must examine the ‘primed’ and ‘unprimed’ generators separately, because we do not know yet what effect θ^{∗} has on d_{i}. For example, the following calculations verify the claimed expressions for d_{i}y and d_{i}y′.

Cor(c^{′i−1}d′)y = Cor(c^{′i−1}d′Res(y)) = 0,
Cor(c^{′i−1}d′)y′ = Cor(c^{′i−1}d′Res(y′)) = −Cor(c^{′i−1})Y′.

Similar calculations may be used to determine d_{i}Y and d_{i}Y′, and then we may apply the Bockstein to these relations (substituting from earlier relations for terms involving c_{i}) to obtain expressions for d_{i}x, d_{i}x′, d_{i}X and d_{i}X′.

The expressions for d_{i}d_{j} and c_{i}d_{j} may be determined similarly. As an example we determine Cor(c^{′i−1}d′)Cor(c^{′j−1}d′), where without loss of generality we may assume that i > j. If we write ȳ′ = Res(y′), x̄′ = Res(x′), then the action of conjugation by A on H^{1}(B⟨B, C⟩; 𝔽_{p}) fixes ȳ′ and sends d′ to d′ +ȳ′. It follows from the restriction-corestriction formula that

ResCor(c^{′j−1}d′) = ∑_{k=0}^{p−1}(c′ + kx̄′)^{j−1}(d′ + kȳ′).

Applying Newton’s formula as in the proof of Proposition 3 we see that this expression is zero for j < p − 1, and that ResCor(c^{′p−2}d′) = −ȳ′x̄^{′p−2}. It only remains to determine Cor(c^{′p−1}d′)Cor(c^{′p−2}d′), which may be determined as below.

Cor(c^{′p−1}d′)Cor(c^{′p−2}d′) = −Cor(c^{′p−1}d′ȳ′x̄^{′p−2})
= Cor(c^{′p−1})Y′x̄^{′p−2}
= (c_{p−1} + x^{p−1})Y′x^{′p−2}
= x^{p−1}x^{′p−2}Y′ − x^{′2p−3}Y′.

All that remains to be calculated is the effect of automorphisms of P (3) on d_{i}. The multiplicative relations demonstrated above imply that in the spectral sequence d_{i} yields the element [uc_{i−1}] in E_{∞}^{2i−2,1}. An automorphism of P (3) that restricts to the centre as the map C ↦ C^{j} extends to an endomorphism of P̃ that wraps the S^{1} subgroup j times around itself. This endomorphism induces an endomorphism of the spectral sequence which sends u to ju, and c_{i} to j^{i}c_{i}. It follows that d_{i} is sent to j^{i}d_{i} modulo E_{∞}^{2i−1,0}. Since β(d_{i}) = c_{i} for i < p and β(d_{p}) = 0, we also know that d_{i} is sent to j^{i}d_{i} modulo the kernel of β. The result now follows, since Ker(β) ∩ E_{∞}^{2i−1,0} is trivial. ■

We now state and prove a similar assertion for the case when p = 3. This result has also been obtained by Milgram and Tezuka [13], who show that the whole ring is detected by the restriction maps to proper subgroups. For p > 3 there are elements (for example XX′, c_{i} for i < p − 1) that restrict trivially to all proper subgroups of P (3). One way to explain the difference between the presentations of H^{∗}(BP (3); 𝔽_{p}) in the cases p > 3 and p = 3 is in terms of the different properties of Massey products. If A is any space, and v ∈ H^{1}(A; 𝔽_{p}), then ⟨v, v, v⟩ = 0 if p > 3, whereas ⟨v, v, v⟩ = β(v) if p = 3. This may be verified using explicit cochain calculations for the ‘universal example’ which is the case when A = BC_{p}, or see [8].

Theorem 7. Let p = 3. Then H^{∗}(BP (3); 𝔽_{3}) is generated by elements y, y′, x, x′, Y, Y′, X, X′, z, with

deg(y) = deg(y′) = 1, deg(x) = deg(x′) = deg(Y) = deg(Y′) = 2,
deg(X) = deg(X′) = 3, deg(z) = 6,
\begin{aligned}β(y) & = x, \\ β(Y) & = X, \end{aligned} \begin{aligned}β(y′) & = x′, \\ β(Y′) & = X′, \end{aligned}

subject to the following relations:

yy′ = 0, xy′ = x′y,
yY = y′Y′ = xy′, yY′ = y′Y,
Y Y′ = xx′,  Y^{2} = xY′,  Y^{′2} = x′Y,

\begin{aligned}yX & = xY − xx′, \\ Xy′ & = x′Y − xY′, \\ XY & = x′X, \\ XY′ & = −X′Y, \end{aligned}  \begin{aligned}y′X′ & = x′Y′ − xx′, \\ X′y & = xY′ − x′Y, \\ X′Y′ & = xX′, \\ xX′ & = −x′X, \end{aligned}

XX′ = 0,  x(xY′ + x′Y) = −xx^{′2},  x′(xY′ + x′Y) = −x′x^{2},
\begin{aligned}x^{3}y′ − x^{′3}y & = 0, \\ x^{3}Y′ + x^{′3}Y & = −x^{2}x^{′2}, \end{aligned}  \begin{matrix}x^{3}x′ − x^{′3}x = 0, \\ x^{3}X′ + x^{′3}X = 0, \end{matrix}

We define y, y′ ∈ H^{1}(BP (3); 𝔽_{3}) by the cocycles y(A^{r}B^{s}C^{t}) = r and y′(A^{r}B^{s}C^{t}) = s. The equation yy′ = 0 implies that we may define unique elements in H^{2}(BP (3); 𝔽_{3}) by forming the Massey product of any three elements of H^{1}(BP (3); 𝔽_{3}), and we define Y, Y′ by Y = ⟨y, y, y′⟩, Y′ = ⟨y′, y′, y⟩. We also define x = β(y), x′ = β(y′), X = β(Y), X′ = β(Y′), and note that x, x′ satisfy x = ⟨y, y, y⟩, x′ = ⟨y′, y′, y′⟩. The effect of automorphisms of P (3) on the generators y, y′, x, x′, Y, Y′, X, X′ is determined by the above definitions. An automorphism of P (3) which restricts to the centre as C ↦ C^{j} sends z to jz, and we may define z to be the restriction from P̃ of the generator of the same name. We also note that

Res_{P} ; P (3)(c_{2}) = −xY′ − x′Y − x^{2} − x^{′2}.

Proof. Much of the proof is exactly as in the case when p > 3, except that the expressions for Y Y′, Y^{2} and Y^{′2} require a different proof. Again we consider the spectral sequence for BP (3) as an S^{1} bundle over P̃, which has E_{2}^{∗,∗} ≅ Λ[u] ⊗ H^{∗}(BP̃; 𝔽_{p}), and d_{2}(u) = ±yy′. The proof that x, x′, Y and Y′ form a basis for H^{2} by calculating restrictions to the subgroups ⟨A, C⟩ and ⟨B, C⟩ works exactly as in Theorem 6. Many of the relations may be proved exactly as in Theorem 6, although the fact that ⟨y, y, y⟩ is equal to x rather than 0 makes the relations look different. For example, yY = y⟨y, y, y′⟩ ≡ ⟨y, y, y⟩y′ = xy′, where the congruence is modulo {0}. The relations containing y′Y′, y′Y, yX and y′X′

follow similarly. As in the proof of Theorem 6, any Z ∈ ⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩ satisfies

Zy = −xY + xx′ and Zy′ = x′Y − xY′. Similarly, any Z′ ∈ ⟨(x′, x), ( \begin{matrix}y \\ −y′ \end{matrix} ) , y′⟩ satisfies

Z′y′ = −x′Y′ + xx′ and Z′y = xY′ − x′Y. We deduce that X, X′ satisfy the relations claimed in H^{4}, and that X, X′,xy, xy′, x′y′,Y y′ form a basis for H^{3}. The relations xY′y = x^{′2}y and x′Y′y = x^{2}y′ follow easily from the relations we have already proven, now we apply the Bockstein to them, and obtain x(xY′+x′Y) = −xx^{′2} and x′(xY′+x′Y) = −x′x^{2}. It follows that the relation in H^{4} yielding the spectral sequence relation x[uy′] = x′[uy] must be 0 = xY′ + x′Y + c_{2} + x^{2} + x^{′2}. We deduce that x^{2}, xx′, x^{′2}, xY, xY′, x′Y, and x′Y′ form a basis for H^{4}. Now we shall return to the other relations we wish to prove in H^{4}.

The subspace of H^{4} of elements restricting trivially to ⟨A, C⟩ and ⟨B, C⟩ contains Y Y′, and has basis xx′, xY′, x′Y. Hence there is some expression Y Y′ = λxx′ + λ′xY′ + λ′′x′Y. The automorphism θ of P (3), which has the effect of exchanging the ‘primed’ and ‘unprimed’ generators fixes Y Y′, so λ′ and λ′′ must be equal. Multiplying both sides of the expression for Y Y′ by y gives xx′y = λxx′y − λ′x^{′2}y, and hence λ = 1, λ′ = 0 as required. The expressions for Y^{2} and Y^{′2} may be deduced from that for Y Y′ by considering the effect of various automorphisms of P (3).

We apply the Bockstein to the expressions for Y^{2}, Y Y′, Y^{′2} and Res_{P} ; P (3)(c_{2}) respectively to obtain the relations XY = −xX′, X′Y′ = −x′X, XY′ = −X′Y and xX′ = −x′X. The spectral sequence argument suggested in Theorem 6 shows that XY′ is not in xH^{3} + x′H^{3}, and so no new generators are needed in H^{5}. It is easily checked that XX′ is annihilated by both x and x′, which implies that XX′ = 0. We already have the relations x^{3}y′ = x^{′3}y, x^{3}x′ = x^{′3}x, and as in the case p > 3, we prove the relation x^{3}Y′ + x^{′3}Y = −x^{2}x^{′2} by applying P^{1} to the relation 0 = x′Y + xY′ + c_{2} + x^{2} + x^{′2}, and we apply the Bockstein to this relation to obtain x^{3}X′ + x^{′3}X = 0.

It may be checked that all products of degree at least 6 of elements of degree at most three may be expressed in the form

f_{1} + f_{2}Y + f_{3}Y′  for even total degree
f_{1}y + f_{2}y′ + f_{3}X + f_{4}X′  for odd total degree

where f_{i} is a polynomial in x and x′. The relations we have given between such elements are sufficient to imply the relations that hold between the corresponding elements of the spectral sequence, hence our presentation of the ring H^{∗}(BP (3); 𝔽_{3}) is complete. The effect of automorphisms on z follows from its definition as a restriction from P̃, and we already have the required expression for Res_{P} ; P (3)(c_{2}). ■

It is unreasonable to expect that by choosing different generators we could make the statement of Theorem 7 look more like the statement of Theorem 6. For example, for p > 3, H^{2}(BP (3); 𝔽_{p}) may be expressed as the direct sum of two Aut(P (3))-invariant subspaces:

H^{2}(BP (3); 𝔽_{p}) = ⟨x, x′⟩ ⊕ ⟨Y, Y′⟩,

whereas for p = 3 the subspace ⟨x, x′⟩ has no Aut(P (3))-invariant complement.

In each case, the action of the Steenrod algebra on the generators of H^{∗}(BP (3); 𝔽_{p}) is determined (up to a scalar multiple in the case of P^{1}(d_{4})) as in Proposition 3, except for the action of P^{1} on X and X′. This is described below.

Proposition 8. With notation as in Theorems 6 and 7,

\begin{aligned}P^{1}(X) & = x^{p−1}X + zy \\ P^{1}(X′) & = x^{′p−1}X′ − zy′. \end{aligned}

Proof. The spectral sequence operation _{B}P^{1} defined by Araki and Vasquez ([1], [16]) on the E_{∞} page of the spectral sequence for BP (3) as an S^{1}-bundle over BP̃ sends ux′ to ux^{′p}, so we deduce that P^{1}(X′) ≡ x^{′p−1}X′ modulo the image of the restriction from P̃. Let K be a subgroup of P (3) of index p, and let H^{∗}(BK; 𝔽_{p}) = 𝔽_{p}[x̄, c] ⊗ Λ[ȳ, d], where β(d) = c, β(ȳ) = x̄, and d, considered as a morphism from K to 𝔽_{p}, sends C to 1. Then if we let Res = Res_{K}^{P (3)} we have Res(X′) = λ(x̄d − cȳ), since X′ is in the image of the Bockstein and the image of the Bockstein in H^{3}(BK) is generated by x̄d − cȳ. We obtain

Res(P^{1}(X′) − x^{′p−1}X′) = −λ(c^{p} − x̄^{p−1}c)ȳ.

If P is an expression of degree 2p + 1 involving only y, y′, x, and x′, then Res(P) is a multiple of x̄^{p}ȳ, and if Q is in the span of zy and zy′, then Res(Q) is a multiple of (c^{p} −x̄^{p−1}c)ȳ. We know that P^{1}(X′)−x^{′p−1}X′ = P +Q for some such choices of P and Q, and we deduce that for all K, Res(P) = 0. Thus Res(βP) = 0, and hence βP is a multiple of x^{p}x′ − x^{′p}x, so is zero in H^{∗}(BP (3); 𝔽_{p}). The Bockstein is injective on the subspace of H^{2p+1} generated by x, x′, y, and y′, so we deduce that P = 0. Using explicit cochain calculations as in the proof of Theorem 6 we can determine λ in the case K = ⟨A, C⟩ and ȳ = Res(y) (resp. K = ⟨B, C⟩ and ȳ = Res(y′)), and conclude that Q = −zy′. The result for P^{1}(X) follows from this result or may be deduced similarly. ■ Acknowledgements. The author thanks his research supervisor, C. B. Thomas, also D. J. Benson, D. J. Green and P. H. Kropholler. The author is indebted to J. F. Adams, who first suggested the use of Massey products in these calculations.

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