Mathematische Zeitschrift Manuscript-Nr. (will be inserted by hand later)

A(p) generators for H^{∗}V and Singer’s homological transfer

M. D. Crossley

Centre de Recerca Matemàtica, Institut d’Estudis Catalans, Apartat 50, E-08193, Bellaterra, Barcelona, Spain

e-mail : crossley@crm.es

Received: 11 June 1996 / Accepted: February 19 1997

Summary. We list explicitly a minimal set of generators for the cohomology of an elementary abelian p-group, V, of rank 1 or 2, as a module over the mod p Steenrod algebra, for an odd prime p. Following Singer, we then construct a transfer map to the vector space spanned by such generators, where V now has arbitrary rank, from the homology of the Steenrod algebra. We show that this map takes images in the subspace of GL(V)-invariants and that it is an isomorphism for V having rank 1 or 2.

Subject Classifications: 55S10 55Q45

1. Introduction and Statement of Results

Let A(p) denote the mod p Steenrod algebra and V an elementary abelian pgroup of rank k. We are interested in two problems of a computational nature. The first is the well known problem of calculating Ext_{A(p)}^{s,t}(Z/p, Z/p), the cohomology of the Steenrod algebra. The second problem is that of finding a minimal set of generators for H^{∗}V = H^{∗}(BV; F_{p}) as a module over A(p). This is equivalent to the problem of finding a basis for the F_{p}-vector space

QH^{∗}(k) = H^{∗}V/Ā(p)H^{∗}V = Z/p ⊗_{A(p)} H^{∗}V = Tor_{0,∗}^{A(p)}(Z/p, H^{∗}V)

where Ā(p) is the augmentation ideal of A(p) and, in the right hand expressions, Z/p has the trivial A(p) action.

The importance of the first problem needs no comment, not least because the object in question forms the E_{2} term of the Adams spectral sequence for the stable homotopy groups of spheres.

The importance of the second problem is not so well established but is becoming so. Among other areas, it has implications in modular representation theory [15] and cobordism [9]. However in this paper we are concerned with another of its applications – to the solution of the first problem.

Our main result is the following

Theorem 1. For any prime p, there is a morphism of bi-graded co-algebras

φ : ⨁_{n,k} Tor_{n,n+k}^{A(p)}(Z/p, Z/p) ⟶ ⨁_{n,k}[QH^{n}(k)]^{GL^{k}},

where [QH^{n}(k)]^{GL^{k}} denotes the submodule of elements that are invariant under the action of GL_{k} = GL(k, F_{p}). For k ≤ 2 the component

φ_{k} : ⨁_{n} Tor_{n,n+k}^{A(p)}(Z/p, Z/p) ⟶ ⨁_{n}[QH^{n}(k)]^{GL^{k}},

is an isomorphism.

The co-algebra structure on the bi-graded sum ⊕_{n,k}[QH^{n}(k)]^{GL^{k}} is derived from that on ⊕_{k}H^{∗}V_{k} (V_{k} having rank k), which comes from the natural isomorphisms H^{∗}V_{k} ≈ H^{∗}V_{i} ⊗ H^{∗}V_{j} (i + j = k).

Theorem 1 is then an extension of Propositions 4.2 and 9.14 of [11] which form the p = 2 case of the theorem. We should mention also Boardman’s result [1] that φ_{3} is an isomorphism for p = 2. Of course, it was shown in [11] that the mod 2 transfer map is not an isomorphism in general – φ_{5} has non-zero kernel in degree 14 (which makes one wonder whether φ_{4} is an isomorphism or not)

– and it is reasonable to expect worse behaviour at odd primes. Nevertheless, this map ought to shed some light on the ‘Ext’ groups, Ext_{A(p)}^{s,t}(Z/p, Z/p) which form the range of the dual map. For now, however, we are largely in the dark. For example, it is still unknown why φ_{k} should be an isomorphism at all – the only proofs that we have rely on explicit calculation ([11], [1] and this article) – so we do not know what underlying phenomena are at work here.

Henceforth we will consider only odd primes, that is, p > 2.

Since our proof of Theorem 1 is computational, we need the next result, which itself is not without interest, concerning the structure of QH^{n}(k). Unfortunately, in order to state the theorem, we need to establish some facts about H^{∗}V.

Recall that H^{∗}V = Λ(V^{∗})⊗S(V^{∗}). Λ(V^{∗}) is an exterior algebra on generators of degree 1 while S(V^{∗}) is a symmetric algebra on generators of degree 2. In either case the generators can be identified with a basis for V^{∗}, the dual of V (which can be considered as a vector space over F_{p}). We will now label these generators once and for all. When k = rank V = 1, we denote the exterior generator by u and the symmetric generator by x, requiring that x = βu (β ∈ A(p) being the Bockstein operator). When k = 2, we denote the exterior generators by u, v and the symmetric generators by x, y, requiring that x = βu and y = βv.

Theorem 2. QH^{n}(1) is 0 unless n = 0 or n = 2(i+1)p^{r}−1 where 1 ≤ i ≤ p−1, r ≥ 0, in which case QH^{n}(1) has dimension 1 and a basis given by 1 (if n = 0) or x^{(i+1)p^{r}−1}u (otherwise).

QH^{n}(2) has a basis as given in Table 1 for n > 0, while QH^{0}(2) = F_{p}.

A(p) generators for H^{∗}V and Singer’s homological transfer 3

Table 1.

Degree, n  Basis for QH^{n}(2)
2t, 1 ≤ t ≤ p − 2  {x^{i}y^{t−1−i}uv | 0 ≤ i ≤ t − 1}
2t + 1, 0 ≤ t ≤ p − 1  {x^{i−1}y^{t+1−i}u + x^{i}y^{t−i}v | 0 ≤ i ≤ t + 1}
2t + 1, p ≤ t ≤ 2p − 3  {x^{i−1}y^{t+1−i}u + x^{i}y^{t−i}v | t + 2 − p ≤ i ≤ p − 1}
2((i + 1)p + j + 1)p^{s} − 2  {x^{(k+1)p^{s}−1}y^{(ip+p+j−k)p^{s}−1}uv | min(i + 1, j) ≤ k ≤ p − 1}
0 ≤ i, j ≤ p − 1, s ≥ 0  ∪{x^{(k+1)p^{s+1}−1}y^{(ip−kp+j+1)p^{s}−1}uv | 1 ≤ k ≤ i}
2((i + 1)p^{r} + (j + 1)p^{s}) − 2 {x^{(k+1)p^{s+1}−1}y^{(i+1)p^{r}+(j+1−pk−p)p^{s}−1}uv | 1 ≤ k ≤ p − 1}
1 ≤ i, j + 1 ≤ p − 1,  ∪{x^{(j+1)p^{s}−1}y^{(i+1)p^{r}−1}uv, x^{(i+1)p^{r}−1}y^{(j+1)p^{s}−1}uv}

r − 1 > s ≥ 0

2(i + 1)p^{s} − 1  {x^{(i+1)p^{s}−1−k}y^{k}u + x^{(i+1)p^{s}−k}y^{k−1}v | 1 ≤ k ≤ p − 1}
1 ≤ i ≤ p − 1, s ≥ 1  ∪{x^{(i+1)p^{s}−1}u, y^{(i+1)p^{s}−1}v}
2(p^{2} + ip + j + 1)p^{s} − 2  {x^{(k+1)p^{s+1}−1}y^{(p^{2}+(i−k−1)p+j+1)p^{s}−1}uv | i ≤ k ≤ j}

1 ≤ i ≤ j ≤ p − 2, s ≥ 0

Otherwise  0

That QH^{n}(1) is as claimed is an easy exercise. The statement about QH^{n}(2) is much harder to prove. However, in Theorem 1.2 of [2], we computed the space M(2) whose dual has, as its basis, a minimal set of A(p)-generators for S(V^{∗}), considered as a submodule of H^{∗}V, when k = 2. The methods of [2] were to define a right action of A(p) on homology, using the formula

< ωθ, ζ >=< ω, θζ >

where θ ∈ A(p), ω is a class in homology and ζ is a class in cohomology, and to calculate which homology classes were annihilated by all elements of Ā(p). We can extend these results to obtain the space of annihilated elements in H_{∗}(V) by a brief consideration of the (dualised) action of β. We see that ω is annihilated if ω = wūv̄ for some w ∈ M(2) (where ū, v̄ are the dual exterior generators) or if ω = wū + zv̄ for some w, z ∈ M(2) satisfying wβū + zβv̄ = 0. Then, having calculated the subring of annihilated elements in H_{∗}V, we can compare this space with M(2) and then use the monomial bases for S(V^{∗}) (considered as H^{∗}(CP^{∞} × CP^{∞})) given in [3] to provide convenient bases for H^{∗}V. This programme leads to Theorem 2 without difficulty.

The structure of the paper is as follows : in section 2 we sketch a description of QH^{n}(2) as a GL_{2} module in order to determine the range of φ_{2}. In section 3 we construct the transfer maps φ_{k} and show that their images consist of GL_{k} invariants. Finally in section 4 we show that φ_{1} and φ_{2} are isomorphisms.

I would like to thank Nori Minami for suggesting this topic and for his comments on an earlier version of this article. I am also indebted to Bill Singer for a number of helpful conversations and comments, and to Don Davis for MR 80h 55014. I gratefully acknowledge the hospitality of the Centre de Recerca Matemàtica in Barcelona and the financial support of the CRM and the DGI-CYT. Finally I wish to express my gratitude to the referee for pointing out a number of inaccuracies and for suggesting a number of improvements.

2. QH^{∗}(2) as a GL_{2} module

Since we claim that φ_{2} takes values in QH^{∗}(2)^{GL^{2}}, we need to determine this submodule of invariants. However, with the results of [2], it is not hard to deduce complete composition series for QH^{∗}(2). So, for the sake of completeness, we will list all composition factors occurring in QH^{∗}(2), not just the trivial ones.

We present these in the form of a table (Table 2), designed to be read in conjunction with Table 1.

Table 2.

Degree, n  Composition factors in QH^{n}(2)
2t, 1 ≤ t ≤ p − 2  S^{t−1}(V^{∗}) ⊗ Δ
2t + 1, 0 ≤ t ≤ p − 1  S^{t+1}(V^{∗}) if t < p − 1, S^{1}(V^{∗}), S^{p−2}(V^{∗}) if t = p − 1
2t + 1, p ≤ t ≤ 2p − 3  S^{2p−3−t}(V^{∗}) ⊗ Δ^{⊗t−p+2}
2((i + 1)p + j + 1)p^{s} − 2  S^{j−i−2}(V^{∗}) ⊗ Δ^{⊗i+2}, S^{p−1+i−j}(V^{∗}) ⊗ Δ^{⊗j+1} if i ≤ j
0 ≤ i, j ≤ p − 1, s ≥ 0  S^{i−j−2}(V^{∗}) ⊗ Δ^{⊗j+2}, S^{i−j}(V^{∗}) ⊗ Δ^{⊗j+1},
S^{p−1+j−i}(V^{∗}) ⊗ Δ^{⊗i+1} if j < i
2((i + 1)p^{r} + (j + 1)p^{s}) − 2 S^{i−j}(V^{∗}) ⊗ Δ^{⊗j+1}, S^{p−1+j−i}(V^{∗}) ⊗ Δ^{⊗i+1} if i ≥ j
1 ≤ i, j + 1 ≤ p − 1,  S^{p−1+i−j}(V^{∗}) ⊗ Δ^{⊗j+1}, S^{j−i}(V^{∗}) ⊗ Δ^{⊗i+1} if i < j

r − 1 > s ≥ 0

2(i + 1)p^{s} − 1  S^{i+1}(V^{∗}), S^{p−2−i}(V^{∗}) ⊗ Δ^{⊗i+1}

1 ≤ i ≤ p − 1, s ≥ 1

2(p^{2} + ip + j + 1)p^{s} − 2  S^{j−i}(V^{∗}) ⊗ Δ^{⊗i+1}

1 ≤ i ≤ j ≤ p − 2, s ≥ 0

Otherwise  0

Comments

1. The representations we have used to describe QH^{∗}(2) are essentially those shown in [4] to give a full set of simple GL_{2} modules over F_{p}. These are S^{k}(V^{∗}), the symmetrised k-fold tensor product of copies of V^{∗}, Δ, the dual of the standard 1-dimensional determinant representation, and products of the form S^{k}(V^{∗}) ⊗ Δ^{⊗j}, with 0 ≤ k < p, 0 < j < p. In row 4 we have used the convention that S^{k}(V^{∗}) is 0 if k < 0. Thus the first composition factor of the i ≤ j case of row 4 is only present if i ≤ j − 2.

2. Note that, in each degree, generally only the first listed composition factors occur as submodules; the others can only be found as quotients. The only exceptions to this rule occur in row 5 when j = i and in row 6 when i = p−2. In these cases, QH^{∗}(2) splits as a direct sum of the two factors.

3. A note on the derivation of these results : We see from Theorem 2 that in even degrees, QH^{∗}(2) is obtained from the dual of M(2) (see [2]) by multiplying all elements by uv. As the subspace < uv > is isomorphic to Δ as a GL_{2} space, it follows that composition factors for QH^{2n}(2) are obtained by dualising those of M^{n−1}(2) (given in [2]) and ‘tensor’-ing each term with Δ. In the few odd degrees where QH^{∗}(2) is non-zero, the result follows either by inspection in low degrees, or by comparison with row 5 in high degrees.

4. To complete the picture, QH^{n}(1) is isomorphic to Δ^{⊗i} when n = 2(i+1)p^{r}−1 or Δ^{⊗0} if n = 0.

The problem of finding invariants is now reduced to that of finding trivial submodules in the above table. These usually appear as S^{0}(V^{∗}) ⊗ Δ^{⊗p−1}, for example, the first such submodule (in positive degrees) occurs in row 3 when t = 2p − 3 and the second appears in row 4 when i + 2 = j = p − 1.

Corollary 1. QH^{n}(1)^{GL^{1}} is zero unless n = 0 or n = 2(p − 1)p^{r} − 1 for some r ≥ 0, when it has dimension 1.

QH^{n}(2)^{GL^{2}} is zero unless n is equal to

0,

4(p − 1) − 1,

2(p^{2} − p)p^{s} − 2  for s ≥ 0,
2(p^{2} + p − 2)p^{s} − 2  for s ≥ 0,
2(p − 1)(p^{r} + p^{s}) − 2  for r − 2 ≥ s ≥ 0,
2(p − 1)p^{s} − 1  for s ≥ 1,
or 2(2p^{2} − p − 1)p^{s} − 2 for s ≥ 0,

when it has dimension 1.

3. Construction of φ_{k}

In this section we construct the maps φ_{k} and show that they take values in the submodules of invariants. We will follow closely the programme set out in [11], but assume that p is odd throughout.

Singer’s construction is the paragon of elegant simplicity : choose an element, e_{1}, of Ext_{A(p)}^{1,1}(Z/p, H^{∗}Z/p) (the class of a short exact sequence) and define φ by taking cap products with powers of e_{1}.

Consider the split short exact sequence of F_{p} vector spaces

0 ⟶ H^{∗}Z/p ⟶^{i} T ⟶^{π} Z/p ⟶ 0

where T = H^{∗}Z/p⊕Z/p < x^{−1}u >. We make T into an A(p)-module as follows. If S ⊂ H^{∗}Z/p is the multiplicative subset of powers of x then Proposition 2.1 of [14] assures us that the localisation S^{−1}H^{∗}Z/p has a unique A(p)-algebra structure extending that of H^{∗}Z/p. T is then a subspace of this localisation which is closed with respect to the A(p)-action (but not the product structure) and thus inherits the structure of A(p)-module. Explicitly, the action is given by setting P^{q}(x^{−1}u) = \binom{−1}{q}x^{q(p−1)−1}u and β(x^{−1}u) = 1, while the summand H^{∗}Z/p has its usual A(p) action.

The maps i and π are then A(p)-module homomorphisms and the sequence is still exact, but the A(p) action does not respect the splitting. Let e_{1} denote the class in Ext_{A(p)}^{1,1}(Z/p, H^{∗}Z/p) represented by this short exact sequence.

By the isomorphism H^{∗}V = (H^{∗}Z/p)^{⊗k}, we can consider e_{k} = e_{1} × ⋯ × e_{1} as an element of Ext_{A(p)}^{k,k}(Z/p, H^{∗}V). (The reader who is unfamiliar with such constructions as the cross product in homological algebra is referred to [11] for a detailed summary or [6] for a thorough exposition). We then define

φ_{k} : Tor_{k,n+k}^{A(p)}(Z/p, Z/p) ⟶ Tor_{0,n}^{A(p)}(Z/p, H^{∗}V) = QH^{n}(k)

to be the result of taking cap products with e_{k}, that is, φ_{k}(x) = x ⌢ e_{k}.

Proposition 1. im φ_{k} ⊂ [QH^{∗}(k)]^{GL^{k}}.

Proof. We will prove this by showing that e_{k} is invariant under the action of GL_{k} on Ext_{A(p)}^{k,k}(Z/p, H^{∗}V). It is of course sufficient to check that e_{k} is invariant under the action of a set of generators for GL_{k}. Such a set is given by taking all permutations on k symbols, all transformations t_{1} ↦ λt_{1} for some λ ∈ F_{p} − {0} and the transformation t_{1} ↦ t_{1}, t_{2} ↦ t_{1} +t_{2}. By definition, e_{k} is invariant under permutations.

Consider the transformation t_{1} ↦ λt_{1} which we will consider as an element λ ∈ GL_{1}. The image of e_{1} under the action of this element is given by the bottom row of the following commutative diagram

0 ⟶ H^{∗}Z/p ⟶^{i} T ⟶^{π} Z/p ⟶ 0

λ↓  ↓  ⟨

0 ⟶ H^{∗}Z/p ⟶ P ⟶ Z/p ⟶ 0

where the left hand vertical map denotes the action of λ ∈ GL_{1} on H^{∗}Z/p. The left hand square is a pushout, which comes with a map P ⟶ Z/p making the right hand square commute. (The top line is a representative for e_{1}.)

However, by the universal property of pushouts, it follows that if we can replace P, and the three homomorphisms going from or to P, by any other module Q and triple of homomorphisms, without losing commutativity, then this altered bottom row will be equivalent to the original.

Thus, any short exact sequence

0 ⟶ H^{∗}Z/p ⟶ Q ⟶ Z/p ⟶ 0

and homomorphism T ⟶ Q, which fits into a commutative diagram

0 ⟶ H^{∗}Z/p ⟶^{i} T ⟶^{π} Z/p ⟶ 0

λ↓  ↓  ⟨

0 ⟶ H^{∗}Z/p ⟶ Q ⟶ Z/p ⟶ 0

will also be a representative for λ(e_{1}).

But, if we take the short exact sequence

0 ⟶ H^{∗}Z/p ⟶^{i} T ⟶^{π} Z/p ⟶ 0,

together with the homomorphism T → T given by λ on the summand H^{∗}Z/p and the identity on the summand Z/p < x^{−1}u > (one checks that this is an A(p)- linear map), then we see that this fits in the above diagram. Hence this exact sequence (which is, by definition, a representative of e_{1}) is also a representative for λ(e_{1}) and so λ(e_{1}) = e_{1}.

Finally we consider the transformation t_{1} ↦ t_{1}, t_{2} ↦ t_{1} + t_{2} which we will consider as an element ρ of GL_{2}. By the same argument as above, we see that ρ(e_{2}) = e_{2} because there is a commutative diagram

0 ⟶ H^{∗}Z/p ⊗ H^{∗}Z/p ⟶^{i⊗1} T ⊗ H^{∗}Z/p ⟶^{π⊗i} T ⟶^{π} Z/p ⟶ 0

ρ↓  ρ↓  ⟨  ⟨

0 ⟶ H^{∗}Z/p ⊗ H^{∗}Z/p ⟶^{i⊗1} T ⊗ H^{∗}Z/p ⟶^{π⊗i} T ⟶^{π} Z/p ⟶ 0

This completes the proof of the proposition.

4. φ_{2} is an isomorphism

Finally, in this section, we will complete the proof of Theorem 1 by showing that φ_{1} and φ_{2} are isomorphisms. Of course, in the light of [11], we can (and do) restrict our attention to the case where p is odd.

It will often be more convenient to work dually, with the homomorphism

φ_{k}^{∗} : (QH^{∗}(k)^{GL^{k}})^{∗} ⟶ Ext_{A(p)}^{k,∗+k}(Z/p, Z/p)

since the dual spaces have an algebra structure which, psychologically, is easier to work with than the co-algebra structure present in the Tor groups. In this vein, we quote the following theorem on the modules Ext^{s,t} = Ext_{A(p)}^{s,t}(Z/p, Z/p).

Theorem 3 (Theorem 3.4.2 of [10], Theorem 3.0.1 of [5]). Let p be an odd prime.

1) Ext^{1,∗} has a vector space basis given by a_{0} ∈ Ext^{1,1} and h_{i} ∈ Ext^{1,qp^{i}}, (i ≥ 0), q being the ubiquitous coefficient q = 2(p − 1).

2) Ext^{2,∗} has a vector space basis given by :

a_{0}^{2} ∈ Ext^{2,2},
ρ =< h_{0}, h_{0}, a_{0} > ∈ Ext^{2,2q+1},
λ_{i} =< h_{i}, ⋯^{p factors}, h_{i} > ∈ Ext^{2,qp^{i}p} (i ≥ 0),
μ_{i} =< h_{i}, h_{i}, h_{i+1} > ∈ Ext^{2,q(p+2)p^{i}} (i ≥ 0),
h_{i}h_{j} ∈ Ext^{2,qp^{i}+qp^{j}} (0 ≤ i ≤ j − 2),
a_{0}h_{i} ∈ Ext^{2,qp^{i}+1} (i ≥ 1),
ν_{i} =< h_{i}, h_{i+1}, h_{i+1} > ∈ Ext^{2,q(2p+1)p^{i}} (i ≥ 0).

Now, comparing with Corollary 1, we see that Ext^{k,n+k} and (QH^{n}(k)^{GL^{k}})^{∗} are isomorphic as vector spaces for k ≤ 2. This sets the scene for the proof that φ_{1}^{∗} and φ_{2}^{∗} are themselves isomorphisms. This proof falls naturally into four parts :

Proposition 2. φ is a morphism of (bi-graded) co-algebras.

And so φ^{∗} is multiplicative.

Lemma 1. φ_{1}^{∗} is an isomorphism.

From which it follows, by multiplicativity, that φ_{2}^{∗} is an isomorphism in those degrees where Ext^{2,∗} is spanned by products of elements of Ext^{1,∗}.

Lemma 2. φ_{2}^{∗} maps onto the elements μ_{0}, ν_{0}, ρ and λ_{0}.

Lemma 3. φ_{2}^{∗} maps onto the elements μ_{i}, ν_{i} and λ_{i} for all i.

The proof of our Proposition 2 is a straightforward generalisation to odd primes of that of Singer’s Proposition 4.2, [11].

Lemmas 1 and 2 are proved by direct methods while Lemma 3 hinges on a result of Minami [7].

In order to carry out some direct computations, we need to establish a few facts about φ_{k}^{∗}. Let

⋯ ⟶ C_{2} ⟶ C_{1} ⟶ C_{0} ⟶ Z/p ⟶ 0

be a minimal, free resolution of Z/p over A(p), with C_{i} = ⊕_{S_{i}}A(p), for some (graded) set of generators, S_{i}. It is well known that S_{0} consists of one element, 1, of degree 0, that S_{1} consists of {β, P^{1}, P^{p}, P^{p^{2}}, ⋯} in the obvious degrees (1, 2(p−1), 2p(p−1), 2p^{2}(p−1), ···) and that S_{2} is in 1-1 (graded) correspondence with a minimal set of generators for the ideal of Adem relations. An explicit description of S_{2} can be found in [8], extending Wall’s classic work [13], and we will quote some of the results of [8] shortly.

Because the resolution is minimal, we have Ext^{s,t} = hom_{A(p)}^{t}(C_{s}, Z/p) and Ext^{k,n}(Z/p, H^{∗}V) = hom_{A(p)}^{n}(C_{k}, H^{∗}V) (see §6 of [12] for example). In particular, e_{k} ∈ Ext^{k,k}(Z/p, H^{∗}V) corresponds to a homomorphism ε_{k} : C_{k} → H^{∗}V (of degree k). Since e_{k} is invariant under the action of GL_{k}, it follows that ε_{k} can be considered as a map C_{k} → (H^{∗}V)^{GL^{k}}.

An explicit description of φ_{k}^{∗} can now be given: Let f ∈ (QH^{∗}(k)^{GL^{k}})^{∗}, then φ_{k}^{∗}(f) corresponds to the homomorphism

C_{k} ⟶^{ε_{k}} (H^{∗}V)^{GL^{k}} ⟶ QH^{∗}(k)^{GL^{k}} ⟶^{f} Z/p

where the second map is the restriction of the natural projection from H^{∗}V to QH^{∗}(k) = H^{∗}V/Ā(p)H^{∗}V.

It follows that φ_{k}^{∗} is non-zero (which, for k ≤ 2, is equivalent to φ_{k}^{∗} being an isomorphism) if and only if the image of ε_{k} is not contained in Ā(p)H^{∗}V. In the degrees in which we employ direct methods, Ā(p)H^{∗}V is easily calculated and it will not be difficult to check that im ε_{k} ⊄ Ā(p)H^{∗}V.

We have now assembled the necessary facts to prove Lemma 1. The proof is achieved in a straightforward manner following the recipe set out implicitly above: describe ε_{1} as a homomorphism C_{1} → H^{∗}(Z/p), calculate its image and Ā(p)H^{∗}(Z/p) and observe that the latter does not contain the former.

For Lemma 2 however, we need the explicit description of S_{2} offered by Papastavridis [8]. Since we are only concerned with a handful of degrees (those containing μ_{0}, ν_{0}, ρ and λ_{0}), we will not list the general result. In degrees 4p−3, 2(p^{2} + p − 2), 2(p^{2} − p) and 2(2p^{2} − p − 1), S_{2} contains the following elements :

P^{1}βP^{1} − βP^{2} − P^{2}β,
P^{p}P^{2} + P^{p+2} − P^{p+1}P^{1},
P^{1}P^{p−1},
P^{p}P^{p+1} − P^{2p+1} − P^{2p}P^{1}.

Now we can proceed as in Lemma 1, describing ε_{2} as a homomorphism C_{2} → H^{∗}V, calculating im ε_{2} and Ā(p)H^{∗}V (but only in the particular degrees that pertain to Lemma 2) and showing that (in these degrees) the former is not contained in the latter. The details are almost as straightforward as in the preceding case and so are omitted.

Finally then, we turn to Lemma 3. To prove this, we note from Table 1 of §2.6 of [5] that μ_{i}, ν_{i} and λ_{i} are obtained from μ_{i−1}, ν_{i−1} and λ_{i−1} respectively, by taking representative cocycles, raising these to the power p, and taking their cohomology classes. Then Theorem 4.4 of [7] states that this ‘p-th power’ operation, corresponds, under the (dual) transfer map, to the linear operator f : H_{∗}V → H_{∗}V defined as follows. If V = Z/p then f is given by

f(x_{n}ū^{ϵ}) = ϵx_{pn+p−1}ū

and linearity, where ū ∈ H_{1}(Z/p) is dual to the exterior generator u ∈ H^{1}(Z/p), x_{n} ∈ H_{2n}(Z/p) is dual to x^{n} ∈ H^{2n}(Z/p) and ϵ is equal to 0 or 1. If V has rank k then we identify H_{∗}V with H_{∗}(Z/p) ⊗ ⋯ ⊗ H_{∗}(Z/p) (k factors) and define f in the obvious manner, as the k-fold tensor product of the above map. (N.B. This f is denoted by P^{0} in [7].)

It can be seen from Corollary 3.6 of [2] that f preserves the subring P H_{∗}(k) of H_{∗}V dual to QH^{∗}(k) and so restricts to a map f̄ : P H_{∗}(k) → P H_{∗}(k). Then μ_{i} = φ_{2}^{∗}(f̄^{i}(m_{0})) where m_{0} is the element satisfying μ_{0} = φ_{2}^{∗}(m_{0}), whose existence was shown in Lemma 2, and f̄^{i} is the i-fold iteration of f̄. Thus μ_{i} is in the image of φ_{2}^{∗} and the same argument can be employed to show that ν_{i} and λ_{i} are also in this image. Thus φ_{2}^{∗} is onto which completes the proof of Lemma 3 and hence also that of Theorem 1.

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