THE COHOMOLOGY OF CERTAIN GROUPS

Ian James Leary

January, 1992

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Contents. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Circle Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Cohomology Rings of Various p-Groups . . . . . . . . . . . . . . . . . . 20 The Size of the Chern Subring and its Closure . . . . . . . . . . . . . . . . . 67 Further Calculations at the Prime Three . . . . . . . . . . . . . . . . . . . . 79 The Integral Cohomology of the Held Group . . . . . . . . . . . . . . . . . . 94 Yagita’s Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 The Davis Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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Introduction. This dissertation comprises six chapters which are a improved version of the author’s PhD thesis (submitted in November 1990), together with a short chapter describing more other recent work not directly related to the rest of the thesis. Most of the work contained in this dissertation consists of calculations in the integral and mod-p cohomology rings of specific finite groups. A number of questions concerning group cohomology are resolved by means of these examples, as I shall explain below. I have tried to describe my results in sufficient detail that they may provide a tool for further calculations, which accounts for the length of the statements of some of the theorems. Of the seven chapters of this dissertation, all except the first and sixth involve, to some degree, calculations for specific groups.

Chapter one consists mainly of an explanation of a technique used throughout the calculational chapters. The technique was suggested by J. Huebschmann and P. H. Kropholler, but its subsequent development in this chapter, including its close connection with questions concerning Chern classes and their corestrictions, is my own work. The technique involves embedding a finite group in a compact Lie group whose identity component is a circle. The reason why I have chosen the ‘topologist’s notation’ for the cohomology of a group (as the cohomology of its classifying space) for this dissertation is to avoid confusion between the cases when I consider the cohomology of the circle as a topological space and as a Lie group. The remainder of chapter one consists of a generalisation whereby a finite group is embedded in a Lie group with identity component any torus, and a classification of Lie groups containing the circle as a subgroup of index p^{n} for n ⩽ 3. The difficult cases of this classification are when p = 2 (so that the circle subgroup need not be central) and are not used in later chapters, but are included for completeness.

Chapter six of this dissertation is not very closely related to the other chapters, and might better be described as an appendix, except that I have chosen to reserve that title for the brief discussion of p-groups of small order (which is almost entirely unoriginal). Chapter six concerns an integer invariant of finite groups, due to Yagita [Ya], which is

3 related to free actions of the groups on products of spheres. The original invariant is defined in terms of images of the integral cohomology of the group, and the aim of the chapter is to investigate a similar definition using images of the Chern subring. Since the new invariant may be calculated directly from the complex representation ring of the group, it is not surprising that the invariant bears a relation to linear actions analogous to the relation of the original invariant to free actions.

Chapter two is the first of three chapters devoted to p-groups for odd primes p. The groups considered are the non-abelian groups expressible as central extensions of a cyclic group by the elementary abelian group of order p^{2}. There are two isomorphism classes of such group of given order. I determine, for each such group, its integral cohomology ring and the action of its outer automorphism group upon this ring. These calculations are referred to frequently throughout chapters three, four and five (often without comment). For the non-metacyclic groups I also make similar calculations for the mod-p cohomology rings and determine the action thereon of the Steenrod algebra. Similar calculations could be made for the metacyclic groups, which may also be expressed as split extensions of a cyclic group by a group of order p, but I have not included these. Diethelm has shown that there are (for odd p) only two isomorphism types of ring that occur as mod p cohomology rings of split metacyclic p-groups [Die], so one really ought to examine the actions of the automorphism group and the Steenrod algebra for all split metacyclics, rather than just those with a cyclic subgroup of index p. Neither the method used here, nor Diethelm’s method (which involves consideration of the spectral sequence for the ‘defining extension’) seems suited to this task. I feel that the best approach to this problem will involve the spectral sequence for these groups considered as extensions of their derived subgroups by their abelianisations. Some of the results of chapter two contain as special cases results of other authors; I verify Lewis’ calculations of the integral cohomology rings of non-abelian groups of order p^{3}, Thomas’ result that Chern classes generate the even degree cohomology of each of the groups considered in chapter two, and Thomas’ calculations of the integral

4 cohomology rings of the metacyclic p-groups with a cyclic subgroup of index p. Various preprints of Huynh-Mui dated from 1981 onwards concern the calculation of the modp cohomology ring of the group of order p^{3} and exponent p, but the methods used are different from those used here, and I believe that Huynh-Mui does not give a complete description of the relations between the generators that he uses.

Chapters three and four consist mainly of calculations in the integral cohomology of the p-groups of p-rank two and nilpotency class three. It is these groups (in the case p ⩾ 5) that are shown to provide counterexamples to the conjecture of C. B. Thomas, that for all p-groups of p-rank two Chern classes generate the even degree cohomology, and p-group counterexamples to the conjecture of M. F. Atiyah, that Chern classes generate the E_{∞}- page of the Atiyah-Hirzebruch spectral sequence. These results have also been obtained by N. Yagita using Brown-Peterson cohomology. I also show that for p = 3 Chern classes do generate the even degree cohomology of these groups, and that for p ⩾ 5 corestrictions of Chern classes suffice. These results do not seem to be obtainable from the work of Yagita. In chapter four I calculate the cohomology rings of some of these groups for p = 3, and thus exhibit non-isomorphic groups of order 3^{5} with isomorphic integral cohomology rings. The best result of a similar nature for p ⩾ 5 in chapter four is the existence of non-isomorphic groups of order p^{6} with isomorphic integral cohomology groups.

Chapter five consists of an extended application of the results of chapter two to the study of the Held group, one of the sporadic simple groups [He]. The p-torsion in the integral cohomology of the Held group is determined for all odd primes p, by calculating the image of the restriction map to a Sylow-p subgroup. This provides enough information to determine the E_{2}-page of the Rector spectral sequence [Re]. I intend to examine this spectral sequence in the near future.

Chapter seven of this dissertation contains a description of a method due to Davis [Da] for constructing contractible simplicial complexes with groups acting with finite stabilisers. Among other things this method can yield important cohomological information concerning

5 Coxeter groups. This chapter does not contain any original results however.

I hope that the notation used in this dissertation is not unusual. My notation for cohomology is largely similar to that of [Br] and [Th3], except that I refer to H^{∗}(BG) instead of H^{∗}(G) for reasons explained above. Throughout chapter five much of the notation I use is an approximation to that of [Co]. I follow other authors in using Greek letters to stand for integral cohomology elements and Roman letters to stand for mod-p elements. If g, h are elements of a group G, then h^{g} = g^{−1}hg and [g, h] = g^{−1}g^{h}. If H is a subgroup of G, then c_{g} is the map from H to H^{g} given by h ↦ h^{g}, and c_{g}^{∗} is the induced map in cohomology. I speak of extensions of normal subgroups by quotient groups, and say that these are central if the normal subgroup is contained in the centre of the big group.

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1. The Circle Construction. In this section we describe a method for examining the cohomology of a finite group by embedding it in a compact Lie group of dimension one. This method was suggested by P. Kropholler and J. Huebschmann, and has also been used by B. Moselle [Hu2], [Hu3], [Mo]. We derive general properties of the method, which demonstrate why it is particularly relevant to the study of the Chern subring. We also describe a generalisation of the original method, and classify the compact Lie groups of dimension one and p^{n} components, where n ⩽ 3.

Definition and Properties of G̃. Given a finite group G and a central cyclic subgroup C, we fix an embedding of C into S^{1}, and define

G̃ = S^{1} ×_{C} G.

Then we have a commutative diagram:

C ⟶ G ⟶ Q
↓ ↓ ↓
S^{1} ⟶ G̃ ⟶ Q

If M is a G-module on which C acts trivially, we may consider M as a G̃-module by letting S^{1} act trivially, and the Lyndon-Hochschild-Serre spectral sequence for the second extension is often simpler than that for the first. To find H^{∗}(BG; M), given H^{∗}(BG̃; M), we use the Serre spectral sequence of the fibration

S^{1}/C ≅ G̃/G⟶BG⟶BG̃.  (1)

This spectral sequence has E_{2}^{i,j} = 0 for j > 1, so the only possible non-zero differential is d_{2}. The above was first suggested to the author by P. Kropholler. A similar idea occurs in J. Huebschmann’s papers [Hu2], [Hu3].

7

The Circle Construction

We shall examine the cases in which M is ℤ or 𝔽_{p}, with the trivial G-action. If we let R stand for either ℤ or 𝔽_{p}, then

H^{∗}(S^{1}; R) = Λ[ξ],

the exterior algebra over R on one generator ξ of degree 1, and

H^{∗}(BS^{1}; R) = R[τ],

the polynomial algebra over R on one generator τ of degree 2. The E_{2}^{∗,∗} page of our spectral sequence is given by

E_{2}^{∗,∗} ≅ H^{∗}(BG̃; R) ⊗ Λ[ξ],

except that the product structure is changed by a sign. The effect of this sign change is to make ξ anticommute with elements of H^{∗}(BG̃; R). We shall call this ring the anticommutative ring generated by H^{∗}(BG̃; R) and ξ. It is clear that

d_{2}(ξ) = \overline{τ},

where \overline{τ} is the inflation to G̃ of a generator τ for H^{2}(BG̃/G; R) ≅ H^{2}(BS^{1}; R). This follows by naturality of the spectral sequence with respect to maps of fibrations, in particular for the following commutative diagram.

S^{1}/C ⟶ BG ⟶ BG̃
↓ ↓ ↓
G̃/G ⟶ B{1} ⟶ BG̃/G

Here, all the vertical maps are induced by the quotient map

G̃ ⟶ G̃/G,

and the lower fibration is, up to homotopy, the path-loop fibration over BS^{1}. We may also think of \overline{τ} as the first Chern class of BG considered as a circle or U(1) bundle over BG̃.

8

The Circle Construction

This follows because the first Chern class of the standard S^{1} bundle over BS^{1} generates H^{2}(BS^{1}; R). With integer coefficients there is another interpretation for \overline{τ}. Recall that there is for any compact Lie group K a natural isomorphism between H^{2}(BK; ℤ) and Hom(K, S^{1}). If we use this isomorphism to regard \overline{τ} as a homomorphism from G̃ to S^{1}, its kernel is the subgroup G.

It appears that we may obtain another filtration of H^{∗}(BG; R) by examining the Eilenberg-Moore spectral sequence for the pullback square:

BG ⟶ BG̃
↓ ↓
{∗} ⟶ BG̃/G

The E_{2}^{∗,∗} page of this spectral sequence is

E_{2}^{∗,∗} ≅ Tor_{R[τ]}^{∗,∗}(R, H^{∗}(BG̃; R)).

To calculate E_{2}^{∗,∗}, we may construct a 2-stage free R[τ] resolution of R by

0⟶R[τ]⟶^{×τ}R[τ]⟶R

and we see that

Tor_{R[τ]}^{i,∗}(R, H^{∗}(BG̃; R)) ≅ \{ \begin{matrix}H^{∗}(BG̃; R)/\overline{τ}H^{∗}(BG̃; R) & for i = 0 \\ Ker × \overline{τ} : H^{∗}(BG̃; R) → H^{∗}(BG̃; R) & for i = 1 \end{matrix}

It is now easy to see that this spectral sequence collapses, and that its E_{2}^{∗,∗} page gives the same filtration of H^{∗}(BG; R) as the Lyndon-Hochschild-Serre spectral sequence for the fibration (1). Of course, these spectral sequences are just alternative ways to view the Gysin sequence for the S^{1} bundle BG. We now prove various results concerning G̃. We assume in the statements that G is expressed as a central extension of C by Q, and that G̃ is the corresponding extension of S^{1} by Q. LEMMA 1·1. Given G as above, consider the spectral sequence of the fibration (1) with coefficients in the trivial G-module R.

9

The Circle Construction

a) For R = ℤ, the spectral sequence does not collapse.

b) For R = 𝔽_{p}, the spectral sequence collapses iff C is not contained in G^{p}G′.

Proof. By G^{p} we mean the subgroup of G generated by pth powers of elements of G. If we let ξ be a generator of H^{1}(S^{1}; R), then as above, in each case E_{2}^{∗,∗} is the anticommutative ring generated by H^{∗}(BG̃; R) and ξ, and the spectral sequence collapses if and only if d_{2}(ξ) = 0. G is finite, so H^{1}(BG; ℤ) = 0, so in case a) d_{2} : E_{2}^{0,1} → E_{2}^{2,0} must be injective.

Hom(G, 𝔽_{p}) ≅ H^{1}(BG; 𝔽_{p}) ≅ E_{3}^{0,1} ⊕ E_{3}^{1,0},
Hom(G, 𝔽_{p}) ≅ H^{1}(BG; 𝔽_{p}) ≅ E_{2}^{1,0},

and the map E_{2}^{1,0} ↠ E_{3}^{1,0} corresponds to restriction from G̃ to G. E_{3}^{0,1} is either C_{p} or 0, so the spectral sequence collapses iff there are homomorphisms from G to 𝔽_{p} which cannot be extended to G̃. A homomorphism f : G → 𝔽_{p} can be extended to G̃ iff f(C) = 0, and this will be true for all f iff CG′/G′ is contained in (G/G′)^{p} iff C is contained in G^{p}G′. ■

LEMMA 1·2. For n ⩾ 2, the inclusion of G in G̃ restricts to an isomorphism from Γ_{n}(G) to Γ_{n}(G̃), and from G^{(n−1)} to G̃^{(n−1)}.

Proof. Any element of G̃ can be expressed as a product xg, where x is central, and g is in the image of G. In any group, if x and x′ are central, then for all g, g′ we have [xg, x′g′] = [g, g′]. ■

COROLLARY 1·3. G is nilpotent (resp. soluble) if and only if G̃ is, and if so they have the same nilpotency class (resp. soluble length). ■

LEMMA 1·4. The following conditions on G and G̃ are equivalent:

a) S^{1} ↣ G̃ ↠ Q is split;

b) The extension class of G in H^{2}(BQ; C) is in the kernel of the Bockstein from H^{2}(BQ; C) to H^{3}(BQ; ℤ) associated to the short exact sequence ℤ ↣ ℤ ↠ C;

10

The Circle Construction

c) There is an extension \overline{G} of ℤ by Q and a map \overline{G} ↠ G such that the following diagram commutes:

ℤ ⟶ \overline{G} ⟶ Q
↓ ↓ ↓
C ⟶ G ⟶ Q.

Proof. Using the classification of extensions up to equivalence, see for example [Th3], we see that a) holds if and only if the extension class of G in H^{2}(BQ; C) maps to zero under the map to H^{2}(BQ; S^{1}) induced by the inclusion of C in S^{1}. To verify that this is equivalent to b), we consider the following map of short exact sequences:

ℤ ⟶ ℤ ⟶ C
↓ ↓ ↓
ℤ ⟶ ℝ ⟶ S^{1}.

The Bockstein is natural for maps of short exact sequences, so we obtain the following commutative diagram:

H^{2}(BQ; C) ⟶ H^{3}(BQ; ℤ)
↓ ↓
H^{2}(BQ; S^{1}) ⟶ H^{3}(BQ; ℤ).

The group Q is finite, so the lower map is an isomorphism, and hence a) and b) are equivalent. The equivalence of b) and c) follows from the cohomology long exact sequence for the coefficient sequence ℤ ↣ ℤ ↠ C. ■

LEMMA 1·5. Any complex representation of G extends to one of G̃.

Proof. Given ρ : G → Aut(V), an irreducible representation of G, ρ must restrict to C as scalar multiplication, because an eigenspace for C in V would be a G submodule. Hence ρ may be extended to S^{1}, then to G̃ by defining:

ρ(g) = ρ(x)ρ(h)  where g = xh for x ∈ S^{1} and h ∈ i(G).  ■

The close connection between the representation ring of a group and the K theory of its classifying space suggest the following :

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The Circle Construction

LEMMA 1·6. K^{∗}(BG̃) restricts onto K^{∗}(BG). Similarly, if K(n)^{∗} is the nth Morava K theory, and K(n)^{od}(BG) is trivial, then K(n)^{∗}(BG̃) restricts onto K(n)^{∗}(BG). Proof. The generalised Atiyah-Hirzebruch spectral sequence for the fibration

BG → BG̃ → BS^{1}

has E_{2}^{∗,∗} page H^{i}(BS^{1}; H^{j}(BG)), converging to a filtration of H^{i+j}(BG̃), and the fibre map is the restriction. If H is K theory, or if H is Morava K theory and G satisfies the condition above, then E_{2}^{i,j} is trivial if i or j is odd, so the spectral sequence collapses. The statement that this condition holds for all finite G is known as Ravenel’s conjecture. ■ LEMMA 1·7. The subring Ch(G̃) of the integral cohomology ring of G̃ generated by Chern classes maps onto Ch(G) under the restriction. Similarly \overline{Ch}(G̃) maps onto \overline{Ch}(G), where \overline{Ch}(H) is the subring of the integral cohomology generated by corestrictions of Chern classes. Proof. The first statement follows immediately from lemma 1·5. For the second statement, we note that a subgroup of G̃ of finite index must contain S^{1}, so is of the form H̃ for some subgroup H of G, so the image of \overline{Ch}(G̃) is contained in \overline{Ch}(G). To show that the image is all of \overline{Ch}(G), we note that the following diagram commutes:

Ch(H̃) ⟶^{Cor} H^{ev}(BG̃)
↓^{Res} ↓^{Res}
Ch(H) ⟶^{Cor} H^{ev}(BG)

This is because H̃ ∩ G = H, and H̃G = G̃, so we only need one double coset in the restriction-corestriction formula (see [Br]). ■ LEMMA 1·8. H^{n+1}(BG̃; ℤ) restricts onto H^{n+1}(BG; ℤ) if and only if multiplication by c(G), the first Chern class of BG as an S^{1} bundle over BG̃, is injective on H^{n}(BG̃; ℤ). Proof. Consider the spectral sequence for the fibration

G̃/G⟶BG⟶BG̃.
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The Circle Construction

The cokernel of the restriction map from H^{n+1}(BG̃; ℤ) to H^{n+1}(BG; ℤ) is isomorphic to E_{∞}^{n,1}, which is isomorphic to the kernel of multiplication by c(G) in H^{n}(BG̃; ℤ). ■ THEOREM 1·9. H^{ev}(BG) = Ch(G) (resp. H^{ev}(BG) = \overline{Ch}(G)) if and only if H^{ev}(BG̃) = Ch(G̃) (resp. H^{ev}(BG̃) = \overline{Ch}(G̃)) and multiplication by c(G) is injective on H^{od}(BG̃). Proof. By lemma 1·8 the condition on c(G) is equivalent to H^{ev}(BG̃) maps onto H^{ev}(BG) by the restriction. The two conditions on G̃ therefore imply that Ch(G̃) (resp. \overline{Ch}(G̃)) restricts onto H^{ev}(BG)), but by lemma 1·7 the image is Ch(G) (resp. \overline{Ch}(G)). Conversely, by lemma 1·7, the condition on G implies that H^{ev}(BG̃) restricts onto H^{ev}(BG).

If H^{ev}(BG̃) restricts onto H^{ev}(BG) = Ch(G), we prove by induction that Ch^{2n}(G̃) = H^{2n}(BG̃; ℤ). For any compact Lie group H, H^{2}(BH; ℤ) = Ch^{2}(H), so without loss of generality we may assume that n is at least 2. Lemma 1·7 tells us that Ch^{2n}(G̃) maps onto Ch^{2n}(G), so Ch^{2n}(G̃) maps onto E_{3}^{2n,0} in the spectral sequence for the S^{1} bundle BG. Therefore, given x ∈ H^{2n}(BG̃; ℤ), we may pick y ∈ H^{2n−2}(BG̃; ℤ) and z ∈ Ch^{2n}(G̃) such that x = z + c(G)y. Hence x ∈ Ch^{2n}(G̃). The proof for the case of \overline{Ch}(G̃) is similar. ■ A Generalisation of the Circle Construction. Let G an extension of A ≅ (C_{m})^{n} by Q such that the ‘action map’ φ : Q → GL_{n}(ℤ_{/m}) lifts to a map \overline{φ} from Q to GL_{n}(ℤ). We may use this map to define an action of Q on the n dimensional torus, 𝕋^{n}, which restricts to the original action on a subgroup of 𝕋^{n} isomorphic to A, which we will identify with A. We may then form the semidirect product 𝕋^{n} ⋊ G, that is the manifold 𝕋^{n} × G with multiplication defined by

(x, g)(y, h) = (xy^{φ̄π(g^{−1})}, gh).

The set B = {(a^{−1}, a)|a ∈ A} is a normal subgroup, and we define

Ĝ = 𝕋^{n} ⋊_{A} G = 𝕋^{n} ⋊ G/B.

G is a subgroup of Ĝ, so there is a fibration

𝕋^{n}/A ≅ Ĝ/G ⟶ BG ⟶ BĜ,
13

The Circle Construction

but the fundamental group of BĜ (which is of course isomorphic to Q) will in general act non-trivially on the cohomology of the coset space Ĝ/G. In fact the action of π_{1}(BG) on H^{1}(Ĝ/G; ℤ) is exactly the action of Q on ℤ^{n} given by φ̄. If we write l( ) for the soluble length, and Q is soluble, then clearly the following inequalities hold.

l(Q) ⩽ l(G) ⩽ l(Ĝ) ⩽ l(Q) + 1

There is however no analogue of lemma 1·2, and Ĝ is not in general nilpotent. Ĝ depends on the choice of lifting φ̄, which will not be unique. Even if we take G to be a group of order 4 expressed as an extension of C_{2} by C_{2}, we may obtain Ĝ isomorphic to either S^{1} × C_{2} or the orthogonal group O_{2} depending on our choice of φ̄ from C_{2} to {±1}.

There is an analogue of lemma 1·4, where we replace C by A, and ℤ by ℤ^{n} considered as a ℤQ module via φ̄, with exactly the same proof. Not all representations of G will extend to Ĝ, although in each particular case it seems easy to describe the ones that will. Lemma 1·6 breaks down because G is not necessarily normal in Ĝ. In spite of all these shortcomings I believe that Ĝ will prove useful in the study of various groups. For example, there is a family of 3-groups of 3-rank two, which may be presented as split extensions of C_{3^{t}} ⊕ C_{3^{t}} by C_{3}, where the action is induced from a non-trivial action of C_{3} on ℤ^{2}. These groups are nilpotent of class 2t, so are unsuited to study via central extensions. Classification of Circle groups on p^{n} components. Here we classify Lie groups G with maximal connected subgroup is S^{1} and group of components of order dividing p^{3}. Aut(S^{1}) has order two, so for p odd S^{1} is central in G. Central extensions of S^{1} by H are classified up to equivalence by H^{2}(BH; S^{1}), which can be identified with H^{3}(BH; ℤ) via the Bockstein for the sequence ℤ ↣ ℝ ↠ S^{1}. In each of the cases we examine, either H^{3}(BH; ℤ) is trivial, or the action of Aut(H) is transitive on the non-zero elements, so there is at most one non-split extension. The groups H for which H^{3}(BH; ℤ) is non-trivial are (C_{p})^{2}, (C_{p})^{3}, C_{p^{2}} × C_{p}, the dihedral group D_{8} and P_{2}, the non-abelian group of exponent p (see appendix). For odd p, section 2 contains sufficient

14

The Circle Construction

information to check the third cohomology group of the groups H. The case when p = 2 may be calculated similarly, or see [CE] for Q_{8} and [Ev2] for D_{8}. The split extensions are just direct products S^{1} × H, and the non-split extensions are as described below. The comments indicate why they are not isomorphic to the split extensions.

H ≅ (C_{p})^{2}. H^{3} ≅ 𝔽_{p}. The non-split extension is not abelian, and may be obtained from either of the non-abelian groups G of order p^{3} as G̃.

H ≅ C_{p^{2}} × C_{p}. H^{3} ≅ 𝔽_{p}. As above the non-split extension is not abelian, and may be obtained from any of the non-abelian extensions of C_{p} by C_{p^{2}} × C_{p} using the circle construction.

H ≅ (C_{p})^{3}. H^{3} ≅ 𝔽_{p}^{3}. The non-split extension is the direct product of C_{p} and the nonsplit extension of S^{1} by (C_{p})^{2}. This follows from the fact that any element of H^{3}(C_{p}^{3}) will restrict trivially to some C_{p} subgroup.

H ≅ P_{2}. H^{3} ≅ 𝔽_{p} ⊕ 𝔽_{p}. The non-split extension has nilpotency class three, and may be obtained from any of the groups of order p^{4} and nilpotency class three via the circle construction.

H ≅ D_{8}. H^{3} ≅ 𝔽_{2}. The generator restricts non-trivially to either C_{2} × C_{2} subgroup (this may be verified readily using the spectral sequence for this extension), so the non-split extension has a non-abelian subgroup of index two.

Our approach to the classification of the non-central extensions is slightly different. If G is a non-central extension of S^{1} by a group of order 2^{n}, let K be the centraliser in G of S^{1}. Then K has index 2, and the isomorphism type of K as a group equipped with a map from C_{2} to Out(K) is an invariant of the group. For fixed K such a structure is determined by an involution in Out(K), and an isomorphism between structures specified by involutions t_{1} and t_{2} is determined by an element s of Out(K) such that st_{1} = t_{2}s. Thus the first stage of our classification of non-central extensions of S^{1} by a group of order 2^{n} is to take

15

The Circle Construction

each K, a central extension of S^{1} by a group of order 2^{n−1}, and find all conjugacy classes of involutions in Out(K) which restrict non-trivially to S^{1}. We shall classify only groups with 2, 4, and 8 components, so we need consider only one case in which K is non-abelian, that when K is the non-split central extension of S^{1} by (C_{2})^{2}. First we shall consider the case when K is abelian. In this case Out(K) = Aut(K), and if we pick a representative for each of our conjugacy classes of involution, these define distinct ℤC_{2}-module structures on K containing the submodule Ŝ^{1}, that is S^{1} with the non-trivial C_{2}-action. For each such M we calculate H^{2}(BC_{2}; M) and the action of Aut_{ℤC_{2}}(M) (which is isomorphic to the centraliser in Aut(K) of the involution used to define M) upon it. Finally we decide whether or not elements in distinct Aut(M) orbits really do give non-isomorphic groups. If K is not abelian, we pick a representative for each conjugacy class of involution in Out(K) restricting non-trivially to S^{1}, and use each of these to define a ℤC_{2}-module structure M′ on the centre Z(K) of K. Either there are no extensions of K by C_{2} corresponding to this involution, or such extensions are classified up to equivalence by H^{2}(BC_{2}; M′) (see [Br]), depending on the vanishing of an obstruction in H^{3}(BC_{2}; M′). (In the only case we need the obstruction group is trivial.) Once again we are left to determine whether or not inequivalent extensions really give non-isomorphic groups. There are sufficiently few cases to examine when K is non-abelian that ad hoc methods suffice. In the abelian case the following lemma, suggested to the author by R. E. Borcherds, is useful.

LEMMA 1·10. If H is a finite group with a fixed map from C_{p} to Aut(H), then a non-split extension of H by C_{p} and the split extension cannot be isomorphic.

Proof. If G is an extension of H by C_{p}, it splits if and only if there is an element of order p in G \ H, so a non-split extension has the same number of elements of order p as H, whereas a split extension has more. ■ This lemma also applies to the case when H is a compact abelian Lie group of dimension one, because these contain only finitely many elements of order p.

16

The Circle Construction

The non-central extensions of S^{1} by H are described below. We consider S^{1} as the complex numbers of unit modulus, with automorphism z ↦ z̄.

H of order 2. In this case M must be S^{1} with the non-trivial C_{2} action. H^{2}(BC_{2}; M) ≅ 𝔽_{2}, and we obtain two groups.

\begin{aligned}D̂ & = ⟨S^{1}, T | T^{2} = 1, z^{T} = z̄⟩ \\ Q̂ & = ⟨S^{1}, T | T^{2} = −1, z^{T} = z̄⟩ \end{aligned}

D̂ is isomorphic to O_{2}, and Q̂ is isomorphic to the subgroup of SU_{2} of elements of the forms

( \begin{matrix}z & 0 \\ 0 & z̄ \end{matrix})  and  ( \begin{matrix}0 & z \\ −z̄ & 0 \end{matrix} ).

H of order 4. Aut(S^{1} × C_{2}) ≅ C_{2} × C_{2}, so there are two C_{2} structures on S^{1} × C_{2} restricting non-trivially to S^{1}. Considering C_{2} as a subgroup of ℂ^{×}, the two actions are

\begin{matrix}M_{1} & (z, a) & ↦ (z̄, a) \\ M_{2} & (z, a) & ↦ (z̄a, a) \end{matrix}

H^{2}(BC_{2}; M_{1}) ≅ (𝔽_{2})^{2}, but two elements are exchanged by Aut(M_{1}), and we obtain three groups, with presentations

⟨S^{1}, A, T | A^{2} = T^{2} = 1 z^{A} = z A^{T} = A z^{T} = z̄⟩
⟨S^{1}, T | T^{4} = 1 z^{T} = z̄⟩
⟨S^{1}, A, T | A^{2} = 1 T^{2} = −1 z^{A} = z A^{T} = A z^{T} = z̄⟩
H^{2}(BC_{2}; M_{2}) ≅ 𝔽_{2}, and so we obtain two groups.
⟨S^{1}, A, T | A^{2} = T^{2} = 1 z^{A} = z A^{T} = −A z^{T} = z̄⟩
⟨S^{1}, T | T^{4} = −1 z^{T} = z̄⟩

Of these five groups, three have quotient isomorphic to C_{2} × C_{2}, and two C_{4}. 17

The Circle Construction

We do not give presentations for the twenty-one non-central extensions of S^{1} by groups of order eight, but we list the possible ‘modules’ (we shall refer to a non-abelian K equipped with a choice of involution in Out(K) as a C_{2}-‘module’), and then tabulate the groups that arise for each of them. In the abelian cases lemma 1·10 together with the calculation of the isomorphism types of the finite quotients suffices to show that extensions whose classes are in distinct Aut(M) orbits in H^{2}(BC_{2}; M) do give rise to distinct groups.

Aut(S^{1} × C_{4}) is a group of order 16 containing seven conjugacy classes of involution, four of which restrict non-trivially to S^{1}. Considering C_{4} as contained in ℂ^{×}, the following four modules are representatives for the distinct isomorphism types.

\begin{matrix}M_{1} & (z, a) & ↦ (z̄, a) \\ M_{2} & (z, a) & ↦ (z̄a, a) \\ M_{3} & (z, a) & ↦ (z̄, a^{−1}) \\ M_{4} & (z, a) & ↦ (z̄a^{2}, a^{−1}) \end{matrix}

Aut(S^{1} × C_{2}^{2}) is isomorphic to C_{2} × S_{4}, so has five conjugacy classes of involutions, three of which restrict non-trivially to S^{1}. The following are representatives of each class.

\begin{matrix}M_{5} & (z, a, b) & ↦ (z̄, a, b) \\ M_{6} & (z, a, b) & ↦ (z̄, b, a) \\ M_{7} & (z, a, b) & ↦ (z̄ab, a, b) \end{matrix}

There is also the non-split central extension of S^{1} by C_{2}×C_{2}, which we shall refer to as Q̃, since it could be obtained from the quaternion group by applying the circle construction. It may be presented as follows.

Q̃ = ⟨S^{1}, A, B|A^{2} = B^{2} = 1 z^{A} = z^{B} = z [A, B] = −1⟩

Inn(Q̃) is isomorphic to (C_{2})^{2}, Out(Q̃) is isomorphic to C_{2} × S_{3}, and Aut(Q̃) is the split extension of (C_{2})^{2} by C_{2} × S_{3} where C_{2} acts trivially and S_{3} acts as GL_{2}(𝔽_{2}). It follows that any involution in Out(Q̃) will give rise to a split extension of Q̃ by C_{2}. There are two

18

The Circle Construction

conjugacy classes of involution in Out(Q̃) which restrict non-trivially to Aut(S^{1}), and we may choose as lifts to Aut(Q̃) of representatives of them the following elements.

\begin{aligned}t_{1} & : z ↦ z̄ \\ t_{2} & : z ↦ z̄ \end{aligned}  \begin{matrix}A & ↦ A \\ A & ↦ B \end{matrix}  \begin{matrix}B & ↦ B \\ B & ↦ A \end{matrix}

For each of these the ℤC_{2}-module structure of Z(Q̃) is isomorphic to Ŝ^{1}, the non-trivial ℤC_{2}-module structure on S^{1}. Since H^{2}(BC_{2}; Ŝ^{1}) has order 2 there are in each case two equivalence classes of extensions of Q̃ by C_{2}. In each case these two groups can be shown to be distinct by comparing the orders of elements in the components of the group not contained in Q̃.

The results of the above calculations are contained in figure 1·1. Note that in each case there is a split extension, and the isomorphism type of the finite quotient of the split extension is listed first.

Module  |H^{2}(BC_{2}; M)|  Extensions  Quotients
M_{1}  4  3  C_{2} × C_{4}, C_{2} × C_{4}, C_{8}
M_{2}  2  2  C_{2} × C_{4}, C_{8}
M_{3}  4  3  D_{8}, D_{8}, Q_{8}
M_{4}  2  2  D_{8}, Q_{8}
M_{5}  8  3  (C_{2})^{3}, (C_{2})^{3}, C_{2} × C_{4}
M_{6}  2  2  D_{8}, D_{8}
M_{7}  2  2  (C_{2})^{3}, C_{2} × C_{4}
Ŝ^{1} = Z(Q̃) and t_{1}  2  2  (C_{2})^{3}, (C_{2})^{3}
Ŝ^{1} = Z(Q̃) and t_{2}  2  2  D_{8}, D_{8}

1·1. The non-central extensions of S^{1} by groups of order eight.

We summarise our calculations in the following theorem.

19

The Circle Construction

THEOREM 1·11. For p an odd prime there are 8 (resp. 3, 1) groups consisting of p^{3} (resp. p^{2}, p) circles. There are 29 (resp. 8, 3) groups consisting of 8 (resp. 4, 2) circles, but in only 8 (resp. 3, 1) of these is the S^{1} subgroup central. ■ Remarks. It is desirable to have a stronger result than lemma 1·10, but informed opinion seems to be that for a fixed map from Q to Aut(N) it may be possible for the split extension and a non-split extension to be isomorphic groups. No examples seem to be known for Q and N both finite, but J. C. Rickard suggested the following. PROPOSITION 1·12. Let N be ∏_{n=1}^{∞} C_{4} × ∏_{n=1}^{∞} C_{2}, and let Q be C_{2} acting trivially on N. Then

H^{2}(BQ; N) ≅ ∏_{n=1}^{∞} H^{2}(BC_{2}; C_{4}) × ∏_{n=1}^{∞} H^{2}(BC_{2}; C_{2}),

and any extension G whose class restricts to zero in ∏_{n=1}^{∞} H^{2}(BC_{2}; C_{4}) is isomorphic to the split extension. Proof. The split extension and all such G are isomorphic to N. ■

20

2. The Cohomology Rings of Various p-Groups. In this section we shall calculate the integral and mod-p cohomology rings of an infinite family of p-groups for odd p, which we shall call P (n). The group P (n) is defined for each n ⩾ 3, has order p^{n}, and may be presented as below.

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

P (3) is the non-abelian group of order p^{3} and exponent p, which we also refer to as P_{2}, and P (4) is the second group on Burnside’s list of groups of order p^{4} (see appendix). The centre of P (n) is the subgroup generated by C of index p^{2}. Applying the circle construction described in section 1 to the whole centre we obtain the same group for all n, the unique non-abelian group consisting of p^{2} circles, which we shall call P̃. The metacyclic groups with a cyclic subgroup of index p which we shall refer to as M(n), where p^{n} is the order, also occur as subgroups of P̃, but their cohomology rings have been determined by other means [Th3], so we consider them only briefly. The Calculation of H^{∗}(BP̃; ℤ) We now begin our calculation of H^{∗}(BP̃; ℤ) by examining the spectral sequence with integer coefficients for P̃ considered as an extension of S^{1} by C_{p} ⊕ C_{p}. The E_{2} page is readily seen to be generated by elements α, β ∈ E_{2}^{2,0}, γ ∈ E_{2}^{3,0} and τ ∈ E_{2}^{0,2} subject only to the relations pα = pβ = 0, pγ = 0 and γ^{2} = 0. Note that τ has infinite order. Since E_{2}^{i,j} is trivial for j odd, we see that all the even differentials must vanish. The behaviour of the differentials is summarised in the following lemma. LEMMA 2·1. In the above spectral sequence there are exactly two non-zero differentials, d_{3} and d_{2p−1}. d_{3}(τ) is a non-zero multiple of γ, and E_{4} is generated by the classes of the elements α, β, pτ, …, pτ^{p−1}, τ^{p} and τ^{p−1}γ (see figure 2·1). All of these generators are universal cycles except for τ^{p−1}γ, which is mapped by d_{2p−1} to a non-zero multiple of α^{p}β − β^{p}α. The E_{∞} page is generated by the elements α, β, pτ, …, pτ^{p−1}, τ^{p} subject only to the relations they satisfy as elements of E_{2}, and the relation α^{p}β = β^{p}α.

21

The Cohomology Rings of Various p-Groups

\begin{matrix}ζ & − & ζα,ζβ & − & \begin{matrix}ζα^{2},ζαβ, \\ ζβ^{2} \end{matrix} \\ − & − & − & − & − \\ χ_{p−1} & − & − & η & − & ηα,ηβ \\ χ_{2} \\ − \\ χ_{1} \\ − \\ 1 & − & α,β & − & α^{2},αβ,β^{2} & − & \begin{matrix}α^{3},α^{2}β, \\ αβ^{2},β^{3} \end{matrix} \end{matrix}

2·1. The E_{4} page of the spectral sequence of lemma 2·1.

Proof. The derived subgroup of P̃ consists of the subgroup of its central S^{1} of order p, so there can be no homomorphism from P̃ to S^{1} that restricts to an isomorphism from the centre to S^{1}. It follows by considering the natural isomorphism H^{2}(BG; ℤ) ≅ Hom(G, S^{1}) that the element τ cannot survive to E_{∞}, so we must have d_{3}(τ) a non-zero multiple of γ. This determines d_{3} completely. It may be checked that E_{4} is isomorphic to the subring of E_{2} generated by α, β, pτ, …, pτ^{p−1}, τ^{p} and τ^{p−1}γ. All these elements must be universal cycles, with the possible exception of τ^{p−1}γ, because the groups in which their images under d_{n} lie are already trivial. (The E_{4} page of the spectral sequence is depicted in figure 2·1, where χ_{i} = pτ^{i}, ζ = τ^{p}, and η = τ^{p−1}γ.) The only remaining potentially

22

The Cohomology Rings of Various p-Groups

non-zero differential is d_{2p−1}(τ^{p−1}γ). To complete this proof it suffices to show that in the E_{∞} page the relation α^{p}β = β^{p}α must hold.

Let Q be the quotient of P̃ by its S^{1} subgroup, and take generators α′, β′ for H^{2}(BQ; ℤ) and γ′ for H^{3}(BQ; ℤ). The statement that γ does not survive to E_{∞} in the spectral sequence is equivalent to the statement that γ′ is mapped to zero by the inflation map from Q to P̃. Now we calculate φ(γ′), where φ is the integral cohomology operation δ_{p}P^{1}π_{∗}, where π_{∗} is the map induced by the change of coefficients from ℤ to 𝔽_{p}, P^{1} is a reduced power, and δ_{p} is the Bockstein for the sequence ℤ ↣ ℤ ↠ 𝔽_{p}. Taking y, y′ ∈ H^{1}(BQ; 𝔽_{p}) such that δ_{p}(y) = α′, and δ_{p}(y′) = β′, we see that

φ(γ′) = δ_{p}P^{1}π_{∗}(γ′) = δ_{p}P^{1}(β_{p}(y)y′ − β_{p}(y′)y) = δ_{p}(β_{p}(y)^{p}y′ − β_{p}(y′)^{p}y) = α^{′p}β′ − β^{′p}α′.

It follows that

α^{p}β − β^{p}α = Inf(α^{′p}β′ − β^{′p}α′) = Inf(φ(γ′)) = φInf(γ′) = 0.  ■

We are now ready to state our theorem on H^{∗}(BP̃). THEOREM 2·2. Let p be an odd prime, and let P̃ be the group defined above. Then H^{∗}(BP̃; 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}

23

The Cohomology Rings of Various p-Groups

Chern classes of representations of P̃ generate the whole ring. An automorphism of P̃ sends χ_{i} to χ_{i} (resp. (−1)^{i}χ_{i}) and ζ to ζ (resp. −ζ) if it fixes (resp. reverses) S^{1}. The effect of an automorphism on α, β may be determined from their definition. Considered as elements of Hom(P̃, S^{1}), α has kernel ⟨S^{1}, B⟩ and sends A to e^{2πi/p}, and β has kernel ⟨S^{1}, A⟩ and sends B to e^{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 generator τ. Similarly, ζ = c_{p}(ρ), where ρ is an irreducible representation of P̃ restricting to S^{1} as p copies of the representation ξ with c_{1}(ξ) = τ. Proof. First we note that in the E_{∞} page of the above spectral sequence all the group extensions that we need to examine are extensions of finite groups by the infinite cyclic group, so are split. The elements α and β defined in the statement above clearly yield generators for E_{∞}^{2,0}, and the relations between them are exactly the relations that hold between the corresponding elements in the spectral sequence. Let β′ in H^{2}(BH) be the restriction to H of β, and take any choice of τ′ as in the statement. We may show by considering β′ and τ′ as homomorphisms from H to S^{1} that conjugation by A^{i} induces the map on H^{2}(BH) that fixes β′ and sends τ′ to τ′ − iβ′. Now applying the formula for Res_{K}^{G}Cor_{H}^{G} (see for example [Br]) it follows that χ_{i} restricts to S^{1} as pτ^{i}, so yields a generator for E_{∞}^{0,2i}.

Any irreducible representation of P̃ has degree 1 or p, because P̃ has an abelian subgroup of index p. Let ρ be the representation of P̃ induced from a 1-dimensional representation of H with first Chern class τ′. ρ restricts to S^{1} as p copies of the representation with first Chern class τ, so its total Chern class restricts to S^{1} as (1 + τ)^{p}, and so c_{p}(ρ) yields a generator for E_{∞}^{0,2p}, and c_{i}(ρ) = 1/p\binom{p}{i}χ_{i} + P_{i}(α, β) for some polynomial P_{i}. We shall show later that P_{i} = 0.

24

The Cohomology Rings of Various p-Groups

The restriction to H of α is trivial, so by Fröbenius reciprocity

αCor_{P} ; H(τ^{′i}) = Cor_{P} ; H(Res_{P} ; H(α)τ^{′i}) = 0,

and the expressions given for αχ_{i} follow. By calculating α(βχ_{i}) = β(αχ_{i}), we may deduce that βχ_{i} = 0 for i < p−1 and βχ_{p−1} = λ(α^{p−1}β −β^{p})−α^{p−1}β for some scalar λ. To show that λ = 1 we use the restriction map to H, and the formula for corestriction followed by restriction.

\begin{aligned}Res_{P} ; H(βχ_{p−1}) & = β′ ∑_{i=0}^{p−1}(τ′ + iβ′)^{p−1} \\ & = β′ ∑_{j=0}^{p−1} τ^{′p−1−j}β^{′j} ∑_{i=0}^{p−1} i^{j} \end{aligned}

Newton’s formula tells us that

∑_{i=1}^{p−1} i^{j} ≡ \{ \begin{matrix}0 (p) & for j ≢ 0 (p − 1) \\ 1 (p) & for j ≡ 0 (p − 1) \end{matrix}

so Res_{P} ; H(βχ_{p−1}) = −β^{′p}, and the required relation follows.

We now know Res_{P} ; S^{1}(χ_{i}χ_{j}), αχ_{i}χ_{j}, and βχ_{i}χ_{j}, which together imply the relations given for χ_{i}χ_{j}. To complete the proof of the theorem we must determine the effect of automorphisms of P̃ on the χ_{i}. We know that an automorphism sends c_{i}(ρ) to itself or (−1)^{i} times itself depending whether or not it reverses the sense of S^{1}, so it will suffice to show that χ_{i} = 1/p\binom{p}{i}c_{i}(ρ). The character of ρ is zero except on S^{1}, so if θ is a 1dimensional representation of P̃ restricting trivially to S^{1}, then ρ ⊗ θ is isomorphic to ρ. If we apply the formula expressing c.(ρ ⊗ θ) in terms of c.(ρ) and c.(θ) (see [At]) we obtain

c_{i}(ρ) = c_{i}(ρ ⊗ θ) = ∑_{j=0}^{i} \binom{p − i + j}{j}c_{1}(θ)^{j}c_{i−j}(ρ).

and hence inductively

c_{i}(ρ)c_{1}(θ) = \{ \begin{matrix}0 & for i < p − 1 \\ −c_{1}(θ)^{p} & for i = p \end{matrix} .

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The Cohomology Rings of Various p-Groups

Since α and β are possible values for c_{1}(θ) the required result follows. We may show inductively that χ_{i} is in the subring generated by Chern classes because χ_{1} is, and χ_{1}χ_{i−1}, 1/p\binom{p}{i}χ_{i} are coprime multiples of χ_{i}. ■

We are now ready to state our theorem on the integral cohomology of P (n). THEOREM 2·3. Let p be an odd prime and let P (n) be as defined above. Then H^{∗}(BP (n); ℤ) is generated by elements α, β, μ, ν, χ_{1}, …, χ_{p−1}, ζ, with

deg(α) = deg(β) = 2 deg(μ) = deg(ν) = 3 deg(χ_{i}) = 2i deg(ζ) = 2p

subject to the following relations:

pα = pβ = 0 pμ = pν = 0 p^{n−3}χ_{1} = 0 p^{n−2}χ_{i} = 0 p^{n−1}ζ = 0
αμ = βν
α^{p}β = β^{p}α α^{p}μ = β^{p}ν

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}

μχ_{i} = \{ \begin{matrix}0 \\ −β^{p−1}μ \end{matrix} νχ_{i} = \{ \begin{matrix}0 & for i < p − 1 \\ −α^{p−1}ν & 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}

μν = \{ \begin{matrix}0 & for n > 3 \\ λχ_{3} & for n = 3, p > 3, λ ∈ 𝔽_{p}^{×} \\ 3λζ & for n = 3, p = 3, λ = ±1 \end{matrix}

Chern classes of representations of P (n) generate H^{ev}(BP (n); ℤ). Under an automorphism of P (n) which restricts to the centre as C ↦ C^{j}, χ_{i} is mapped to j^{i}χ_{i}, and ζ is mapped to j^{p}ζ. The effect of automorphisms on α and β is determined by the natural isomorphism H^{2}(BG; ℤ) ≅ Hom(G, ℝ/ℤ), under which

\begin{matrix}α : A & ↦ 1/p \\ B & ↦ 0 \\ C & ↦ 0 \end{matrix} \begin{matrix}β : A & ↦ 0 \\ B & ↦ 1/p \\ C & ↦ 0 \end{matrix} \begin{matrix}χ_{1} : A & ↦ 0 \\ B & ↦ 0 \\ C & ↦ 1/p^{n−3}. \end{matrix}
26

The Cohomology Rings of Various p-Groups

An automorphism of P (n) which sends α to n_{1}α+n_{2}β, β to n_{3}α+n_{4}β and restricts to the centre as C ↦ C^{j} sends μ to j(n_{4}μ+n_{3}ν) and ν to j(n_{2}μ+n_{1}ν). If γ′ in H^{2}(B⟨B, C⟩; ℤ) is such that it maps to the following element of Hom(⟨B, C⟩, ℝ/ℤ)

\begin{matrix}γ′ : B & ↦ 0 \\ C & ↦ 1/p^{n−2} \end{matrix} ,

then χ_{i} is defined as follows:

χ_{i} = \{ \begin{matrix}Cor_{⟨B,C⟩}^{P (n)}(γ^{′i}) & for i < p − 1 \\ Cor_{⟨B,C⟩}^{P (n)}(γ^{′p−1}) − α^{p−1} & for i = p − 1. \end{matrix}

These are, up to scalar multiples, equal to c_{i}(ρ), where ρ is a p-dimensional irreducible representation of P (n), whose restriction to ⟨C⟩ is a sum of p copies of the representation θ, with c_{1}(θ) = Res_{⟨C⟩}^{⟨B,C⟩}(γ′). In fact, c_{i}(ρ) = 1/p\binom{p}{i}χ_{i}. Also, we may define ζ = c_{p}(ρ).

Proof. We examine the spectral sequence for BP (n) as an S^{1}-bundle over BP̃. E_{2}^{∗,0} is isomorphic to H^{∗}(BP (n); ℤ) and E_{2}^{∗,∗} is freely generated by E_{2}^{∗,0} and an element ξ of infinite order in E_{2}^{0,1}. We know that H^{2}(BP (n)) ≅ Hom(P (n), S^{1}) ≅ C_{p^{n−3}} ⊕ C_{p} ⊕ C_{p}, so d_{2}(ξ) must be ±p^{n−3}χ_{1}. If we wanted to calculate the cohomology of the metacyclic groups M(n) described above, the differential in this spectral sequence would send ξ to ±p^{n−3}χ_{1} + γ for some non-zero γ in ⟨α, β⟩. It is now easy to see that E_{∞} is generated by the elements α, β, μ = βξ, ν = αξ, χ_{1}, …, χ_{p−1} and ζ subject to the relations they satisfy as elements of E_{2}^{∗,∗} together with p^{n−3}χ_{1} = 0, p^{n−2}χ_{i} = 0, and p^{n−1}ζ = 0. For each m, the filtration of H^{m}(BP (n)) given by the E_{∞} page is trivial, so we may use the same symbols to denote elements of H^{m}(BP (n)), and the relations that hold in E_{∞} determine all the relations that hold in H^{m}(BP (n)) except for the product of the two odd dimensional generators.

We know that pμν = 0, and the relation αμ = βν implies that αμν = βμν = 0, and so μν must be a multiple of p^{n−3}χ_{3} for p ⩾ 5 (resp. 3^{n−2}ζ for p = 3). Note that these elements restrict to zero on all proper subgroups of P (n). In the case of P (3), Lewis [Lew]

27

The Cohomology Rings of Various p-Groups

shows that μν is not zero by considering the spectral sequence for P (n) considered as an extension of a maximal subgroup by C_{p}. A similar method will work in general, but we offer an alternative proof that involves expressing μ and ν as Bocksteins of elements of H^{2}(BP (n); 𝔽_{p}). This proof is contained in lemma 2·4 and corollary 2·5.

The effect of automorphisms on χ_{i} and ζ is easily seen to be as claimed from their alternative definitions as Chern classes. To determine the effect of automorphisms on μ and ν, we note that an automorphism of P (n) restricting to the centre as C ↦ C^{j} extends to an endomorphism of P̃ which wraps the central circle j times around itself, so induces a map of the above spectral sequence to itself sending ξ to jξ. This completes the proof of theorem 2·3 modulo lemma 2·4 and its corollary. ■

We now examine the spectral sequence with 𝔽_{p} coefficients for the central extension C_{p^{n−2}} ↣ P (n) ↠ C_{p} ⊕C_{p}. Take generators so that H^{∗}(BC_{p} ⊕C_{p}; 𝔽_{p}) ≅ 𝔽_{p}[x, x′]⊗Λ[y, y′], where β_{p}(y) = x, β_{p}(y′) = x′, and H^{∗}(BC_{p^{n−2}}; 𝔽_{p}) ≅ 𝔽_{p}[t]⊗Λ[u], where β_{p}(u) = t for n = 3 (resp. β_{p}(u) = 0 for n ⩾ 4). Then the E_{2} page is isomorphic to 𝔽_{p}[x, x′, t] ⊗ Λ[y, y′, u], and the first two differentials are as described in the following lemma.

LEMMA 2·4. With notation as above, identify the elements x, x′, y, y′ in the spectral sequence with their images in H^{∗}(BP (n); 𝔽_{p}) under the inflation map.

1) Let n ⩾ 4. Then d_{2} is trivial, and d_{3}(t) is a non-zero multiple of xy′ − x′y. The elements x, x′, yy′, u′y, u′y′, form a basis for H^{2}(BP (n)), where u′ is any element of H^{1}(BP (n)) restricting to C_{p^{n−2}} as u.

2) Let n = 3. Then d_{2}(u) is a non-zero multiple of yy′, d_{2}(t) = 0, and E_{3} is generated by y, y′, x, x′, [uy], [uy′] and t subject to the relation yy′ = 0 and those implied by the relations in E_{2}. In particular [uy]y′ = −[uy′]y but this element is non-zero. As in the case n ⩾ 4, d_{3}(t) is a non-zero multiple of xy′ − x′y. Let Y, Y′ be elements of H^{2}(BP (3)) such that x, x′, Y, Y′ form a basis for H^{2}(BP (3)), and let X = β_{p}(Y), X′ = β_{p}(Y′). Then yY′, xy, xy′, x′y′, X, X′, form a basis for H^{3}(BP (3)) and xX, xX′, x′X′, x^{2}y, x^{2}y′, xx′y′, x^{′2}y′, Y X′, form a basis for H^{5}(BP (3)). The E_{4} page of this spectral sequence is depicted in figure

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2·2, where boldface numerals indicate numbers of generators required in the case p > 3, and plain numerals the case p = 3.

\begin{matrix}− & − & 1 1 & 1 1 & 0 2 & 2 2 \\ − & [t^{2}y],[t^{2}y′] & − & 3 4 & − & 4 6 \\ − & − & [uy][ty′] & 1 1 & − & 2 2 \\ − & [ty],[ty′] & − & \begin{matrix}x[ty],x[ty′], \\ x′[ty′] \end{matrix} & − & 4 4 \\ − & [uy],[uy′] & [uy]y′ & \begin{matrix}x[uy],x[uy′], \\ x′[uy],x′[uy′] \end{matrix} & − & 6 6 \\ 1 & y,y′ & x,x′ & \begin{matrix}xy,xy′, \\ x′y′ \end{matrix} & \begin{matrix}x^{2},xx′, \\ x^{′2} \end{matrix} & \begin{matrix}x^{2}y,x^{2}y′, \\ xx′y′,x^{′2}y′ \end{matrix} \end{matrix}

2·2. The E_{4} page of the spectral sequence of lemma 2·4.

Proof. 1). In this case H^{1} has order p^{3}, so u must survive. The element xy′ − x′y is the image under π_{∗} of a generator for H^{3}(B(C_{p} ⊕ C_{p}); ℤ), so must be killed by some differential. We have already shown that it cannot be killed by d_{2}, so the only possibility is that t survives until E_{3} and kills it. The rest of the statement follows easily.

2). In this case H^{1} has order p^{2}, so d_{2}(u) must be non-zero. It is true in general that if G is a central extension of C_{p} by Q, then in the corresponding spectral sequence with 𝔽_{p} coefficients d_{2} : E_{2}^{0,1} → E_{2}^{2,0} must kill the extension class. This follows by naturality, since one may regard the extension class as defining a homotopy class of maps from BQ to K(C_{p}, 2) such that BG is the BC_{p}-bundle induced by the path-loop fibration over K(C_{p}, 2). Since all subgroups of P (3) of order p^{2} are copies of C_{p} ⊕ C_{p}, the extension class of P (3) must restrict to zero on all cyclic subgroups, so must be a multiple of yy′. The transgression commutes with the Bockstein so d_{2}(t) = 0 and d_{3}(t) = β_{p}d_{2}(u).

Given the values of these differentials it is routine to compute the E_{4} page of the 29

The Cohomology Rings of Various p-Groups

spectral sequence. If we write E_{r}^{n} = ⨁_{i+j=n} E_{r}^{i,j}, then {[uy], [uy′], x, x′} forms a basis for E_{4}^{2} = E_{∞}^{2}, and {[ty], [ty′], [uy]y′, xy, xy′, x′y′} forms a basis for E_{4}^{3} = E_{∞}^{3}. The spectral sequence operation _{F} β introduced by Araki [Ar] and Vasquez [Va] maps [uy] to [ty] and [uy′] to [ty′], so if Y and Y′ are chosen to yield the generators for E_{4}^{1,1} their Bocksteins yield generators for E_{4}^{1,2}. A basis for E_{4}^{5} is given by the eight elements of the statement, which we know to be universal cycles, and the elements [t^{2}y], [t^{2}y′]. E_{4}^{4} consists of universal cycles, and the universal coefficient theorem tells us that H^{5} has order p^{8}, so [t^{2}y] and [t^{2}y′] cannot be universal cycles. ■

The author has determined all the differentials in the above spectral sequence, using theorems 2·14 and 2·15. The non-trivial differentials are d_{2}, d_{3}, d_{4}, d_{2p−2}, and d_{2p−1}. The E_{∞} page of this spectral sequence in the case p = 7 is depicted in figure 2·4 (on page 65).

COROLLARY 2·5. In H^{∗}(BP (n); ℤ) the product μν is non-zero if and only if n = 3.

Proof. In the notation of lemma 2·4 it suffices to determine δ_{p}(u′y)δ_{p}(u′y′) in the case n ⩾ 4, and δ_{p}(Y)δ_{p}(Y′) in the case n = 3. In the case when n = 3,

δ_{p}(Y)δ_{p}(Y′) = δ_{p}(Y β_{p}(Y′)) = δ_{p}(Y X′).

The kernel of δ_{p} : H^{5}(BP (3); 𝔽_{p}) → H^{6}(BP (3); ℤ) is equal to π_{∗}(H^{5}(BP (3); ℤ)), which is generated by xX, xX′ and x′X′, so by lemma 2·4 δ_{p}(Y X′) is non-zero.

In the case when n = 4, H^{i}(BP (4); ℤ) has exponent p for i = 2, 3, so π_{∗} is injective from these groups, and Kerβ_{p} : H^{2}(BP (4)) → H^{3}(BP (4)) is equal to β_{p}(H^{1}(BP (4))). β_{p}(yy′) = xy′−x′y = 0, so we may choose the element u′ in lemma 2·4 so that β_{p}(u′) = λyy′ for some non-zero λ. Then we have

δ_{p}(u′y)δ_{p}(u′y′) = δ_{p}(u′yβ_{p}(u′y′)) = δ_{p}(u′y(λyy′y′ − u′x′)) = 0.

The case when n ⩾ 5 is similar but simpler, since u′ may be chosen so that δ_{p}(u′) = p^{n−4}χ_{1}, which implies that β_{p}(u′) = 0. ■

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Remarks. The author can not think of a method to determine the non-zero constant λ in the equation for μν in the case λ = 3. If we replace the generators α and ν by λ^{−1}α and λ^{−1}ν respectively and leave the other generators unchanged, the new generating set will satisfy exactly the same relations as the old ones, except that the constant λ is replaced by 1.

Theorem 2·3 contains independent proofs of Thomas’ result that the even degree subring of H^{∗}(BP (n); ℤ) is generated by Chern classes [Th2], and Lewis’ calculation of H^{∗}(BP (3); ℤ) [Lew]. Our notation differs slightly from that of Lewis. We have renumbered the generators χ_{i} (note that χ_{1} vanishes for n = 3). Also our χ_{p−1} and Lewis’ χ_{p−2} are related by the formula

χ_{p−2}^{Lewis} = χ_{p−1} + α^{p−1} + β^{p−1}.

Our result disagrees with that of AlZubaidy [Al2]. We now use a similar method to obtain the result of Lewis [Lew] and Thomas [Th1] concerning the metacyclic groups M(n), which may be presented as follows.

M(n) = ⟨A, B | A^{p} = B^{p^{n−1}} = 1 [B, A] = B^{p^{n−2}}⟩

THEOREM 2·6. Let M(n) be the metacyclic p-group defined above. Then H^{∗}(BM(n); ℤ) is generated by elements α, χ_{1}, …, χ_{p−1}, ζ, η, with

deg(α) = 2 deg(χ_{i}) = 2i deg(ζ) = 2p deg(η) = 2p + 1

subject to the following relations:

pα = 0 p^{n−2}χ_{i} = 0 p^{n−1}ζ = 0 pη = 0

αχ_{i} = \{ \begin{matrix}0 \\ −α^{p} \end{matrix} ηχ_{i} = \{ \begin{matrix}0 & for i < p − 1 \\ −α^{p−1}η & 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} & i = j = p − 1 \end{matrix}

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The even-dimensional generators are multiples of Chern classes of irreducible representations of M(n). Considered as elements of Hom(G, ℝ/ℤ), α sends A to 1/p and B to 0, and χ_{1} sends A to 0 and B to 1/p^{n−2}. Under an automorphism of M(n) which sends α to n_{1}α + n_{2}p^{n−3}χ_{1} and restricts to the centre as B^{p} ↦ B^{jp}, χ_{i} is mapped to j^{i}χ_{i}, ζ to j^{p}ζ, and η to jn_{1}η. Let H be the subgroup of M(n) generated by B. If β′ is the element of H^{2}(BH; ℤ) mapping B to 1/p^{n−1}, then we may define

χ_{i} = \{ \begin{matrix}Cor_{H}^{M(n)}(β^{′i}) & for i < p − 1, \\ Cor_{H}^{M(n)}(β^{′p−1}) − α^{p−1} & for i = p − 1. \end{matrix}

Proof. As in the case of P (n), we consider the spectral sequence for BM(n) as an S^{1}- bundle over BP̃. The E_{2} page is as in theorem 2·3, and we may take d_{2}(ξ) = β − p^{n−3}χ_{1}. Since A is in the kernel of β this is consistent with the relation A^{p} = 1 in M(n). The E_{∞} page is seen to be generated by the elements α, χ_{i}, ζ, and ξα(α^{p−1} − β^{p−1}), subject to the relations they satisfy as elements of E_{2} together with p^{n−2}χ_{i} = 0, p^{n−1}ζ = 0. These relations completely determine those that hold in H^{∗}(BM(n); ℤ). The action of Aut(M(n)) is determined as in theorem 2·3. ■ Remarks. Ths result confirms Thomas’ calculation of H^{∗}(BM(n); ℤ) [Th1] which generalised Lewis’ calculation for H^{∗}(BM(3); ℤ) [Lew], but note that our generator in degree 2p − 2 is not the usual one. Their method is simpler than the one used here (if one is not also interested in P (n)), but does not seem to yield the action of Aut(M(n)), which the author believes to be a new result. The method involves considering M(n) as an extension of C_{p^{n−1}} by C_{p}. The C_{p^{n−1}} subgroups of M(n) are not characteristic however, so Aut(M(n)) does not act on this extension, or on the corresponding spectral sequence.

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The Cohomology Rings of Various p-Groups

Calculation of H^{∗}(BP̃; 𝔽_{p}).

The Spectral Sequence. We proceed as for the corresponding calculations in integral cohomology. We start by examining the spectral sequence with mod-p coefficients for the central extension:

S^{1} ↣ P̃ ↠ C\overline{A} ; p ⊕ C\overline{B} ; p.

If we write

H^{∗}(BC\overline{A} ; p ⊕ C\overline{B} ; p; 𝔽_{p}) = 𝔽_{p}[x, x′] ⊗ Λ[y, y′],

where β(y) = x and β(y′) = x′, and

H^{∗}(BS^{1}; 𝔽_{p}) = 𝔽_{p}[t]

then we see that:

E_{2}^{i,j} = 𝔽_{p}[x, x′, t] ⊗ Λ[y, y′].

We must have d_{3}(t) = λ(xy′ − x′y) for some non-zero λ, and so E_{4}^{∗,∗} is generated by elements y, y′, x, x′, [t^{p}], [tyy′], …, [t^{p−1}yy′], and [t^{p−1}(xy′ − x′y)], subject to the relations that they satisfy in E_{2}^{∗,∗}, together with the relations

t^{i}xy′ = t^{i}x′y for i ≢ −1 (p).

If we write z for [t^{p}], c_{i} for [t^{i−1}yy′], and d for [t^{p−1}(xy′ − x′y)], then E_{4}^{∗,∗} is as shown in figure 2·3.

Using the universal coefficient theorem, and our calculation of H^{∗}(BP̃; ℤ), we find that

|H^{2i}(BP̃; 𝔽_{p})| = |H^{2i−1}(BP̃; 𝔽_{p})| = p^{i+2} for i = 1, …, p − 1,

and we deduce that c_{i} is a universal cycle for i ⩽ p − 1, and hence E_{2p−1}^{∗,∗} ≅ E_{4}^{∗,∗}.

The mod-p reduction of ζ ∈ H^{2p}(BP̃; ℤ) restricts non-trivially to S^{1}, so z is a universal cycle. Applying P^{1}, and βP^{1} to xy′ − x′y, we obtain:

\begin{aligned}0 & = P^{1}(xy′ − x′y) = x^{p}y′ − x^{′p}y, \\ 0 & = β(x^{p}y′ − x^{′p}y) = x^{p}x′ − x^{′p}x. \end{aligned}

33

\begin{matrix}z & zy,zy′ & \begin{matrix}zx,zx′, \\ zyy′ \end{matrix} & \begin{matrix}zxy,zxy′, \\ zx′y′ \end{matrix} & ... \\ − \\ − & − & c_{p} & d & c_{p}x,c_{p}x′ & dx,dx′ & ... \\ − \\ − & − & c_{p−1} \\ − & − & c_{2} \\ − \\ 1 & y,y′ & x,x′ & \begin{matrix}xy,xy′, \\ x′y′ \end{matrix} & \begin{matrix}x^{2},xx′, \\ x^{′2},yy′ \end{matrix} \end{matrix}

2·3. The E_{4} page of the spectral sequence discussed above.

We know that z, z(λy + μy′) are universal cycles, so the only way the above relations can be introduced into E_{∞}^{∗,∗} is if

\begin{aligned}d_{2p−1}(c_{p}) & = λ(x^{p}y′ − x^{′p}y) \\ d_{2p−1}(d) & = λ′(x^{p}x′ − x^{′p}x) \end{aligned}

for some non-zero scalars λ, λ′.

Now it is easy to see that E_{2p}^{∗,∗} ≅ E_{∞}^{∗,∗}, and is generated by elements y, y′, x, x′, z, c_{2}, …, c_{p−1}, with bidegrees as above, subject to the relations:

y^{2} = y^{′2} = 0 xy′ = x′y

x^{p}y′ = x^{′p}y x^{p}x′ = x^{′p}x
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c_{i}y = c_{i}y′ = 0  c_{i}x = c_{i}x′ = 0  for all i
c_{i}c_{j} = 0 for all i, j.

Multiplicative Relations. We use the notation π_{∗} and δ_{p} introduced during our integral calculations for the maps

\begin{matrix}π_{∗} : H^{n}(BG; ℤ) & ⟶ H^{n}(BG; 𝔽_{p}) \\ δ_{p} : H^{n}(BG; 𝔽_{p}) & ⟶ H^{n+1}(BG; ℤ). \end{matrix}

It is easy to see that we may choose the elements π_{∗}(α), π_{∗}(β), π_{∗}(χ_{i}), and π_{∗}(ζ) respectively to yield the generators x, x′, c_{i}, and z of E_{∞}^{∗,∗}. We define generators of H^{∗}(BP̃; 𝔽_{p}) by:

\begin{aligned}x & = π_{∗}(α) \\ x′ & = π_{∗}(β) \\ c_{i} & = π_{∗}(χ_{i}) for i = 2, …, p − 1 \\ z & = π_{∗}(ζ). \end{aligned}

Then elements y, y′ are uniquely defined by the equations:

β(y) = x β(y′) = x′.

The relations between these generators follow from the spectral sequence and the above definitions, and we obtain the following theorem, which has also been obtained independently by Moselle [Mo]. THEOREM 2·7. 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}
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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} .

An automorphism of P̃ sends c_{i} to c_{i} (resp. (−1)^{i}c_{i}) and z to z (resp. −z) if it fixes (resp. reverses) S^{1}. The effect of an automorphism on y, y′, x, x′ may be determined by their definitions. Using the natural isomorphism H^{1}(BP̃; 𝔽_{p}) ≅ Hom(P̃, 𝔽_{p}), we define:

\begin{matrix}y : A & ↦ 1 (p) \\ B & ↦ 0 (p) \end{matrix} \begin{matrix}y′ : A & ↦ 0 (p) \\ B & ↦ 1 (p), \end{matrix}

and then define x and x′ by the equations:

x = β(y) x′ = β(y′).

We may define c_{i} (resp. z) to be the image under π_{∗} of χ_{i} (resp. ζ), as defined during the statement of theorem 3. π_{∗}(χ_{1}) is λyy′ for some non-zero λ. Proof. We have done most of this already. We should check that the two definitions of x, as π_{∗}(α) and β(y), agree. It suffices for this to check that δ_{p}(y) = α, because β = π_{∗}δ_{p}. This is true by naturality of the Bockstein for maps of short exact sequences of coefficient modules, in particular for the map:

ℤ ⟶^{×p} ℤ ⟶ 𝔽_{p}
↓ ↓^{×1/p} ↓
ℤ ⟶ ℚ ⟶ ℚ/ℤ.

Also, we note that δ_{p} : H^{2i+1}(BP̃; 𝔽_{p}) → H^{2i+2}(BP̃; ℤ) is injective for all i, and δ_{p}(c_{i}y) = χ_{i}x, so the relations involving c_{i} and y, y′ follow. ■

Much of the action of the Steenrod algebra A_{p} on H^{∗}(BP̃; 𝔽_{p}) is determined implicitly by theorem 2·7. For example c_{i} is expressible as Cor_{P} ; H(t^{i}) (−x^{p−1} when i = p−1), so P^{1}(c_{i}) is easily calculated. An example of this calculation occurs in the proof of theorem 2·14. The only further piece of information required to specify the action of A_{p} is P^{1}(z), which does not follow immediately from theorem 2·7, but may be detemined as follows.

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PROPOSITION 2·8. With notation as in theorem 2·7,

P^{1}(z) = zc_{p−1}.

Proof. Throughout this proof K shall stand for any of the p + 1 subgroups of P̃ of index p (each of which is isomorphic to S^{1}⊕C_{p}), and Res = Res_{P} ; K. Let H^{∗}(BK; 𝔽_{p}) = 𝔽_{p}[t, x̄]⊗Λ[ȳ], where t restricts to S^{1} as the mod-p reduction of the standard generator of H^{2}(BS^{1}; 𝔽_{p}) (called τ in the proof of theorem 2·2), and β(ȳ) = x̄. There are p − 1 possible choices for ȳ (and hence for x̄), and p choices for t. If ρ is the representation of P̃ mentioned in the statement of theorem 2·2, then

Res(π_{∗}c.(ρ)) = ∏_{i=0}^{p−1}(1 + t + ix̄) = 1 − x̄^{p−1} + t^{p} − x̄^{p−1}t,

and this expression is independent of the choice of t and x̄. It follows that

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

It follows that Res(P^{1}(z)) = x̄^{2p−2}t − x̄^{p−1}t^{p}. A typical element of H^{4p−2}(BP̃; 𝔽_{p}) is of the form λzc_{p−1} + zP (x, x′) + Q(x, x′). We shall show that zc_{p−1} is the only such element restricting correctly to each K.

First, note that Res(Q(x, x′)) = λ′x̄^{2p−1}, and that λ′ = 0 for K = ⟨S^{1}, AB^{i}⟩ (resp. K = ⟨S^{1}, B⟩) iff x′ − ix (resp. x) divides Q. Hence x^{p}x′ − x^{′p}x must divide Q, so Q is zero in H^{∗}(BP̃). To complete the proof it suffices to show that for each λ′′ ∈ 𝔽_{p}, no polynomial P (x, x′) homogeneous of degree p − 1 can restrict to all K as λ′′x̄^{p−1}. For every choice of K except ⟨S^{1}, A⟩ we may choose x̄ to be Res(x′). We have p such K, and x will restrict to each of them as a distinct multiple of x̄. Hence P as above would have to satisfy P (X, 1) − λ′′ = 0 for all X ∈ 𝔽_{p}, but P (X, 1) cannot have p roots. ■

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Calculation of H^{∗}(BP (n); 𝔽_{p}) for n ⩾ 4.

The Spectral Sequence. We follow the method used in the corresponding calculations for integral cohomology, and consider the spectral sequence of the fibration:

S^{1}/Z(P (n)) ⟶ BP (n) ⟶ BP̃(n).

This has

E_{2}^{i,j} ≅ H^{i}(BP̃; H^{j}(S^{1}; 𝔽_{p})),

so if we set

H^{∗}(S^{1}; 𝔽_{p}) = Λ[u],

then E_{2}^{∗,∗} is the anticommutative ring generated by H^{∗}(BP̃; 𝔽_{p}) and u. We may apply lemma 1·1, and we deduce that the spectral sequence collapses iff n > 3. For n > 3 the relations in H^{∗}(BP (n); 𝔽_{p}) follow easily from those in H^{∗}(BP̃; 𝔽_{p}). In the case n = 3, d_{2}(u) = λ(yy′) for some non-zero λ, and so E_{∞}^{∗,∗} requires many generators which appear in E_{∞}^{∗,1}. There is a corresponding increase in the number of elements required to generate H^{∗}(BP (3); 𝔽_{p}) and consequently in the number of relations we must calculate. We are forced to use new techniques, which shall be introduced in the next chapter.

THEOREM 2·9. Let n be at least 4. Then H^{∗}(BP (n); 𝔽_{p}) is generated by elements u, y, y′, x, x′, c_{2}, …, c_{p−1}, z with

deg(u) = 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}

β(u) = \{ \begin{matrix}y′y & for n = 4 \\ 0 & for n > 4 \end{matrix}

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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} .

Under an automorphism of P (n) which restricts to the centre as C ↦ C^{j}, c_{i} is mapped to j^{i}c_{i} and z is mapped to jz. We define u, y, y′ by regarding them as elements of Hom(P (n), 𝔽_{p}) then

\begin{matrix}u : A & ↦ 0 \\ B & ↦ 0 \\ C & ↦ 1 \end{matrix}  \begin{matrix}y : A & ↦ 1 \\ B & ↦ 0 \\ C & ↦ 0 \end{matrix}  \begin{matrix}y′ : A & ↦ 0 \\ B & ↦ 1 \\ C & ↦ 0 \end{matrix} .

This determines the effect of automorphisms on u, y, y′ and on x = β(y), x′ = β(y′). We may define

c_{i} = π_{∗}(χ_{i}) and z = π_{∗}(ζ).

Proof. Almost all of this follows from our work above. We may, of course choose u to be any element of H^{1}(BP (n); 𝔽_{p}) not in the span of y, y′. With our definition it follows that δ_{p}(u) = p^{n−4}χ_{1}, so we obtain

β(u) = \{ \begin{matrix}λyy′ & for some λ ≠ 0 for n = 4 \\ 0 & for n > 4. \end{matrix}

It can be checked by explicit calculation with 1- and 2-cochains that in the case n = 4

β(u) = y′y = −yy′,

and hence also that for n = 4

π_{∗}(χ_{1}) = y′y.

We leave the details to the interested reader. ■

Note that the action of A_{p} on H^{∗}(BP (n); 𝔽_{p}) for n ⩾ 4 is determined completely by information contained in theorem 2·9 and proposition 2·8.

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The Cohomology Rings of Various p-Groups

The Massey Product. This section is not original, but it introduces techniques we shall use to construct various explicit elements in H^{∗}(BP_{2}; 𝔽_{p}), and to find relations between them. Throughout this section R will be a commutative ring on which π_{1}(X) acts trivially, and C˙ (X; R) the singular cochain complex of X with coefficients in R. Any cochains or cohomology classes represented by a single letter will be homogeneous, and we shall write, for example (−1)^{u} for (−1)^{deg(u)}. In particular, (−1)^{uv} shall mean (−1)^{deg(u)deg(v)}, not (−1)^{deg(uv)}. Definition. Let u, v, w be elements of C˙ (X; R), with

[uv] = 0 [vw] = 0 in H^{∗}(X; R).

Then choose a ∈ C^{u+v−1}(X; R) and b ∈ C^{v+w−1}(X; R) with

δa = uv δb = vw,

and define the Massey product ⟨[u], [v], [w]⟩ by

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

The Massey product is only well-defined modulo uH^{v+w−1}(X; R) + wH^{u+v−1}(X; R) because of the freedom of choice of a and b. It is easily seen to be linear in each of its arguments. Before stating some more properties enjoyed by the Massey product we state a relation due to Hirsch [Hi] between the cup-0 and cup-1 products. We recall that a cup-0 product is a product on cochains inducing the cup product on cohomology, for example the standard product defined by the Alexander-Whitney formula. A cup-1 product is a natural cochain transformation of degree −1

⌣_{1}: (C˙ ( )^{⊗2}) ⟶ C˙ ( ),

satisfying the following ‘coboundary formula’ for all (homogeneous) cochains a and b:

δ(a ⌣_{1} b) = −δa ⌣_{1} b − (−1)^{a}a ⌣_{1} δb + ab + (−1)^{ab}ba.
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The Cohomology Rings of Various p-Groups

THEOREM 2·10. (Hirsch). With the standard choice of cup-0 product, there is a choice of cup-1 such that for all cochains a, b, c

(ab) ⌣_{1} c = (−1)^{a}a(b ⌣_{1} c) + (−1)^{bc}(a ⌣_{1} c)b.

Note that Hirsch uses Steenrod’s original definition [St] of the cup-1 product, which satisfies a slightly different coboundary formula to the one stated above. This explains the difference between our statement of theorem 2·10 and Hirsch’s original statement. If we write ⌣_{S} for Steenrod’s definition of the cup-1 product, then we may define a choice of cup-1 in the modern sense by

a ⌣_{1} b = (−1)^{a+b−1}(a ⌣_{S} b),

and this is a choice for cup-1 which satisfies the identity of theorem 2·10. LEMMA 2·11. The Massey product satisfies the following identities, which are valid whenever all the terms are defined, for any u, v, w, x, y ∈ H^{∗}(X; R):

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

Proof. The verification of these is fairly straightforward. The proofs of (3) and (4) use theorem 2·10. For example, to prove (3), if we let u, v, w also stand for representative cocycles, then pick a, b and c such that

δa = uv δb = vw δc = wu,
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The Cohomology Rings of Various p-Groups

then

(−1)^{wu}⟨u, v, w⟩ + (−1)^{uv}⟨v, w, u⟩ + (−1)^{vw}⟨w, u, v⟩

contains the element

(−1)^{wu+u}ub − (−1)^{wu}aw + (−1)^{uv+v}vc − (−1)^{uv}bu + (−1)^{vw+w}wa − (−1)^{vw}cv
= − (−1)^{wu}(uv) ⌣_{1} w − (−1)^{uv}(vw) ⌣_{1} u − (−1)^{vw}(wu) ⌣_{1} v
=δ((u ⌣_{1} v) ⌣_{1} w).  ■

Matrix Massey Products. This generalisation of the Massey product is due to May [Ma]. We consider now homogeneous ‘vectors’ of cocycles (u_{i}), (v_{ij}), (w_{j}) of degrees u, v, w respectively, where 1 ⩽ i ⩽ l, and 1 ⩽ j ⩽ m, with the property that

∑_{i}[u_{i}v_{ij}] = 0 for all j
∑_{j}[v_{ij}w_{j}] = 0 for all j.

We choose (a_{j}) and (b_{i}) such that

δa_{j} = ∑_{i} u_{i}v_{ij} δb_{i} = ∑_{j} v_{ij}w_{j},

then define the matrix Massey product ⟨[u_{i}], [v_{ij}], [w_{j}]⟩ by

\begin{aligned}⟨[u_{i}], [v_{ij}], [w_{j}]⟩ & = (−1)^{u} ∑_{i} u_{i}b_{i} − ∑_{j} a_{j}w_{j} \\ & ∈ H^{u+v+w−1}(X; R)/(∑_{i} u_{i}H^{v+w−1}(X; R) + ∑_{j} w_{j}H^{u+v−1}(X; R)). \end{aligned}

The obvious generalisations of the properties claimed in lemma 2·11 are valid for the matrix Massey product, with very similar proofs. For example suppose that we are given (u_{i}), (v_{ij}), (w_{jk}), (x_{k}) for 1 ⩽ i ⩽ l, 1 ⩽ j ⩽ m, 1 ⩽ k ⩽ n with

∑_{i}[u_{i}v_{ij}] = 0 for all j
∑_{j}[v_{ij}w_{jk}] = 0 for all i, k
∑_{k}[w_{jk}x_{k}] = 0 for all j.
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The Cohomology Rings of Various p-Groups

We wish to show that

∑_{k}⟨[u_{i}], [v_{ij}], [w_{jk}]⟩[x_{k}] + (−1)^{u} ∑_{i}[u_{i}]⟨[v_{ij}], [w_{jk}], [x_{k}]⟩ ≡ 0 mod ∑_{i,k} u_{i}H^{∗}x_{k}.

We choose (a_{j}), (b_{ik}), (c_{j}) such that

δa_{j} = ∑_{i} u_{i}v_{ij} δb_{ik} = ∑_{j} v_{ij}w_{jk} δc_{j} = ∑_{k} w_{jk}x_{k},

then

∑_{k}⟨[u_{i}], [v_{ij}], [w_{jk}]⟩[x_{k}] − (−1)^{u} ∑_{i}[u_{i}]⟨[v_{ij}], [w_{jk}], [x_{k}]⟩ contains

(−1)^{u} ∑_{i,k} u_{i}b_{i,k}x_{k} −∑_{j,k} a_{j}w_{jk}x_{k} +(−1)^{u+v} ∑_{i,j} u_{i}v_{ij}c_{j} −(−1)^{u} ∑_{i,k} u_{i}b_{i,k}x_{k} = −δ(∑_{j} a_{j}c_{j}).

We now give an example of a calculation using Massey products, which we shall refer to in our later work. LEMMA 2·12. Let p be a prime not equal to 2, and let y generate H^{1}(BC_{p}; 𝔽_{p}). Then ⟨y, y, y⟩ is a unique element of H^{2}(BC_{p}; 𝔽_{p}), and

⟨y, y, y⟩ = \{ \begin{matrix}0 & for p > 3 \\ β(y) & for p = 3. \end{matrix}

Proof. We apply the ‘Jacobi identity’ (lemma 2·11, equation (3)) and obtain

⟨y, y, y⟩ + ⟨y, y, y⟩ + ⟨y, y, y⟩ ≡ 0 mod 0.

Hence for p > 3 we obtain ⟨y, y, y⟩ = 0. β(y) generates H^{2}(BC_{p}; 𝔽_{p}), and ⟨y, y, y⟩ is defined modulo 0, so for p = 3 it only remains to find the constant λ in the equation

⟨y, y, y⟩ = λβ(y).

To find λ we resort to the definition in terms of cochains in the bar resolution. For any G, and any cocycle y ∈ C^{1}(BG; 𝔽_{p}), the 1-cochain a defined by

a([g]) = −\frac{1}{2}y([g])^{2} satisfies δa([g|h]) = y([g])y([h]),
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The Cohomology Rings of Various p-Groups

so for any such y, a choice for ⟨y, y, y⟩ is

⟨y, y, y⟩([g|h]) = −y([g])y([h])(y([g]) + y([h])).

Now let G = ⟨A⟩ ≅ C_{3}, and define y by

y : [A^{r}] ↦ r mod (3)

Now define ȳ ∈ C^{1}(BG; ℤ) by

y : [A^{r}] ↦ r mod (3), where 0 ⩽ r ⩽ 2.

Then b = \frac{1}{3}δȳ is a cocycle representing β(y), and satisfies

b([A^{r}|A^{s}]) = \{ \begin{matrix}0 & for 0 ⩽ r + s ⩽ 2 \\ 1 & for 3 ⩽ r + s ⩽ 5, \end{matrix}

where we require 0 ⩽ r, s ⩽ 2. Now it remains to solve for λ in the equation over 𝔽_{3}:

λb([A^{r}|A^{s}]) + rs(r + s) = f(r) + f(s) − f(r + s),

where f is some function from 𝔽_{3} to itself. We obtain λ = 1, as required. ■

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The Cohomology Rings of Various p-Groups

Calculation of H^{∗}(BP_{2}; 𝔽_{p}). In this section we use the information given by the spectral sequence for the fibration

S^{1} ⟶ BP_{2} ⟶ BP̃  (1)

together with the properties of Massey products stated in the previous section to determine H^{∗}(BP_{2}; 𝔽_{p}). To show that a certain product in H^{5}(BP_{2}; 𝔽_{p}) is non-zero we briefly consider the spectral sequence (discussed in lemma 2·4) for the central extension

⟨C⟩ ⟶ P_{2} ⟶ C\overline{A} ; p ⊕ C\overline{B} ; p.

We now examine the spectral sequence for the fibration (1). As before E_{2}^{∗,∗} is the anticommutative ring generated by H^{∗}(BP̃; 𝔽_{p}) and u, a generator of H^{1}(S^{1}; 𝔽_{p}). By naturality of the spectral sequence for changes of coefficients, we see that

d_{2}(u) = λyy′

for some non-zero λ. Hence we see that

\begin{aligned}E_{∞}^{∗,0} & ≅ H^{∗}(BP̃; 𝔽_{p})/yy′H^{∗}(BP̃; 𝔽_{p}) \\ E_{∞}^{∗,1} & ≅ Ker × yy′ : H^{∗}(BP̃; 𝔽_{p}) → H^{∗}(BP̃; 𝔽_{p}), \end{aligned}

and it follows that 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. E_{∞}^{∗,0} is the subring of H^{∗}(BP̃; 𝔽_{p}) given by the image of the restriction from P̃. Throughout this section we shall use the same symbol for an element of H^{∗}(BP̃; 𝔽_{p}) and its image under the restriction. We shall write H^{∗} for H^{∗}(BP_{2}; 𝔽_{p}), unless there is danger of confusion.

We see immediately from the spectral sequence that

H^{1}H^{1} = {0},
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The Cohomology Rings of Various p-Groups

that H^{2} has order p^{4}, and that we need to introduce two new generators in H^{2} not in the image of the restriction. The Massey product of any three elements of H^{1} is defined, and is a unique element of H^{2}. We define

\begin{aligned}Y & = ⟨y, y, y′⟩ \\ Y′ & = ⟨y′, y′, y⟩, \end{aligned}

and claim that x, x′, Y and Y′ generate H^{2}. This is a consequence of the following lemma LEMMA 2·13. Define \overline{y}, d ∈ H^{1}(B⟨A, C⟩; 𝔽_{p}) and \overline{y}′, d′ ∈ H^{1}(B⟨B, C⟩; 𝔽_{p}) by

\begin{matrix}\overline{y} : A^{r}C^{s} & ↦ r \\ d : A^{r}C^{s} & ↦ s \end{matrix} \begin{matrix}\overline{y}′ : B^{r}C^{s} & ↦ r \\ d′ : B^{r}C^{s} & ↦ s, \end{matrix}

so that

\overline{y} = Res_{⟨A,C⟩}^{P_{2}}(y), \overline{y}′ = Res_{⟨B,C⟩}^{P_{2}}(y′).

If Y and Y′ are defined as above, then

\begin{aligned}Res_{⟨A,C⟩}^{P_{2}}(Y) & = \overline{y}d \\ Res_{⟨B,C⟩}^{P_{2}}(Y) & = 0 \end{aligned} \begin{aligned}Res_{⟨A,C⟩}^{P_{2}}(Y′) & = 0 \\ Res_{⟨B,C⟩}^{P_{2}}(Y′) & = −\overline{y}′d′. \end{aligned}

Proof. Recall from the proof of lemma 2·12 that the 1-cochain a defined by

a([g]) = −\frac{1}{2}y([g])^{2}

satisfies

δa([g|h]) = y([g])y([h]).

Define a 1-cochain b by

b([B^{r}A^{s}C^{t}]) = −t,

so that

δb([B^{r}A^{s}C^{t}|B^{r′}A^{s′}C^{t′}]) = r′s = y([B^{r}A^{s}C^{t}])y′([B^{r′}A^{s′}C^{t′}]).

Now we may choose as cocycle representing Y the following:

Y ([g|h]) = y([g])(\frac{1}{2}y([g])y′([h]) − b([h])).

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The Cohomology Rings of Various p-Groups

y restricts to ⟨B, C⟩ as the zero cocycle, so

Res_{⟨B,C⟩}^{P_{2}}(Y) = 0.

Similarly, y′ restricts to ⟨A, C⟩ as the zero cocycle, and b restricts to a cocycle representing −d. Hence

Res_{⟨A,C⟩}^{P_{2}}(Y) = \overline{y}d.

The results for Y′ are proven similarly. ■ We know that

\begin{aligned}Res_{⟨A,C⟩}^{P_{2}}(x) & = β(\overline{y}) \\ Res_{⟨B,C⟩}^{P_{2}}(x) & = 0 \end{aligned} \begin{aligned}Res_{⟨A,C⟩}^{P_{2}}(x′) & = 0 \\ Res_{⟨B,C⟩}^{P_{2}}(x′) & = β(\overline{y}′), \end{aligned}

so it follows that x, x′, Y, Y′ are linearly independent.

We are now ready to state our theorems determining the ring structure of H^{∗}(BP_{2}; 𝔽_{p}). We consider separately the cases p > 3 in theorem 2·14, and p = 3 in theorem 2·15. THEOREM 2·14. Let p be greater than 3. H^{∗}(BP_{2}; 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
47

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′ ∈ H^{1}(BP_{2}; 𝔽_{p}) by

\begin{matrix}y : A^{r}B^{s}C^{t} & ↦ r \\ y′ : A^{r}B^{s}C^{t} & ↦ s, \end{matrix}

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then define x, x′ by the equations

x = β(y) x′ = β(y′).

The equation yy′ = 0 implies that we may define unique elements Y, Y′ by

Y = ⟨y, y, y′⟩ Y′ = ⟨y′, y′, y⟩

and we also define

X = β(Y) X′ = β(Y′).

The effect of automorphisms of P_{2} on the generators y, y′, x, x′, Y, Y′, X, X′ is determined by the above definitions. An automorphism of P_{2} which restricts to the centre as C ↦ C^{j} sends

\begin{matrix}d_{i} & ↦ j^{i}d_{i} \\ c_{i} & ↦ j^{i}c_{i} \\ z & ↦ jz. \end{matrix}

If we define d′ ∈ H^{1}(B⟨B, C⟩; 𝔽_{p}) by

d′ : B^{r}C^{s} ↦ s,

then define c′ by

c′ = β(d′),

then we may define

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

We may define c_{i} in terms of either d_{i} or χ_{i} ∈ H^{2i}(BP_{2}; ℤ), using the equations

c_{i} = β(d_{i}) = π_{∗}(χ_{i}),
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The Cohomology Rings of Various p-Groups

and we define z by

z = π_{∗}(ζ).

We note that

\begin{aligned}π_{∗}(χ_{2}) & = λ(xY′ + x′Y) \\ π_{∗}(χ_{3}) & = λ′XX′ \end{aligned}

for some non-zero λ, λ′.

Proof. We shall prove these assertions in the order in which they are stated. Throughout this proof we adopt the convention that Cor( ) should stand for Cor_{⟨B,C⟩}^{P_{2}}( ), and similarly Res( ) should stand for Res_{⟨B,C⟩}^{P_{2}}( ).

yy′ = 0 follows from the spectral sequence, as does xy′ = x′y. For the other relations in H^{3}, we note that

yY = y⟨y, y, y′⟩
≡ ⟨y, y, y⟩y′ modulo yH^{1}y′ = {0}
= 0,

and similarly for y′Y′. Also

Y y′ = ⟨y, y, y′⟩y′
≡ y⟨y, y′, y′⟩ modulo yH^{1}y′ = {0}
≡ y⟨y′, y′, y⟩ modulo yyH^{1} + yy′H^{1} = {0}
= yY′.

The spectral sequence implies that yY′ is non-zero.

Before moving on to H^{4} we note that

⟨y, y, Y ⟩ ≡ 0 modulo H^{2}y + H^{2}y′,

which follows from

−⟨y, y, Y ⟩ + ⟨y, ⟨y, y, y⟩, y′⟩ − ⟨⟨y, y, y⟩, y, y′⟩ ≡ 0 modulo H^{2}y + H^{2}y′.
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The Cohomology Rings of Various p-Groups

Now

Y^{2} = Y ⟨y, y, y′⟩
≡ −⟨Y, y, y⟩y′ modulo Y H^{1}y′ = {0}
≡ −⟨y, y, Y ⟩y′ modulo Y H^{1}y′ + yH^{2}y′ = {0}
≡ 0 modulo H^{2}yy′ + H^{2}y^{′2} = {0}

Similarly, Y^{′2} = 0 and (Y + Y′)^{2} = 0, because an automorphism of P_{2} sending y to y − y′ and fixing y′ sends Y to Y + Y′, and we deduce that Y Y′ = 0.

The relations involving yX and y′X′ follow by applying the Bockstein to the relations yY = 0 and y′Y′ = 0 respectively. For the remaining relations in H^{4} we introduce some matrix Massey products. We consider

⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩,

which is defined modulo H^{2}H^{1}.

\begin{aligned}⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩y & ≡ −(x, x′) ( \begin{matrix}⟨y′, y, y⟩ \\ −⟨y, y, y⟩ \end{matrix} ) modulo xH^{1}y + x′H^{1}y = \{0\} \\ & = −xY, \end{aligned}

and

⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩y′ ≡ −(x, x′) ( \begin{matrix}⟨y′, y, y′⟩ \\ −⟨y, y, y′⟩ \end{matrix} ) modulo xH^{1}y′ + x′H^{1}y′ = \{0\}

but

⟨y′, y, y′⟩ ≡ −2Y′ modulo {0}

so

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

Similarly,

\begin{matrix}⟨(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′, \end{matrix}

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hence

X ∈ ⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩ 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 xX′ = −x′X

follow by applying the Bockstein to the relations

Y^{2} = 0 Y^{′2} = 0 Y Y′ = 0 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 ⟨x^{2}, xx′, x^{′2}, c_{2}⟩,

then the relations in H^{6} imply that

xY′ + x′Y = λ′c_{2}

for some λ′. 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}.

It follows from lemma 2·4 that H^{5} is generated by x^{2}y, x^{2}y′, xx′y′, x^{′2}y′, xX, xX′, x′X′ and XY′. Now XY′ ∉ Im(π_{∗}), and since H^{6}(BP_{2}; ℤ) has exponent p, we deduce that

XX′ = −β(XY′) ≠ 0.
52

The Cohomology Rings of Various p-Groups XX′x = 0,

but it is apparent from the spectral sequence that in H^{6},

Ker(×x) = ⟨c_{3}⟩,

so

XX′ = λc_{3} λ ≠ 0

and 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 = λc_{2},

but firstly we must find Res(z). We recall that

z = π_{∗}(ζ) = π_{∗}(c_{p}(ρ)),

and that ρ restricts to ⟨B, C⟩ as the sum of p representations, whose first Chern classes map under π_{∗} to c′, c′ + \overline{x}′, …, c′ + (p − 1)\overline{x}′. Hence we see that

\begin{aligned}Res(z) & = ∏_{i=0}^{p−1}(c′ + i\overline{x}′) \\ & = c^{′p} − \overline{x}^{′p−1}c′. \end{aligned}

Now we may calculate P^{1}(c_{2}) as follows:

P^{1}(c_{2}) = P^{1}(Cor(c^{′2}))

= Cor(2c^{′p+1})

= 2Cor(Res(z)c′ + Res(x^{′p−1})c^{′2}) = 2zyy′ + 2x^{′p−1}c_{2} = 0

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The Cohomology Rings of Various p-Groups

The last of these equations follows because

x′c_{2} = π_{∗}(βχ_{2}) = 0.

Now

0 = P^{1}(xY′ + x′Y)
= x^{p}Y′ + xY^{′p} + x^{′p}Y + x′Y^{p}
= x^{p}Y′ + x^{′p}Y.

Now we apply the Bockstein to obtain the relation

x^{p}X′ + x^{′p}X = 0.

We may verify that in degrees greater than 6 all products of the generators y, y′, x, x′, Y, Y′, X, X′ 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. δ_{p} sends elements of the above form and odd total degree to polynomials in α and β, so any element mapping under δ_{p} to χ_{i} (resp. pζ) will suffice to complete a basis for H^{2i−1} (resp. H^{2p−1}). Hence we may add d_{i} defined as above to our generating set.

Before verifying the relations involving c_{i} and the low dimensional generators, we consider the automorphism θ of P_{2}, given by

\begin{matrix}θ : A & ↦ B \\ B & ↦ A, \end{matrix}

which has the effect of exchanging the ‘primed’ and ‘unprimed’ low dimensional generators of H^{∗}. We know the effect of this automorphism on χ_{i}, and we deduce that it sends

θ^{∗} : c_{i} ↦ (−1)^{i}c_{i}.
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The Cohomology Rings of Various p-Groups

We already know the relations between c_{i} and y, y′, x, x′ because they are exactly the relations that they satisfy in H^{∗}(BP̃; 𝔽_{p}). For c_{i}Y, note that

Cor(c^{′i})Y = Cor(c^{′i}Res(Y))

= Cor(0) = 0,

so

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

For the results concerning c_{i}X we apply the Bockstein to the above results. For the corresponding results concerning the ‘primed’ generators, we apply the automorphism θ described above. The results concerning c_{i}c_{j} follow from the relations in H^{∗}(BP̃; 𝔽_{p}), or from those that hold in H^{∗}(BP_{2}; ℤ).

Now we examine the relations between d_{i} and the low dimensional generators. We do not yet know the effect of the automorphism θ on d_{i}, so we must examine both the ‘primed’ and ‘unprimed’ relations.

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

so

d_{i}y = \{ \begin{matrix}0 & for i < p \\ x^{p−2}Xy = −x^{p−1}Y & for i = p. \end{matrix}

Cor(c^{′i−1}d′)y′ = Cor(c^{′i−1}d′\overline{y}′),

where \overline{y}′ is as defined during the statement of lemma 2·13, but

Cor(c^{′i−1}d′\overline{y}′) = −Cor(c^{′i−1})Y′

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

Hence

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

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The Cohomology Rings of Various p-Groups

Similarly,

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

so

d_{i}Y = 0 for all i
d_{i}Y′ = 0 for all i.

For the relations involving d_{i} and x, x′, X, X′, we apply the Bockstein to the above relations and substitute for terms involving c_{i}. For example,

0 = β(d_{p−1}y) = c_{p−1}y − d_{p−1}x,

so

d_{p−1}x = c_{p−1}y = −x^{p−1}y.

We now use the formula for ResCor:

ResCor = ∑_{j=0}^{p−1} c_{A^{j}}^{∗}

where c_{A^{j}}^{∗} is the map of H^{∗}(B⟨B, C⟩; 𝔽_{p}) induced by conjugation by A^{j}. It is easy to verify that

\begin{matrix}c_{A^{j}}^{∗} : d′ & ↦ d′ + j\overline{y}′ \\ c′ & ↦ c′ + j\overline{x}′. \end{matrix}

Then

ResCor(c^{′i}) = ∑_{j=0}^{p−1}(c′ + j\overline{x}′)^{i}.

Similarly,

ResCor(c^{′i−1}d′) = ∑_{j=0}^{p−1}(c′ + j\overline{x}′)^{i−1}(d′ + j\overline{y}′).

Hence

Cor(c^{′i−1}d′)Cor(c^{′j−1}d′) = Cor(c^{′i−1}d′ ∑_{k=0}^{p−1}(c′ + k\overline{x}′)^{j−1}(d′ + k\overline{y}′))

= Cor(c^{′i−1}d′ ∑_{k=0}^{p−1} k\overline{y}′(c′ + k\overline{x}′)^{j−1})

= Cor(c^{′i−1}d′ ∑_{l=0}^{j−1} \binom{j − 1}{l}\overline{y}′c^{′j−1−l}\overline{x}^{′l} ∑_{k=0}^{p−1} k^{l+1})

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The Cohomology Rings of Various p-Groups

but

∑_{k=1}^{p−1} k^{l+1} ≡ \{ \begin{matrix}−1 & if p − 1 divides l + 1, \\ 0 & otherwise \end{matrix}

We may assume that i > j, so the only non-zero case we need consider is j = p − 1 and l = p − 2.

Cor(c^{′p−1}d′)Cor(c^{′p−2}d′) = −Cor(c^{′p−1}d′\overline{y}′\overline{x}^{′p−2})
= Cor(c^{′p−1})Y′x^{′p−2}
= (c_{p−1} + x^{p−1})Y′x^{′p−2}
= x^{p−1}x^{′p−2}Y′ − x^{′2p−3}Y′,

so

\begin{aligned}d_{p}d_{p−1} & = (Cor(c^{′p−1}d′) + x^{p−2}X)(Cor(c^{′p−2}d′) − x^{p−2}y) \\ & = x^{p−1}x^{′p−2}Y′ + x^{2p−3}Y − x^{′2p−3}Y′ \end{aligned}

and we obtain the relations between the d_{i}.

Similarly

Cor(c^{′i−1}d′)Cor(c^{′j}) = Cor(c^{′i−1}d′ ∑_{l=0}^{j} \binom{j}{l}c^{′j−l}\overline{x}^{′l} ∑_{k=0}^{p−1} k^{l}),

so

Cor(c^{′i−1}d′)Cor(c^{′j}) = \{ \begin{matrix}0 & for j < p − 1 \\ −Cor(c^{′i−1}d′\overline{x}^{′p−1}) & for j = p − 1 \end{matrix}

= \{ \begin{matrix}0 & for i < p − 1 or j < p − 1 \\ −(d_{p−1} + x^{p−2}y)x^{′p−1} & for i = j = p − 1 \\ (d_{p} + x^{p−2}X)x^{′p−1} & for i = p and j = p − 1 \end{matrix}

Hence we obtain the relations for d_{i}c_{j} as claimed.

All that remains to be calculated is the effect of automorphisms of P_{2} on d_{i}. The effect of automorphisms on c_{i} and z follow immediately from the corresponding results for integral cohomology. Let φ be an automorphism of P_{2} that restricts to the centre as

φ : C ↦ C^{j}.

We know that

δ_{p}φ^{∗}(d_{i}) = φ^{∗}δ_{p}(d_{i}) = j^{i}δ_{p}(d_{i}),
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The Cohomology Rings of Various p-Groups

so

φ^{∗}(d_{i}) ≡ j^{i}d_{i} modulo Ker(δ_{p}).

Also d_{i} was defined so that it corresponds, in the E_{∞}^{∗,∗} page of the spectral sequence, to λ[uc_{i−1}] for some non-zero λ. If j ≠ ±1 then φ does not extend to an automorphism of P̃, but it can be extended to an endomorphism \overline{φ} of P̃, defined on S^{1} by

\overline{φ} : v ↦ jv for all v ∈ S^{1} ≅ ℝ/ℤ.

It may be verified that \overline{φ}^{∗}(c_{i}) = j^{i}c_{i}, by first verifying that \overline{φ}^{∗}(χ_{i}) = j^{i}χ_{i}. Note that \overline{φ}^{∗} is an automorphism of H^{∗}(BP̃; 𝔽_{p}) although for j ≠ ±1 it is not an automorphism of H^{∗}(BP̃; ℤ). The map induced by \overline{φ};

\overline{φ} : P̃/P_{2} ↦ P̃/P_{2} ≅ S^{1}

has degree j, so u is sent to ju under the map of H^{∗}(S^{1}; 𝔽_{p}) induced by \overline{φ}.

Hence the element [uc_{i−1}] is sent to j^{i}[uc_{i−1}] by the map of the spectral sequence induced by \overline{φ}, so it follows that

φ^{∗}(d_{i}) ≡ j^{i}d_{i} modulo im(Res_{P} ; P_{2}).

Res_{P} ; P_{2}(H^{2i−1}(BP̃; 𝔽_{p})) ∩ Ker(δ_{p}) = {0},

so we see that

φ^{∗}(d_{i}) = j^{i}d_{i}.  ■

THEOREM 2·15. Let p = 3. Then H^{∗}(BP_{2}; 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 58

\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_{2}; 𝔽_{3}) by

\begin{matrix}y : A^{r}B^{s}C^{t} & ↦ r \\ y′ : A^{r}B^{s}C^{t} & ↦ s, \end{matrix}

then define x, x′ by the equations

x = β(y) x′ = β(y′),

The equation yy′ = 0 implies that we may define unique elements in H^{2}(BP_{2}; 𝔽_{3}) by forming the Massey product of any three elements of H^{1}(BP_{2}; 𝔽_{3}), and we define Y, Y′ by

Y = ⟨y, y, y′⟩ Y′ = ⟨y′, y′, y⟩

and we also define

X = β(Y) X′ = β(Y′),
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The Cohomology Rings of Various p-Groups

and note that x, x′ satisfy

x = ⟨y, y, y⟩ x′ = ⟨y′, y′, y′⟩.

The effect of automorphisms of P_{2} on the generators y, y′, x, x′, Y, Y′, X, X′ is determined by the above definitions. An automorphism of P_{2} which restricts to the centre as C ↦ C^{j} sends

z ↦ jz,

and we may define z to be π_{∗}(ζ). We also note that

π_{∗}(χ_{2}) = −xY′ − x′Y − x^{2} − x^{′2}.

Proof. Many of the relations may be proven exactly as in the case p > 3. The relations involving Y^{2},Y^{′2}, and Y Y′ must be proven differently, and we exhibit a basis for H^{4} before attempting to prove them.

yy′ = 0 and xy′ = x′y follow from the spectral sequence. Now

yY = y⟨y, y, y′⟩
≡ ⟨y, y, y⟩y′ modulo yH^{1}y′ = {0}
= xy′,

and similarly for y′Y′. The relation yY′ = y′Y follows exactly as in the proof of theorem 2·14. For the relations involving yX and y′X′ we apply the Bockstein to the relations yY = xy′ and y′Y′ = xy′. As in the proof of theorem 2·14, any

Z ∈ ⟨(x, x′), ( \begin{matrix}y′ \\ −y \end{matrix} ) , y⟩

satisfies

Zy = −xY + xx′ Zy′ = x′Y − xY′.

Similarly, any

Z′ ∈ ⟨(x′, x), ( \begin{matrix}y \\ −y′ \end{matrix} ) , y′⟩

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satisfies

Z′y′ = −x′Y′ + xx′ 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}. Using the relation for X′y we see that

β(Y′y) = −xY′ − x′Y.

The relations

xY′y = x^{′2}y 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} x′(xY′ + x′Y) = −x′x^{2}.

It follows that the relation in H^{4} yielding the relation in E_{∞}^{3,1} of the spectral sequence,

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}.

Consider the map

(Res_{⟨A,C⟩}^{P_{2}}, Res_{⟨B,C⟩}^{P_{2}}) : H^{4}(BP_{2}; 𝔽_{3}) ⟶ H^{4}(B⟨A, C⟩; 𝔽_{3}) × H^{4}(B⟨B, C⟩; 𝔽_{3}).

Lemma 2·13 tells us that Y Y′ is in the kernel of this map, and that xx′,xY′ and x′Y form a basis for this kernel. Hence

Y Y′ = λxx′ + λ′xY′ + λ′′x′Y,  (∗)
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The Cohomology Rings of Various p-Groups

for some λ, λ′, λ′′. The automorphism θ of P_{2} which exchanges the ‘primed’ and ‘unprimed’ generators fixes Y Y′, so

λ′′ = λ′.

Now we multiply equation (∗) by y, and obtain

xx′y = λxx′y − λ′x^{′2}y.

Hence we see that Y Y′ = xx′. For the remaining relations in H^{4}, we consider the effect of an automorphism θ′ of P_{2} having the following effect on H^{1}:

\begin{matrix}θ^{′∗} : y & ↦ y + y′ \\ y′ & ↦ y′. \end{matrix}

We verify that

\begin{matrix}θ^{′∗} : x & ↦ x + x′ \\ x′ & ↦ x′ \\ Y & ↦ Y + x′ − Y′ \\ Y′ & ↦ Y′ + x′, \end{matrix}

and apply θ^{′∗} to the equation Y Y′ = xx′, obtaining

Y Y′ + x^{′2} − Y^{′2} + x′Y = xx′ + x^{′2},

and hence

Y^{′2} = x′Y.

We obtain the relation for Y^{2} similarly.

We apply the Bockstein to the relations

\begin{aligned}Y^{2} & = xY′ \\ Y^{′2} & = x′Y \\ Y Y′ & = xx′ \\ 0 & = xY′ + x′Y + c_{2} + x^{2} + x^{′2} \end{aligned}

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The Cohomology Rings of Various p-Groups

respectively to obtain the relations

\begin{aligned}XY & = −xX′ \\ X′Y′ & = −x′X \\ XY′ & = −X′Y \\ xX′ & = −x′X. \end{aligned}

Lemma 2·4 implies that XY′ is linearly independent of elements of the form

f_{1}y + f_{2}y′ + f_{3}X + f_{4}X′,

where f_{i} is a polynomial in x and x′, and hence as in theorem 2·14 we see that no new generator is needed in H^{5}. XX′ = β(Y X′) is in the image of π_{∗}, which, in H^{6}, is spanned by z, x^{3},x^{2}x′, xx^{′2}, and x^{′3}. It is easily checked that

XX′x = XX′x′ = 0,

but

Ker(×x) ∩ Ker(×x′) ∩ ⟨z, x^{3}, x^{2}x′, xx^{′2}, x^{′3}⟩ = {0},

and so

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}.
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The Cohomology Rings of Various p-Groups

Noting that Res_{⟨B,C⟩}^{P_{2}}(z) = c^{′3} − \overline{x}^{′2}c′, we have

P^{1}(c_{2} + x^{2} + x^{′2}) = P^{1}(Cor_{⟨B,C⟩}^{P_{2}}(c^{′2}) + x^{′2})
= −Cor_{⟨B,C⟩}^{P_{2}}(c^{′4}) − x^{′4}
= −Cor_{⟨B,C⟩}^{P_{2}}(Res_{⟨B,C⟩}^{P_{2}}(z)c′ + \overline{x}^{′2}c^{′2}) − x^{′4}
= −zyy′ − x^{′2}c_{2} + x^{′2}x^{2} − x^{′4}
= x^{′2}x^{2}

and

\begin{aligned}P^{1}(xY′ + x′Y) & = x^{3}Y′ + x^{′3}Y + xY^{′3} + x′Y^{3} \\ & = x^{3}Y′ + x^{′3}Y − x^{2}x^{′2}, \end{aligned}

so

x^{3}Y′ + x^{′3}Y = −x^{2}x^{′2}.

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_{2}; 𝔽_{3}) is complete. The effect of automorphisms on z follows from its definition as π_{∗}(ζ), and we already have the required expression for π_{∗}(χ_{2}) = c_{2}. ■ Remarks. Using the results of theorems 2·14 and 2·15, the author has determined the differentials in the spectral sequence with 𝔽_{p} coefficients for P_{2} expressed as a central extension of C_{p} by C_{p}. The E_{∞}-page in the case p = 7 is depicted in figure 2·4.

64

\begin{matrix}z & zy,zy′ & zx,zx′ & \begin{matrix}zxy,zxy′, \\ zx′y′ \end{matrix} & \begin{matrix}zx^{2},zxx′, \\ zx^{′2} \end{matrix} & \begin{matrix}zx^{2}y,zx^{2}y′, \\ zxx′y′,zx^{′2}y′ \end{matrix} & \begin{matrix}zx^{3},zx^{2}x′, \\ zxx^{′2},zx^{′3} \end{matrix} \\ − \\ − \\ − & − & d_{7} \\ − \\ − & − & d_{6} & c_{6} \\ − \\ − & − & d_{5} & c_{5} \\ − \\ − & − & d_{4} & c_{4} \\ − & − & − \\ − & − & XY′ & XX′ & − \\ − & X,X′ & − & \begin{matrix}xX,xX′, \\ x′X′ \end{matrix} & − & \begin{matrix}x^{2}X,x^{2}X′, \\ xx′X′,x^{′2}X′ \end{matrix} & − \\ − & Y,Y′ & yY′ & \begin{matrix}xY,xY′, \\ x′Y,x′Y′ \end{matrix} & − & \begin{matrix}x^{2}Y,x^{2}Y′, \\ xx′Y′,x^{′2}Y′ \end{matrix} & − \\ 1 & y,y′ & x,x′ & \begin{matrix}xy,xy′, \\ x′y′ \end{matrix} & \begin{matrix}x^{2},xx′, \\ x^{′2} \end{matrix} & \begin{matrix}x^{2}y,x^{2}y′, \\ xx′y′,x^{′2}y′ \end{matrix} & \begin{matrix}x^{3},x^{2}x′, \\ xx^{′2},x^{′3} \end{matrix} \end{matrix}

2·4. The E_{∞}-page for the extension C_{7} ↣ P_{2} ↠ C_{7} ⊕ C_{7} with 𝔽_{7} coefficients.

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The Cohomology Rings of Various p-Groups

It is unreasonable to expect that by rechoosing generators we could make the statement of theorem 2·15 look more like the statement of theorem 2·14. For example, for p > 3, H^{2}(BP_{2}; 𝔽_{p}) may be expressed as the direct sum of two Aut(P_{2})-invariant subspaces:

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

For p = 3 the subspace ⟨x, x′⟩ has no Aut(P_{2})-invariant complement.

We recall that central extensions of C_{p} by G are classified up to equivalence of extensions by H^{2}(BG; 𝔽_{p}) (see [Br] or [Th3] for details), where E is equivalent to E′ if there is a homomorphism θ making the following diagram commute:

C_{p} ↣ E ↠ G
↓^{Id} ↓^{θ} ↓^{Id}
C_{p} ↣ E′ ↠ G

It is easily seen that a two generator group G of exponent p and order p^{4} must be a central extension of C_{p} by P_{2}. Conversely, such an extension has exponent p iff its extension class restricts to zero on all cyclic subgroups, and in this case G will be a two generator group iff its extension class is non-zero. For p > 3, the elements λY + μY′ restrict to zero on all cyclic subgroups, whereas for p = 3 there is no such non-zero element. We thus have verified a result due to Burnside [Bu], that there are two generator groups of exponent p and order p^{4} only when p > 3.

The action of the Steenrod algebra A_{p} on the 1- and 2-dimensional generators of H^{∗}(BP_{2}; 𝔽_{p}), and on the c_{i} and d_{i} (which are expressed in terms of corestrictions from an abelian subgroup) is apparent. The generator z is the restriction of the generator of H^{2p}(BP̃; 𝔽_{p}) of the same name, so P^{1}(z) is determined by lemma 2·8. The following proposition completes the description of the action of A_{p} on H^{∗}(BP_{2}; 𝔽_{p}). PROPOSITION 2·16. With notation as in theorems 2·14 and 2·15,

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

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The Cohomology Rings of Various p-Groups

Proof. The spectral sequence operation _{B}P^{1} defined by Araki and Vasquez ([Ar], [Va]) on the E_{∞} page of the spectral sequence for BP_{2} 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_{2} 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_{2}} 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_{2}; 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. From lemma 2·13 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. ■

The above information tells us what happens in the Atiyah-Hirzebruch spectral sequence for P_{2}. COROLLARY 2·17. In the Atiyah-Hirzebruch spectral sequence for P_{2} (E_{2}^{i} ≅ H^{i}(BP_{2}; ℤ) converging to a filtration of the representation ring of P_{2}) the only non-zero differential is d_{2p−1}, which sends μ to a multiple of ζβ, and ν to a multiple of ζα. Proof. In this spectral sequence the first potentially non-zero differential is d_{2p−1}, which is δ_{p}P^{1}π_{∗}. ■

67

3. The Size of the Chern Subring and its Closure. We recall that a group G is said to have p-rank n if n is maximal such that G has a subgroup isomorphic to (C_{p})^{n}. In [Bl], Blackburn classified, for odd primes p, the p-groups having no normal subgroup isomorphic to C_{p} ×C_{p} ×C_{p}. It can be shown, independently of Blackburn’s result, that all such groups satisfy the apparently stronger condition of having p-rank at most two, see for example [Go]. The classification is as follows: THEOREM 3·1. (Blackburn) Let p be an odd prime. Then the p-groups of p-rank two are the following groups:

1. The (non-cyclic) metacyclic p-groups.

2. P (n), where n ⩾ 3 as defined in the introduction to section 2.

3. B(n, ε), where n ⩾ 4, there are two groups for each n, depending whether ε is 1 or a quadratic non-residue modulo p. B(n, ε) has order p^{n}, and may be presented as:

⟨A, B, C | A^{p} = B^{p} = C^{p^{n−2}} = [B, C] = 1 [A, C^{−1}] = B [B, A] = C^{εp^{n−3}}⟩

For p > 3 the groups B(4, ε) are the seventh and eighth groups on Burnside’s list of groups of order p^{4}, and for p = 3, B(4, −1) is the tenth group on Burnside’s list ([Bu] or the appendix to this dissertation), and B(4, 1) is the eighth group.

4. In the case p = 3, every 3-group of maximal nilpotency class except C_{3}, C_{9} and the wreath product of C_{3} with itself (the sixth group on Burnside’s list of groups of order 81 [Bu]). ■

Let us say that a group G has property C if Ch(G) = H^{ev}(BG) (recall that Ch(G) is the subring generated by Chern classes of representations of G). Various of the p-groups of p-rank two have been shown to have property C, for example in section 2 we verified Thomas’ result that P (n) has property C [Th2]. Thomas also verified that the split metacyclics have property C and conjectured that all p-groups of p-rank two would have property C [Th1], [Th2]. He also gave the example A_{4} to show that the conjecture could not

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be extended to arbitrary groups of p-rank at most two for all primes. AlZubaidy claimed to have verified the conjecture for p ⩾ 5, see [Al1], [Al2], but some of his proofs are flawed. Recently Tezuka-Yagita have shown that all metacyclic p-groups have property C [TY2], and Huebschmann has shown independently [Hu1] that all finite metacyclic groups have property C. Moselle has suggested that property \overline{C}, that of satisfying \overline{Ch}(G) = H^{ev}(BG) might be more natural, and points out that G has property \overline{C} if and only if each of its Sylow subgroups does [Mo].

We shall show that for p ⩾ 5 the groups B(n, ε) do not have property C, but that for p = 3 the groups B(n, ε) do have property C. We also show that for all odd p the groups B(n, ε) have property \overline{C}. Yagita has shown independently that for p ⩾ 5 the groups B(n, ε) do not have property C [Ya2].

In [At] Atiyah showed that for any finite group G, K^{0}(BG) is the completion of the representation ring of G with respect to a certain topology. The filtration of K^{0}(BG) given by the E_{∞} page of the Atiyah-Hirzebruch spectral sequence (H^{i}(BG; K^{j}(∗)) ⇒ K^{i+j}(BG)) gives rise to a filtration on the representation ring of G. He conjectured that this filtration coincided with another filtration defined algebraically, and remarked that this conjecture is equivalent to the conjecture that Ch(G) maps onto the E_{∞} page of the AHSS. Many counterexamples have been found of composite order [Wei]. We show that for p ⩾ 5 the groups B(n, ε) are counterexamples. These seem to be the first counterexamples of prime power order.

First we state a result (presumably well known) concerning abelian p-groups and property C, and show that p-groups with an elementary abelian maximal subgroup of rank at least 3 do not have property \overline{C}.

PROPOSITION 3·2. Let A be an abelian p-group. Then the following are equivalent: 1) A has p-rank at most two.

2) Ch(A) = H^{ev}(BA).

3) \overline{Ch}(A) = H^{ev}(BA).

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Proof. An abelian group has only 1-dimensional irreducible representations, so Ch(A) and \overline{Ch}(A) are generated in degree two. For any finite group G,

Ch^{2}(G) = H^{2}(BG; ℤ) ≅ Hom(G, U(1)),

so 2) and 3) are equivalent to each other and to the statement that H^{ev}(BA) is generated in degree two. If A is cyclic then H^{∗}(BA; ℤ) is a polynomial algebra on a generator of degree two, and if A is a product of two cyclic groups then the Künneth theorem implies that H^{ev}(BA) is a polynomial algebra on two generators of degree two. For the general case, let

A ≅ C_{p^{n_{1}}} ⊕ C_{p^{n_{2}}} ⊕ … ⊕ C_{p^{nm}} , where n_{1} ⩽ n_{2} ⩽ … ⩽ n_{m}.
Then H^{∗}(BA; 𝔽_{p}) ≅ 𝔽_{p}[x_{1}, …, x_{m}] ⊗ Λ[y_{1}, …, y_{m}],

where the n_{i}-th higher Bockstein maps y_{i} to x_{i}. The image of \overline{Ch}(A) under reduction mod-p is the subring generated by the x_{i}, and the image of H^{∗}(BA; ℤ) is the universal cycles in the Bockstein spectral sequence. (Recall that the Bockstein spectral sequence has E_{2}^{i} ≅ H^{i}(BA; 𝔽_{p}) and converges to 𝔽_{p} concentrated in degree zero, with differentials the higher Bocksteins and B_{i}^{∗} the image under reduction mod-p of elements of order dividing p^{i−1}.) The first non-zero differential in this spectral sequence sends y_{1}y_{2}y_{3} to x_{1}y_{2}y_{3} + εx_{2}y_{1}y_{3} +ε′x_{3}y_{1}y_{2}, where ε, ε′ are either 0 or 1, so this element is in the image of H^{ev}(BA), but not in the image of \overline{Ch}(A). ■

PROPOSITION 3·3. If G is a p-group with a (C_{p})^{n} subgroup of index p for some n ⩾ 3, then \overline{Ch}(G) ≠ H^{ev}(BG). Proof. The subgroup is maximal, so is normal. Call it N, and let Q be the quotient G/N ≅ C_{p}. By proposition 3·2 we may assume that Q acts non-trivially on N. We may identify H^{1} = H^{1}(BN; 𝔽_{p}) with Hom(N, 𝔽_{p}) and then we see that

H^{∗}(BN; 𝔽_{p}) ≅ S(βH^{1}) ⊗ E(H^{1}),
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where S(βH^{1}) and E(H^{1}) are respectively the symmetric and exterior algebras on the vector space H^{1}. Now H^{2} splits as a direct sum of βH^{1} and E^{2}(H^{1}), H^{3} splits as a direct sum of βH^{1} ⊗ H^{1} and E^{3}(H^{1}), and the action of Q on H^{i} respects these splittings. Let y be an element of H^{1} with an orbit of length p so that its images under Q are y + iy′, for some y′ fixed by Q, and let α = δ_{p}(y) and α′ = δ_{p}(y′). Also let w be an element of E^{3}(H^{1}) fixed by Q, and let ξ = δ_{p}(w). Now writing N_{N}^{G} for Evens multiplicative transfer map from N to G [Ev1], we have

Res_{N}^{G}N_{N}^{G}(α + ξ) = ∏_{i=0}^{p−1}(α + iα′ + ξ) = α^{p} − α^{′p−1}α − α^{′p−1}ξ.

Since π_{∗}(α^{′p−1}ξ) does not lie in S(βH^{1}) it follows that α^{′p−1}ξ is not in \overline{Ch}(N), and hence the component of N_{N}^{G}(α + ξ) in degree 2p + 2 cannot be in \overline{Ch}(G). ■

The Groups B(n, ε). Throughout this section we shall consider B(n, ε) as being presented, as in the introduction, by elements A, B, and C subject to the above relations. We note that the centre of B(n, ε) is cyclic generated by C^{p}, and that if we apply the circle construction to the whole centre we obtain a group B̃ which is independent of both n and ε. This is because there is only one circle group on p^{3} components of nilpotency class three. The subgroup generated by A, B, and C^{p} is isomorphic to P (n − 1), with centre also generated by C^{p}, and so we have a commutative diagram of extensions.

P (n − 1) ⟶ B(n, ε) ⟶ C_{p}
↓  ↓  ↓^{id}
P̃  ⟶  B̃  ⟶ C_{p}

We note also that the subgroup generated by B and C is abelian and isomorphic to C_{p^{n−2}} ⊕ C_{p}. The corresponding subgroup of B̃ (that is the one generated by B, C and S^{1}) is also abelian and is isomorphic to S^{1} ⊕C_{p} ⊕C_{p}. The action induced on H^{∗}(BP (n−1); ℤ) by conjugation by C fixes χ_{i}, ζ, α and ν and sends β to β + α, and μ to μ + ν. The action induced on H^{∗}(BP̃; ℤ) is similar.

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LEMMA 3·4. The fixed point subring of H^{∗}(BP̃; ℤ) (resp. H^{ev}(BG; ℤ)) is generated by the elements α, χ_{i}, ζ, and β^{n}(β^{p} − α^{p−1}β) for n ⩾ 0. Proof. The element β^{p} −α^{p−1}β is the product of all conjugates of β so must be fixed. Also α(β^{p}−α^{p−1}β) = 0 and β^{n} is sent to itself modulo multiples of α, so β^{n}(β^{p}−α^{p−1}β) is fixed. ζ^{i}P (α, β) is fixed if and only if P (α, β) is, so we only need to check that there are no more fixed points in the subring generated by α and β. We may take as basis for the degree n piece of this the elements α^{n}, α^{n−1}β, … β^{n} if n ⩽ p and α^{n}, α^{n−1}β, … α^{n−p+1}β^{p−1}, β^{n} if n ⩾ p. In either case we are left to show that there are no fixed points of the form ∑_{i=1}^{m} λ_{i}α^{n−i}β^{i}, where m ⩽ p − 1. Equating coefficients of α^{n−m+1}β^{m−1} in

∑_{i=1}^{m} λ_{i}α^{n−i}β^{i} = ∑_{i=1}^{m} λ_{i}α^{n−i}(β + α)^{i}

we obtain

λ_{m−1} = λ_{m−1} + mλ_{m}

so λ_{m} = 0, and inductively each λ_{i} is zero. ■ LEMMA 3·5. The image of Ch(B̃) under restriction to P̃ (resp. the image of Ch(B(n, ε)) under restriction to P (n − 1)) is generated by χ_{i}, ζ, α, and β^{p} − α^{p−1}β. Proof. By lemma 1·7 it suffices to prove the assertion for the Lie groups. A direct proof for the finite groups would be very similar. B̃ has an abelian subgroup of index p, so has only 1- and p-dimensional irreducible representations. By considering the natural isomorphisms

Ch^{2}(G) = H^{2}(BG; ℤ) ≅ Hom(G, S^{1})

we see that χ_{1} and α span the image of Ch^{2}(B̃). For any p-dimensional representation ρ of P̃ the representation Ind_{B} ; _{P}(ρ) will split into p p-dimensional representations of P̃, any of which will restrict to ρ. Hence χ_{i} and ζ are in the image. If θ is the 1-dimensional representation of P̃ with c_{1}(θ) = β, then

c_{p}(Ind_{B} ; _{P}(θ)) = ∏_{i=0}^{p−1}(β + iα) = β^{p} − α^{p−1}β.

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We do not need any more generators because the image must be generated in degrees at most 2p. In these degrees we already have all of the fixed points by lemma 3·4. ■

Let H be the subgroup of B̃ generated by S^{1} and B, and define elements τ′, β′ of H^{2}(BH; ℤ) ≅ Hom(H, S^{1}) by the following formulae.

\begin{aligned}τ′|_{S^{1}} & = 1_{S^{1}} \\ τ′(B) & = 0 \end{aligned}  \begin{aligned}β′|_{S^{1}} & = 0 \\ β′(B) & = exp(2πi/p) \end{aligned}

Let M be the abelian maximal subgroup of B̃, that is the subgroup generated by S^{1}, B, and C. The restriction map from H^{∗}(BM; ℤ) to H^{∗}(BH; ℤ) is clearly onto. LEMMA 3·6. If φ ∈ H^{2(p+n)}(BM; ℤ) restricts to H as −τ^{′p−1}β^{n+1}, then Cor_{B} ; M(φ) restricts to P̃ as β^{n}(β^{p} − α^{p−1}β). Proof. MP̃ = B̃, and M ∩ P̃ = H, so the restriction-corestriction formula shows us that

Res_{B} ; _{P}Cor_{B} ; M(φ) = Cor_{P} ; HRes_{H}^{M}(φ)

= −Cor_{P} ; H(τ^{′p−1}β^{′n+1})

= −Cor_{P} ; H(τ^{′p−1})β^{n+1}

= −(χ_{p−1} + α^{p−1})β^{n+1}

= β^{n}(β^{p} − α^{p−1}β).  ■

COROLLARY 3·7. For p ⩾ 5, H^{ev}(BB(n, ε)) is not generated by Chern classes. Proof. β^{p+1} − α^{p−1}β^{2} is in the image of the restriction from B(n, ε) to P (n − 1) by lemma 3·6. For p ⩾ 5 it may be verified that it is not expressible in terms of the generators for the image of Ch(B(n, ε)) given in lemma 3·5. For p = 3,

β^{4} − α^{2}β^{2} = χ_{2}^{2} − 3ζχ_{i} − α^{4}.  ■

Yagita has also proved corollary 3·7 [Ya2] using a method involving his calculation of the Brown-Peterson cohomology of BP_{2} [TY].

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COROLLARY 3·8. For p ⩾ 5 and the groups B(n, ε), the Chern subring does not map surjectively to the E_{∞} page of the Atiyah Hirzebruch spectral sequence. Proof. Let K be the abelian maximal subgroup of B(n, ε). Then by lemma 3·5 and lemma 1·7 there is a φ ∈ Ch^{2p+2}(K) satisfying

Res_{P (n−1)}^{B(n,ε)}Cor_{K}^{B(n,ε)}(φ) = β^{p+1} − α^{p−1}β^{2}.

From the naturality properties of the AHSS it follows that Cor(φ) is a universal cycle. We show that it is not the sum of a Chern class and a universal boundary by considering the restriction to the subgroup generated by B.

H^{∗}(B⟨B⟩; ℤ) = ℤ[\overline{β}]/(p\overline{β}),

where \overline{β} is the restriction from P (n − 1) of β. The AHSS for ⟨B⟩ collapses because E_{2}^{i,j} is trivial if i or j is odd. Hence the universal boundaries B_{∞}(B(n, ε)) restrict trivially to ⟨B⟩. By lemma 3·5 the image of Ch(B(n, ε)) under this restriction is generated by \overline{β}^{p} and \overline{β}^{p−1}. So for p ⩾ 5,

Res_{⟨B⟩}^{B(n,ε)}Cor_{K}^{B(n,ε)}(φ) = \overline{β}^{p+1} ∉ Res_{⟨B⟩}^{B(n,ε)}(B_{∞}(B(n, ε)) + Ch(B(n, ε))).  ■

COROLLARY 3·9. H^{∗}(BB̃) restricts onto the fixed point subring of H^{∗}(BP̃). Proof. Immediate from lemma 3·4 and lemma 3·6. ■ COROLLARY 3·10. H^{ev}(BB̃) is generated by corestrictions of Chern classes, and by Chern classes for p = 3. Proof. Consider the spectral sequence with integer coefficients for the extension

P̃⟶B̃⟶C_{p}.

For any extension with quotient C_{p} the E_{2} page of the corresponding spectral sequence is generated by E_{2}^{0,∗}, E_{2}^{1,∗} and E_{2}^{2,0} ≅ 𝔽_{p}. The inflation of a generator for H^{2}(BC_{p}) is a

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The Size of the Chern Subring and its Closure

Chern class, and yields a generator for E_{2}^{2,0}. Also corollary 3·9 implies that E_{2}^{0,∗} consists of universal cycles, and lemma 3·6 implies that corestriction of Chern classes (resp. Chern classes in the case when p = 3) yield generators for the even degree subring of E_{2}^{∗,∗}. ■

We shall now work towards theorem 3·13, which will tell us that H^{ev}(BB(n, ε)) is generated by corestrictions of Chern classes, and will involve the application of theorem 1·9. Corollary 3·10 has verified one of the hypotheses required for this, so it suffices now to check that multiplication by the extension class c(B(n, ε)) in H^{2}(BB̃) is injective on H^{od}(BB̃). This is proven for all n as a corollary of lemma 3·12, but we first present a simpler proof in the case when n ⩾ 5. The reader who objects to redundancy should turn directly to the end of the proof of lemma 3·11.

Since P̃ has index p in B̃ and trivial odd degree cohomology it follows by the usual corestriction-restriction argument that H^{od}(BB̃) has exponent p. It is easily checked that c(B(n, ε)) is of the form p^{n−4} times an element of infinite order plus a non-zero element of the image of the inflation from B̃/P̃ ≅ C_{p}. Thus it will follow that H^{ev}(BB(n, ε)) is generated by corestrictions of Chern classes (resp. Chern classes for p = 3) for n ⩾ 5 provided that multiplication by a generator for Inf(H^{2}(BB̃/P̃; ℤ)) is injective on H^{od}(BB̃). This is a case of the following lemma. LEMMA 3·11. Consider a fibration

F ⟶E⟶^{π}BC_{p},

such that H^{∗}(F) is concentrated in even degrees and H^{∗}(E) maps onto H^{∗}(F)^{C_{p}}, and let γ be a generator for H^{2}(BC_{p}). Then multiplication by π^{∗}(γ) is injective on H^{od}(E). Proof. Consider the spectral sequence for the fibration. The element π^{∗}(γ) yields a generator for E_{2}^{0,2} which we shall call γ, and all elements in even total degree are universal cycles. It suffices to prove that multiplication by γ is an isomorphism from E_{∞}^{2i+1,j} to E_{∞}^{2i+3,j} for all i and j. In fact we shall show inductively that multiplication by the corresponding element of E_{2n}^{2,0} (which we shall also call γ) is an isomorphism from E_{2n}^{i,j} to E_{2n}^{i+2,j}

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The Size of the Chern Subring and its Closure

provided that either i is odd or i ⩾ 2n. This is true for n = 1. Now d_{2n} is trivial because it maps between even and odd values of j. It is possible for d_{2n+1} to be non-trivial, but only from odd total degree to even total degree. Since by induction the odd degree part of E_{2n+1}^{∗,∗} is generated by E_{2n+1}^{1,∗}, d_{2n+1} is completely determined by its values on E_{2n+1}^{1,∗}. We obtain the following commutative diagram, where the horizontal maps are isomorphisms by induction.

E_{2n+1}^{1,j}  ⟶^{×γ}  E_{2n+1}^{3,j}  ⟶^{×γ}  E_{2n+1}^{5,j}  ⟶ ⋯
↓^{d_{2n+1}}  ↓^{d_{2n+1}}  ↓^{d_{2n+1}}
E_{2n+1}^{2n+2,j−2n−1} ⟶^{×γ} E_{2n+1}^{2n+4,j−2n−1} ⟶^{×γ} E_{2n+1}^{2n+6,j−2n−1} ⟶ ⋯

Hence multiplication by γ induces isomorphisms on the kernels and cokernels, and the inductive step is proven. ■

We shall now deduce that H^{ev}(BG) = \overline{Ch}(G) (resp. Ch(G) for p = 3) for any pgroup G of p-rank two and nilpotency class three using the following lemma. LEMMA 3·12. As above, let M be the abelian subgroup of B̃ of index p. Then the restriction from H^{od}(BB̃) to H^{od}(BM) is injective. Proof. First let us fix a presentation for B̃.

B̃ = ⟨S^{1}, A, B, C | S^{1} central,  A^{p} = B^{p} = C^{p} = 1
[A, B] = C [B, C] = 1 [A, C] = e^{2πi/p}⟩

Note that B̃/S^{1} is generated by the images of A and B, which we shall call A and B too, so that our notation coincides with that of section 2. Note that the subgroup M is expressible as an extension of S^{1} by ⟨B, C⟩. We shall show that the map of spectral sequences induced by the following map of extensions is injective on the E_{∞} page.

S^{1} ⟶ M ⟶ ⟨B, C⟩
↓^{Id} ↓^{i} ↓ī
S^{1} ⟶ B̃ ⟶ P_{2}

Let E_{∗}^{∗,∗} stand for the spectral sequence for B̃, and let \overline{E}_{∗}^{∗,∗} for the spectral sequence for M. Notice that \overline{E}_{∗}^{∗,∗} collapses. If we use the notation of section 2 for elements of H^{∗}(BP_{2}; ℤ)

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The Size of the Chern Subring and its Closure

and let H^{∗}(BS^{1}; ℤ) ≅ ℤ[τ], then we see that

E_{2}^{∗,∗} ≅ ℤ[τ] ⊗ H^{∗}(BP_{2}; ℤ).

If we let ρ be a 1-dimensional representation of M such that Res_{S^{1}}^{M}(c_{1}(ρ)) = τ, then

Res_{B} ; S^{1}(c.(Ind_{B} ; M(ρ))) = (1 + τ)^{p}.

Hence pτ^{i} and τ^{p} are universal cycles, the E_{∞} page is periodic vertically with period 2p, and E_{∞}^{∗,∗} ≅ E_{2p}^{∗,∗}. Since B̃′ ∩ S^{1} has order p there can be no homomorphism from B̃ to S^{1} that restricts to S^{1} as an isomorphism, so d_{3}(τ) must be non-zero. The kernel of the map from E_{2}^{3,0} to \overline{E}_{2}^{3,0} is generated by ν, and d_{3}(τ) must lie in this kernel (because \overline{E}_{∗}^{∗,∗} collapses), so without loss of generality d_{3}(τ) = ν.

It may be verified that E_{5}^{∗,∗} is generated by α, β, χ_{2}, χ_{4}, …, χ_{p−1}, ζ, μ, τ^{i}βμ, τ^{i}χ_{2}, τ^{i}χ_{4}, …, τ^{i}χ_{p−2}, τ^{i}(χ_{p−1} + α^{p−1}), pτ^{i}, τ^{p}, and τ^{p−1}ν, where 1 ⩽ i ⩽ p − 1, subject to various relations. For example, we have that [τ^{i}βμ]α = 0 unless i is congruent to −1 modulo p. Figure 3·1 depicts this E_{5} page in the case when p ⩾ 7.

We now claim that τ^{p−1}ν survives until the E_{2p−1} page, and that d_{2p−1}(τ^{p−1}ν) is a non-zero multiple of ζα (which is equal to δ_{p}P^{1}π_{∗}(ν)—see proposition 2·16 and corollary 2·17). To show this, we first consider the spectral sequence with ℤ_{(p)} coefficients for the path-loop fibration over K(ℤ, 3). Let i_{3} be the (image in ℤ_{(p)} coefficients of the) fundamental class in H^{3}(K(ℤ, 3)), and i_{2} that in H^{2}(ΩK(ℤ, 3)), so that d_{3}(i_{2}) = i_{3}. It is readily verified that i_{2}^{p−1}i_{3} survives until the E_{2p−1} page, and that d_{2p−1}(i_{2}^{p−1}i_{3}) is a unit multiple of δ_{p}P^{1}π_{∗}(i_{3}). The claim concerning τ^{p−1}ν now follows by naturality, because ν may be thought of as a map from BP_{2} to K(ℤ, 3) classifying the K(ℤ, 2) bundle BB̃.

Even on E_{5} pages the map from E_{5}^{od,2i} to \overline{E}_{5}^{od,2i} is injective except when i is congruent to −1 modulo p, and its kernel is the submodule generated by τ^{p−1}ν. Thus it will suffice to show that no differential lower than d_{2p−1} can hit anything in the ideal of E_{5}^{∗,0} generated by ζα. This follows because multiplication by α is injective on this ideal but trivial on E_{5}^{od,2i} for 1 ⩽ i < p − 1. ■

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The Size of the Chern Subring and its Closure

\begin{matrix}pτ^{3} & − & − & − & τ^{3}χ_{2} & τ^{3}βμ & − & τ^{3}β^{2}μ & τ^{3}χ_{4} \\ − & − & − & − & − & − & − & − & − \\ pτ^{2} & − & − & − & τ^{2}χ_{2} & τ^{2}βμ & − & τ^{2}β^{2}μ & τ^{2}χ_{4} \\ − & − & − & − & − & − & − & − & − \\ pτ & − & − & − & τχ_{2} & τβμ & − & τβ^{2}μ & τχ_{4} \\ − & − & − & − & − & − & − & − & − \\ 1 & − & α,β & μ & \begin{matrix}α^{2},αβ, \\ β^{2},χ_{2} \end{matrix} & βμ & \begin{matrix}α^{3},α^{2}β, \\ αβ^{2},β^{3} \end{matrix} & β^{2}μ & \begin{matrix}α^{4},α^{3}β, \\ α^{2}β^{2},αβ^{3}, \\ β^{4},χ_{4} \end{matrix} \end{matrix}

3·1. The E_{5}-page of the spectral sequence of lemma 3·12 in the case p ⩾ 7.

THEOREM 3·13. If G is any finite normal subgroup of B̃ of p-rank two with quotient isomorphic to S^{1}, then H^{ev}(BG) = \overline{Ch}(G), and if p = 3 then H^{ev}(BG) = Ch(G). Examples of such G include the p-groups B(n, ε) and for p = 3 the seventh, eighth and tenth groups on Burnside’s list of groups of order 81 ([Bu] or the appendix).

Proof. Let c(G) be the extension class of G in H^{2}(BB̃), that is the element of H^{2}(BB̃) corresponding to the homomorphism from B̃ to S^{1} with kernel G. Since H = G ∩ M is abelian of p-rank at most two, Ch(H) = H^{ev}(BH), so by theorem 1·9 multiplication by c(H) is injective on H^{od}(BM). But now c(H) = Res_{B} ; M(c(G)), and so by lemma 3·12 multiplication by c(G) is injective on H^{od}(BB̃). The result now follows from theorem 1·9, since corollary 3·10 tells us that H^{ev}(BB̃) is generated by corestrictions of Chern classes

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and by Chern classes for p = 3. The groups listed in the statement do have p-rank two, and they are normal subgroups of B̃, because they have nilpotency class three, and a central cyclic subgroup with quotient group P_{2}, so yield B̃ when the circle construction is applied. ■ COROLLARY 3·14. For p ⩾ 5, if G is a p-group of p-rank two, then H^{ev}(BG) is generated by corestrictions of Chern classes. Proof. This is immediate from a combination of Blackburn’s classification [Bl] (or see theorem 3·1), Huebschmann’s results on the metacyclic groups [Hu1], Thomas’ result for the groups P (n) [Th2] (or theorem 2·3), and theorem 3·13. ■

79

4. Further Calculations at the Prime Three. In this section we shall make a more detailed study of the integral cohomology of the group B̃ and the p-groups B(n, ε), particularly when p = 3. In this case the proof of lemma 3·12 describes most of the behaviour of a spectral sequence converging to a filtration of H^{∗}(BB̃; ℤ). Using this spectral sequence we determine the cohomology of B̃, and then calculate the cohomology of B(n, ε) using the spectral sequence for BB(n, ε) as an S^{1}-bundle over BB̃. The main result of the section is that (for p = 3) the integral cohomology ring of B(n, 1) is isomorphic to that of B(n, −1), so that there are two groups of order 3^{5} with isomorphic integral cohomology rings. The author believes that these are the first examples of such p-groups, although non-isomorphic groups of composite order with isomorphic cohomology rings have been known for some time, and Larson has even exhibited metacyclic examples [La]. For p > 3 it seems far more difficult to determine the cohomology of B̃. Nevertheless, and in spite of the title of this section, we present a proof that the cohomology groups (ignoring the ring structure) of B(n, ε) are independent of ε for all odd p, without actually determining these groups. One response to the main result of this section has been “Are you sure that the groups are different?”, so before starting our calculations we show that as ε varies in 𝔽_{p}^{×} there really are two non-isomorphic groups of the form B(n, ε).

LEMMA 4·1. For any fixed odd prime p and n ⩾ 5, define for each ε ∈ 𝔽_{p} a group G(ε) of order p^{n} by the presentation

\begin{aligned}G(ε) = ⟨A, B, C, D | A^{p} = B^{p} = C^{p} = 1 C central, B & = [A, D] \\ C = [A, B] [B, D] & = 1 D^{p^{n−3}} = C^{ε}⟩. \end{aligned}

Then there are three isomorphism classes of such G, depending as ε is 0, a quadratic residue, or a quadratic non-residue mod-p.

Proof. Note that the relations given suffice to express any element of G in the form A^{i}B^{j}C^{k}D^{l}. Note that the subgroup of G generated by A and B is normal, and isomorphic

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to P_{2}, with quotient cyclic of order p^{n−3}. First we claim that if g ∈ ⟨A, B⟩, then the order of gD^{j} is at least the order of D^{j}. First note that

(gD^{j})^{p} = (∏_{k=0}^{p−1} g^{D^{jk}})D^{jp},

and that the expression on the right is not 1 unless D^{jp} is. Now since D^{p} commutes with elements of ⟨A, B⟩, we see that

(gD^{j})^{p^{2}} = D^{jp^{2}}.  (∗)

It follows that if ε ≠ 0, then ⟨A, B⟩ is the subgroup of G of elements of order p, and we deduce that G(0) is not isomorphic to G(ε) for ε ≠ 0 because it contains more elements of order p. From now on, let us consider only the case ε ≠ 0.

The elements A and D generate G, and any automorphism of G must the form

D ↦A^{i}B^{j}C^{k}D^{l}_{A ↦A^{r}B^{s}C^{t}} where l ≠ 0 mod p

where 1 ⩽ r ⩽ p − 1.

If we define D′ to be the image under such an automorphism of D, A′ to be the image of A, B′ = [A′, D′] and C′ = [A′, B′], then the primed elements define a presentation of the form G(ε′). To complete the proof it suffices to show that ε′r^{2} = ε. From equation (∗) it follows that D^{′p^{n−3}} = D^{lp^{n−3}}. Now

B′ = [A^{r}B^{s}C^{t}, A^{i}B^{j}C^{k}D^{l}]
≡ [A^{r}, D^{l}] mod ⟨C⟩
≡ B^{rl} mod ⟨C⟩,
C′ = [A^{r}B^{s}C^{t}, B^{rl}C^{?}] = C^{r^{2}l},

from which it is apparent that D^{′p^{n−3}} = C^{′ε/r^{2}}. ■

We shall now begin our calculation of H^{∗}(BB̃; ℤ) in the case when p = 3 by considering the spectral sequence for B̃ expressed as an extension of S^{1} by P_{2}. This spectral sequence was studied for arbitrary p during the proof of lemma 3·12, where d_{3} and part of d_{2p−1} were calculated. In the case when p = 3 not much more work is required to complete the spectral sequence. For the rest of this chapter we shall consider only the prime three unless otherwise stated.

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LEMMA 4·2. Consider the spectral sequence with integral coefficients for the extension

S^{1} ⟶ B̃ ⟶ P_{2}.

Use notation as in the proof of lemma 3·12, so that

E_{2}^{∗,∗} ≅ ℤ[τ] ⊗ H^{∗}(BP_{2}; ℤ).

Then d_{3}(τ) = ν, and E_{5}^{∗,∗} is generated by α, β, χ_{2}, ζ, μ, 3τ, τ(χ_{2} + α^{2}), τβμ, 3τ^{2}, τ^{2}ν, τ^{2}(χ_{2} +α^{2}), τ^{2}βμ, and τ^{3}. All of these are universal cycles except for τ^{2}ν and τ^{2}βν, which satisfy

d_{5}(τ^{2}ν) = ζα d_{5}(τ^{2}βμ) = ζ(β^{2} + α^{2} + χ_{2}).

There are no other non-zero differentials, so E_{∞}^{∗,∗} is generated by α, β, χ_{2}, ζ, μ, 3τ, τ(χ_{2} + α^{2}), τβμ, 3τ^{2}, τ^{2}(χ_{2} + α^{2}), τ^{3} and τ^{2}(β^{2} − α^{2})μ.

Proof. The claims concerning d_{3} and E_{5}^{∗,∗} were proved in lemma 3·12. Of the generators required for E_{5}^{∗,∗}, all are universal cycles with the possible exception of τ^{2}ν, τ^{2}(χ_{2} + α^{2}) and τ^{2}βμ. It was shown in lemma 3·12 that d_{5}(τ^{2}ν) = ζα. For the remaining cases we again (as in lemma 3·12) compare the spectral sequence E_{∗}^{∗,∗} with the spectral sequence \overline{E}_{∗}^{∗,∗} for M (the abelian maximal subgroup of B̃) using the map induced by the following map of extensions.

S^{1} ⟶ M ⟶ ⟨B, C⟩
↓^{Id} ↓^{i} ↓ī
S^{1} ⟶ B̃ ⟶ P_{2}

Define generators β′, γ′ for H^{2}(B⟨B, C⟩; ℤ) ≅ Hom(⟨B, C⟩, S^{1}) by

\begin{matrix}β′ : B & ↦ ω \\ C & ↦ 1 \end{matrix}  \begin{matrix}γ′ : B & ↦ 1 \\ C & ↦ ω, \end{matrix}

where ω = exp(2πi/3), so that β′ = Res(β). Also define a generator μ for H^{3}(B⟨B, C⟩; ℤ) by μ′ = Res(μ). Now we know that \overline{E}_{∗}^{∗,∗} collapses and that \overline{E}_{2}^{∗,∗} ≅ ℤ[τ′, β′, γ′] ⊗ Λ[μ′].

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To show that d_{5}(τ^{2}(χ_{2} + α^{2})) = 0, we note that it lies in E_{5}^{9,0}, which is generated by β^{3}μ and ζμ, so maps injectively to \overline{E}_{5}^{9,0} (recall that Res(ζ) = γ^{′3} − β^{′2}γ′). The result follows because \overline{E}_{∗}^{∗,∗} collapses. Similarly, we know that d_{5}(τ^{2}βμ) must be in the kernel of Res_{∗}^{∗,∗}. The equation (τ^{2}ν)β^{2} = (τ^{2}βμ)α implies that

αd_{5}(τ^{2}βμ) = ζαβ^{2},

so d_{5}(τ^{2}βμ) = ζβ^{2} +Q for some Q in the kernel of multiplication by α as a map from E_{5}^{10,0} to E_{5}^{12,0}. This kernel is spanned by ζ(α^{2} + χ_{2}) and β^{5} − α^{2}β^{3}, which map under Res_{∗}^{∗,∗} to (β^{′2}γ′ − γ^{′3})β^{′2} and β^{′5} respectively. It follows that d_{5}(τ^{2}βμ) is as claimed. The E_{∞}-page of this spectral sequence is depicted in figure 4·1 (on page 87), but with elements named by the elements of H^{∗}(BB̃; ℤ) that yield them, rather than as in this statement. ■ COROLLARY 4·3. In the spectral sequence of lemma 4·2, the subring of E_{∞}^{∗,∗} generated by τ^{3} and ζ is of the form ℤ[x, y]/(3y), where x = τ^{3} and y = ζ. As a module for this subring E_{∞}^{∗,∗} is generated by 1, 3τ, 3τ^{2}, α^{i+1}, β^{i+1}, αβ^{i+1}, α^{2}β^{i+1}, χ_{2}, β^{i}μ, τβ^{i+1}μ, τ(α^{2} + χ_{2})β^{i}, τ^{2}(β^{2} −α^{2})β^{i}μ and τ^{2}(α^{2} +χ_{2})β^{i}, where i ⩾ 0. Each of these module generators has order three, except that 1, 3τ and 3τ^{2} have infinite order. Multiplication by τ^{3} is injective on E_{∞}^{∗,∗}, and the kernel of multiplication by ζ is generated (as a ℤ[τ^{3}, ζ]-module) by 3τ, 3τ^{2}, α^{i+1}, αβ^{i+1}, α^{2}β^{i+1} and β^{2} + χ_{2}. Proof. Almost immediate from lemma 4·2. ■

An immediate consequence of lemma 4·2 is that H^{∗}(BB̃; ℤ) may be generated by nine elements of even degree and three of odd degree, since E_{∞}^{∗,∗} can be. It has, however, already been shown in corollary 3·10 that six elements suffice to generate H^{ev}(BB̃). We shall now define six elements of even degree and one of odd degree which (as we shall demonstrate in lemma 4·7) generate H^{∗}(BB̃; ℤ). To determine the relations that they satisfy we shall have to calculate their restrictions to M, which are listed as lemma 4·6. Information given by restriction to proper subgroups does not quite suffice to show that the elements generate H^{∗}(BB̃; ℤ), essentially because there are elements of degree four that restrict trivially to

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every proper subgroup. We are forced, therefore, to prove separately in lemma 4·5 that the elements of even degree defined below generate H^{ev}(BB̃), using the same method as in corollary 3·10.

DEFINITION 4·4. Define τ′, β′, γ′ ∈ H^{2}(BM; ℤ) ≅ Hom(M, S^{1}) by

\begin{matrix}τ′ : zB^{i}C^{j} & ↦ z \\ β′ : zB^{i}C^{j} & ↦ ω^{i} \\ γ′ : zB^{i}C^{j} & ↦ ω^{j}, \end{matrix}

where ω = exp(2πi/3). Now define elements α, β, γ, δ_{1}, δ_{2}, δ_{3}, μ of H^{∗}(BB̃; ℤ) as follows. Let α, β and μ be the images under inflation from B̃/S^{1} = P_{2} of the elements with the same names. Let ρ be a 3-dimensional irreducible representation of B̃, whose restriction to M contains the 1-dimensional representation with first Chern class τ′, and define

δ_{3} = c_{3}(ρ).

Define also

δ_{1} = Cor_{B} ; M(τ′)
δ_{2} = Cor_{B} ; M(τ^{′2})
γ = Cor_{B} ; M(τ^{′2}(β′ − γ′)) = δ_{2}β − Cor_{B} ; M(τ^{′2}γ′).  ■

LEMMA 4·5. The elements α, β, γ, δ_{1}, δ_{2} and δ_{3} defined in 4·4 generate H^{ev}(BB̃).

Proof. If we consider β as a homomorphism from B̃ to S^{1}, it has image C_{3} and kernel P̃, with C_{3} acting on P̃ via outer automorphisms. This is, modulo choice of generators, the extension considered in corollary 3·10. Recall that in corollary 3·10 it was shown (for p = 3) that Chern classes generate H^{ev}(BB̃). The method involved checking that their restrictions to P̃ generate H^{∗}(BP̃; ℤ)^{C_{3}}. By a similar argument we may show that the elements of the statement generate H^{ev}(BB̃) by showing that the restrictions of α, γ,

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δ_{1}, δ_{2} and δ_{3} to P̃ (that is, to Ker(β)) generate H^{∗}(BP̃; ℤ)^{C_{3}}, which was described in lemma 3·4. Some caution is required since the notation used in lemma 3·4 for elements of H^{∗}(BP̃; ℤ) clashes with our current notation (which was designed to harmonise with our notation for elements of H^{∗}(BB̃/S^{1}; ℤ) rather than with the cohomology of any subgroup). We claim that the restrictions are as decribed below, where the elements of H^{∗}(BB̃; ℤ) are as defined in 4·4, and their images are named as in lemma 3·4.

Res(α) = α  Res(δ_{1}) = χ_{1}  Res(δ_{2}) = χ_{2} + α^{2}
Res(γ) = β^{3} − α^{2}β Res(δ_{3}) = ζ.

For the elements of degree two this may be checked by considering them as homomorphisms from B̃ to S^{1}. For the elements defined as corestrictions from M to B̃ we use the formula Res_{B} ; _{P}Cor_{B} ; M = Cor_{P} ; M∩_{P}Res_{M∩_{P}}^{M}, which holds because P̃M = B̃. For example,

Res_{B} ; _{P}(δ_{2}β − Cor_{B} ; M(τ^{′2}γ′)) = 0 − Cor_{P} ; M∩_{P}(τ^{′2}γ′)

= −Cor_{P} ; M∩_{P}(τ^{′2})β

= −(χ_{2} + α^{2})β = β^{3} − α^{2}β.

For δ_{3}, we note that the representation ρ restricts to P̃ as the (unique) irreducible representation of degree three that restricts to S^{1} as three copies of the identity, so c_{3}(ρ) must restrict to ζ. It is clear now from lemma 3·4 that the restrictions of α, γ, δ_{1}, δ_{2} and δ_{3} generate H^{∗}(BP̃; ℤ)^{C_{3}}, and the result follows. ■ LEMMA 4·6. The restrictions to the subgroup M of B̃ of the elements defined in 4·4 are as follows, where the equation for Res(μ) is used to define μ′, a generator for H^{3}(BM; ℤ) ≅ 𝔽_{3}.

Res(α) = 0 Res(β) = β′ Res(μ) = μ′ Res(δ_{1}) = 3τ′ + β′ Res(γ) = γ^{′3} − β^{′2}γ′

Res(δ_{2}) = 3τ^{′2} − τ′β′ + β^{′2} − γ′β′ − γ^{′2} Res(δ_{3}) = τ^{′3} + τ^{′2}β′ − τ′γ′β′ − τ′γ^{′2}

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Proof. The claims for α, β and μ are trivial. For δ_{1}, δ_{2} and γ, we use the restrictioncorestriction formula, which in this case tells us that Res_{B} ; MCor_{B} ; M is equal to the sum of conjugation by the distinct powers of A. By considering them as homomorphisms from B̃ to S^{1}, it is easily verified that conjugation by A fixes β and sends τ′ to τ′ − γ′ and γ′ to γ′ − β′. For example,

Res_{B} ; M(δ_{1}) = ∑_{i=0}^{2} c_{A^{i}}^{∗}(τ′) = τ′ + (τ′ − γ′) + (τ′ + γ′ + β′) = 3τ′ + β′.

For δ_{3}, which we recall was defined as c_{3}(ρ), we note that since the restriction to M of ρ contains a summand with Chern class τ′ and is invariant under conjugation by A it must also contain summands with Chern class τ′ − γ′ and τ′ + γ′ + β′. Thus

Res(δ_{3}) = τ′(τ′ − γ′)(τ′ + γ′ + β′).  ■

LEMMA 4·7. The seven elements defined in 4·4 generate H^{∗}(BB̃; ℤ). Their relation to the generators for the E_{∞}-page of the spectral sequence of lemma 4·2 is as follows.

1. The generators α, β and μ yield the spectral sequence elements of the same name.

2. The generator δ_{1} (resp. δ_{2}, δ_{3}) yields the generator 3τ (resp. 3τ^{2}, τ^{3}) for E_{∞}^{0,2} (resp. for E_{∞}^{0,4}, E_{∞}^{0,6}).

3. The subgroup of H^{4}(BB̃; ℤ) generated by α^{2}, αβ, β^{2} and δ_{1}β is mapped bijectively to E_{∞}^{4,0} (which is generated by α^{2}, αβ, β^{2} and χ_{2}).

4. The subgroup of H^{6}(BB̃; ℤ) generated by α^{3}, α^{2}β, αβ^{2}, β^{3} and γ is mapped bijectively to E_{∞}^{6,0} (which is generated by α^{3}, α^{2}β, αβ^{2}, β^{3} and ζ).

5. δ_{2}β (resp. δ_{2}μ) yields the generator τ(α^{2} + χ_{2}) for E_{∞}^{4,2} (resp. −τβμ for E_{∞}^{5,2}).

6. δ_{2}^{2} − δ_{3}δ_{1} (resp. δ_{2}^{2}μ) yields the generator −τ^{2}(α^{2} + χ_{2}) for E_{∞}^{4,4} (resp. τ^{2}(β^{2} − α^{2})μ for E_{∞}^{7,4}).

The E_{∞}-page of this spectral sequence is depicted in figure 4·1. Proof. To show that the elements generate H^{∗}(BB̃; ℤ) it will suffice to show that they yield a set of generators for the E_{∞}-page of the spectral sequence of lemma 4·2. Thus it suffices

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to prove the numbered assertions made above. Many of these assertions follow easily from lemma 4·6. For example the restriction to M of δ_{2} is 3τ^{′2} modulo terms of lower degree in τ′, the restriction from E_{∞}^{0,4} to \overline{E}_{∞}^{0,4} is injective, and hence δ_{2} must yield the generator 3τ^{2}. Similarly γ yields an element of E_{∞}^{6,0} which restricts to \overline{E}_{∞}^{6,0} as γ^{′3} − β^{′2}γ′, while the image of the subgroup of E_{∞}^{6,0} generated by α^{3}, α^{2}β, αβ^{2} and β^{3} in \overline{E}_{∞}^{6,0} is generated by β^{′3}, so γ is independent of this subgroup, and so the five elements of item 4 above yield generators for E_{∞}^{6,0}. All the assertions except that of item 3 can be proved similarly, and we will consider them proved.

\begin{matrix}δ_{3} & − & δ_{3}α,δ_{3}β & δ_{3}μ & ... \\ − & − & − & − & − & − & − & − & − & − \\ δ_{2} & − & − & − & δ_{2}^{2}−3δ_{3}δ_{1} & − & δ_{2}γ,δ_{2}^{2}β & δ_{2}^{2}μ & δ_{2}^{2}β^{2} & δ_{2}^{2}βμ \\ − & − & − & − & − & − & − & − & − & − \\ δ_{1} & − & − & − & δ_{2}β & δ_{2}μ & δ_{2}β^{2} & δ_{2}βμ & δ_{2}β^{3} & δ_{2}β^{2}μ \\ − & − & − & − & − & − & − & − & − & − \\ 1 & − & α,β & μ & \begin{matrix}α^{2},αβ, \\ β^{2},δ_{1}β \end{matrix} & βμ & \begin{matrix}α^{3},α^{2}β, \\ αβ^{2},β^{3}, \\ γ \end{matrix} & β^{2}μ & \begin{matrix}α^{4},α^{2}β^{2}, \\ αβ^{3},β^{4}, \\ γβ \end{matrix} & β^{3}μ,γμ \end{matrix}

4·1. The E_{∞}-page of the spectral sequence of lemmata 4·2 and 4·7.

It is clear that δ_{1}β yields an element of E_{∞}^{4,0}. To prove that it does not lie in the subgroup of E_{∞}^{4,0} generated by α^{2}, αβ and β^{2} is more difficult. From the spectral sequence it can be seen that H^{4}(BB̃; ℤ) is isomorphic to ℤ ⊕ C_{3}^{4}. The products of the elements of definition 4·4 that have degree four are δ_{2}, δ_{1}^{2}, δ_{1}β, δ_{1}α, α^{2}, αβ and β^{2}. In the group

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generated by these elements the subgroup of elements of order at most three is generated by δ_{1}^{2} − 3δ_{2}, δ_{1}β, δ_{1}α, α^{2}, αβ and β^{2}. We now determine some relations between these elements. By Fröbenius reciprocity

δ_{1}α = Cor(τ′)α = Cor(τ′Res(α)) = 0.

Similarly,

δ_{1}^{2} = Cor(τ′)Cor(τ′) = Cor(τ′ResCor(τ′))
= Cor(3τ^{′2} + τ′β′)
= 3Cor(τ^{′2}) + Cor(τ′)β′
= 3δ_{2} + δ_{1}β.

It follows that δ_{1}α and δ_{1}^{2} − 3δ_{2} are dependent on δ_{1}β, α^{2}, αβ, β^{2}. Lemma 4·5 tells us that α, β, γ, δ_{1}, δ_{2} and δ_{3} generate H^{ev}(BB̃), so it must be that δ_{1}β is not contained in the subgroup generated by α^{2}, αβ and β^{2}. ■ THEOREM 4·8. H^{∗}(BB̃; ℤ) is generated by the elements of definition 4·4 subject to the following relations.

3α = 3β = 0  3μ = 0  3γ = 0
\begin{aligned}δ_{1}^{2} & = δ_{1}β + 3δ_{2} \\ δ_{1}μ & = βμ \end{aligned}  \begin{aligned}γδ_{1} & = γβ \\ δ_{1}α & = 0 \end{aligned}  \begin{aligned}αμ & = 0 \\ γα & = 0 \end{aligned}
\begin{aligned}δ_{1}δ_{2} & = δ_{2}β + 9δ_{3} \\ δ_{2}α & = 0 \end{aligned}  \begin{aligned}δ_{1}β^{2} & = β^{3} − α^{2}β \\ α^{3}β & = αβ^{3} \end{aligned}
δ_{2}^{3} − 27δ_{3}^{2} = −δ_{3}(β^{3} − α^{2}β) + δ_{2}^{2}β^{2} + δ_{2}β^{4} − γ^{2} − β^{6} + α^{2}β^{4}

Proof. If we assume that these relations hold, it follows easily from corollary 4·3 and lemma 4·7 that they are all the relations that we require. (Note in particular that the relations given imply that 3(δ_{2}^{2} − 3δ_{3}δ_{1}) = 0.) Thus it remains to demonstrate that these relations hold.

That the elements α, β, γ and μ have order three is immediate from their definitions. The relations claimed in degree 4 were checked during the proof of lemma 4·7. The relations

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αμ = 0 and δ_{1}μ = βμ hold because the restrictions of these equations to M are valid, and the restriction is injective from E_{∞}^{5,0} to \overline{E}_{∞}^{5,0}. The relations δ_{2}α = 0, γα = 0, and γδ_{1} = γβ follow readily by a Fröbenius reciprocity argument. The relation for δ_{1}δ_{2} may also be deduced by using Fröbenius reciprocity as follows.

δ_{2}δ_{1} = Cor(τ^{′2})δ_{1} = Cor(τ^{′2}(3τ′ + β′))
= Cor(3τ^{′3}) + Cor(τ^{′2})β
= Cor(Res(3δ_{3})) + δ_{2}β
= 9δ_{3} + δ_{2}β.

The relation α^{3}β = αβ^{3} holds because the corresponding relation holds in the spectral sequence. We know (because δ_{1}β yields an element of E_{∞}^{4,0}) that δ_{1}β^{2} is in the span of α^{3}, α^{2}β, αβ^{2}, β^{3} and γ, that its restriction to M is β^{′3}, and that it is annihilated by α. Since the kernel of the restriction from E_{∞}^{6,0} is spanned by α^{3}, α^{2}β and αβ^{2} whereas the kernel of multiplication by α is spanned by γ and β^{3} − α^{2}β, these facts suffice to determine δ_{1}β^{2}.

It is clear from corollary 4·3 and lemma 4·7 that all we are left to do is obtain an expression for δ_{2}^{3}. It may be checked that the relation given is correct when restricted to M, and that when it is multiplied by α the valid equation 0 = 0 is obtained. The result follows, since it may be checked that in degree 12 the kernel of restriction to M is generated by δ_{3}α^{3}, δ_{3}α^{2}β, δ_{3}αβ^{2}, α^{6}, α^{2}β^{4} and αβ^{5}, and that multiplication by α is injective on this subgroup. ■

LEMMA 4·9. For any odd prime p, define δ′, β ∈ H^{2}(BB̃; ℤ) ≅ Hom(B̃, S^{1}), where B̃ is presented as in the proof of lemma 3·12, by

\begin{matrix}δ′ : A^{j}B^{k}C^{l}z & ↦ z^{p} \\ β : A^{j}B^{k}C^{l}z & ↦ exp(2πik/p). \end{matrix}

(In the case when p = 3, β is the element β defined in 4·4 and δ′ is δ_{1} − β.) then for any ε, the subgroup of B̃ with extension class p^{n−4}δ′ − εβ is isomorphic to B(n, ε).

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Proof. The assertions relating δ′ and β to δ_{1} and β in the case p = 3 are easily proved. If we write φ = p^{n−4}δ′ − εβ, then as a homomorphism from B̃ to S^{1}, φ has the following form.

φ : A^{j}B^{k}C^{l}z ↦ z^{p^{n−3}}ω^{−εj},

where ω = exp(2πi/p). The kernel of φ (which we recall is one of the equivalent interpretations of the group with characteristic class φ) is the subgroup generated by A, C, and B′ = Bη^{ε}, where η = exp(2πi/p^{n−2}). It is easily checked that the map from this group to B(n, ε) (as presented in the statement of theorem 3·1) sending A to A, C to B^{−1} and B′ to C is an isomorphism. ■

Before calculating H^{∗}(BB(n, ε); ℤ) we require one more lemma which will determine a generator for H^{5}(BB(n, ε); ℤ).

LEMMA 4·10. Let A be the abelian maximal subgroup of B(n, ε), so that A ≅ C_{3^{n−2}} ⊕ C_{3} and A = B(n, ε)∩M. Also take generators \overline{τ}, \overline{γ} and \overline{μ} for H^{∗}(BA; ℤ), where \overline{τ} = Res_{A}^{M}(τ′), \overline{γ} = Res_{A}^{M}(γ′), and \overline{μ} = Res_{A}^{M}(μ′). Note that \overline{γ} and \overline{μ} have order 3 while \overline{τ} has order 3^{n−2}. then there is an element of H^{5}(BB(n, ε); ℤ) that restricts to A as \overline{γ}μ̄, and this element is in the image of the corestriction from a P (n − 1) subgroup to B(n, ε).

Proof. First we claim that if H is the subgroup of A isomorphic to C_{3^{n−3}} ⊕ C_{3}, and μ̃ is a generator for H^{3}(BH; ℤ), then Cor_{H}^{A}(μ̃) = \overline{μ}. To show this, note that if y is a generator for H^{1}(BC_{3^{n−3}}; 𝔽_{3}) (which is isomorphic to Hom(C_{3^{n−3}}, C_{3})), and we consider C_{3^{n−3}} as a subgroup of C_{3^{n−2}}, then the classical description of the transfer (as a map from G/G′ to H/H′ where H ⊂ G) allows us to check that Cor(y) is a generator for H^{1}(BC_{3^{n−2}}; 𝔽_{3}). Now if we define y ∈ H^{1}(BH; 𝔽_{3}) to be the inflation of a generator for H^{1}(BH/(1⊕C_{3}); 𝔽_{3}), and y′ the inflation of a generator for H^{1}(BH/(C_{3^{n−2}} ⊕C_{3}); 𝔽_{3}) then Fröbenius reciprocity implies that Cor_{H}^{A}(yy′) = Cor_{H}^{A}(y)y′ is an element of H^{2}(BA; 𝔽_{3}) not in the image of H^{2}(BA; ℤ). The claim now follows since corestriction commutes with the Bockstein.

Now note that H = A ∩ P (n − 1), and that from the above paragraph we see that we 90

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may take generators τ̃, γ̃ and μ̃ for H^{∗}(BH; ℤ), where τ̃ = Res(\overline{τ}) and γ̃ = Res(\overline{γ}) while \overline{μ} = Cor(μ̃). If we use the notation of theorem 2·3 and lemma 3·4 for elements of the group H^{∗}(BP (n−1); ℤ), then after a suitable choice of generators the action of B(n, ε)/P (n−1) ≅ C_{3} on H^{∗}(BP (n − 1); ℤ) may be taken to fix α, χ_{1} and ν, and send β to β + α and μ to μ + ν. The kernel of the restriction fom P (n − 1) to H in degree two is generated by α (this may be checked by considering H^{2}(BP (n − 1); ℤ) ≅ Hom(P (n − 1), S^{1})), which implies that the kernel of the restriction from P (n − 1) to H in degree three is generated by ν. Thus we see that

Res_{A}^{B(n,ε)}Cor_{P (n−1)}^{B(n,ε)}(βμ) = Cor_{H}^{A}Res_{H}^{P (n−1)}(βμ)
= ±Cor_{H}^{A}(γ̃μ̃)
= ±γ̃μ̃  ■

THEOREM 4·11. For n ⩾ 5 (and p = 3), H^{∗}(BB(n, ε); ℤ) is generated by elements α, δ_{1}, μ, δ_{2}, ν, δ_{3} and γ of degrees 2, 2, 3, 4, 5, 6 and 6 respectively. The generators other than ν may be taken to be the restrictions from B̃ of the generators for H^{∗}(BB̃; ℤ) of the same name. They are subject to the following relations.

3α = 0  3γ = 0  3μ = 0  3ν = 0
3^{n−3}δ_{1} = 0  3^{n−2}δ_{2} = 0  3^{n−1}δ_{3} = 0
\begin{aligned}3δ_{2} & = δ_{1}^{2}(1 − 3^{n−4}ε) \\ 9δ_{3} & = δ_{1}δ_{2}(1 − 3^{n−4}ε) \\ δ_{2}^{3} & = 27δ_{3}^{2} − γ^{2} \end{aligned}  \begin{aligned}δ_{1}α & = 0 \\ δ_{1}μ & = 0 \\ γδ_{1} & = 0 \end{aligned}  \begin{aligned}αμ & = 0 \\ αδ_{2} & = 0 \\ γα & = 0 \end{aligned}
δ_{1}ν = 0  μν = 0  δ_{2}ν = −γμ  γν = δ_{2}^{2}μ

Proof. We consider the spectral sequence for BB(n, ε) as an S^{1}-bundle over BB̃. It follows from lemma 4·9 that the differential in this spectral sequence sends ξ (a generator for E_{2}^{0,1}) to εβ − 3^{n−4}δ_{1} (because n ⩾ 5 we have that 3^{n−4}δ_{1} = 3^{n−4}δ′). The relations

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given between the generators that are in the image of the restriction from B̃ are the relations of theorem 4·8 after the substitution of β by 3^{n−4}εδ_{1}. In the spectral sequence E_{∞}^{∗,1} corresponds to the kernel of multiplication by εβ − 3^{n−4}δ_{1}, which is the ideal of H^{∗}(BB̃; ℤ) generated by δ_{1}β − β^{2} + α^{2}. This ideal is a free module of rank one for 𝔽_{3}[δ_{3}, α], and the product of δ_{1}β − β^{2} + α^{2} with any of δ_{1}, δ_{2}, γ or μ is zero. It follows that we only need one new generator ν in degree five, and that the only extra relations that we need are expressions for δ_{1}ν, δ_{2}ν, γν and μν in terms of the generators in the image of restriction from B̃. We may take ν as in lemma 4·10, so that its restriction to A (the subgroup of B(n, ε) of lemma 4·10) is \overline{γ}μ̄. The restrictions to A of the other generators may be determined easily from lemma 4·6 (which describes Res_{B} ; M), if we note that Res_{A}^{M}(τ′) = \overline{τ}, Res_{A}^{M}(μ′) = \overline{μ}, Res_{A}^{M}(γ′) = \overline{γ} and Res_{A}^{M}(β′) = 3^{n−4}ε\overline{τ}. Since Res_{B} ; M is injective on H^{od}(BB̃) (see lemma 3·12) it follows that Res_{A}^{B(n,ε)} is injective on the image of H^{od}(BB̃). To check the relations given for δ_{1}ν, δ_{2}ν and γν it therefore suffices to check that they yield valid relations when restricted to A. The relation μν = 0 follows by Fröbenius reciprocity, because ν may be taken to be a corestriction from P (n − 1), and the product of any element of H^{5}(BP (n − 1); ℤ) and any element of H^{3}(BP (n − 1); ℤ) is zero. ■ COROLLARY 4·12. For n ⩾ 5 and p = 3, the p-groups B(n, 1) and B(n, −1) defined in theorem 3·1 are non-isomorphic, but have isomorphic integral cohomology rings. Proof. It was shown in lemma 4·1 that these groups are distinct. We may define an isomorphism from H^{∗}(BB(n, −1); ℤ) to H^{∗}(BB(n, 1); ℤ) by sending each of the generators (as listed in theorem 4·11) for B(n, −1) to the generator of the same name for B(n, 1), except that we must send δ_{2} to (1 − 3^{n−4})δ_{2} and δ_{3} to (1 + 3^{n−4})δ_{3}. To check that this does describe an isomorphism, it helps to observe that for either group, the equations

3δ_{2}^{2} = δ_{2}δ_{1}^{2}(1 − 3^{n−4}ε) = 9δ_{3}δ_{1}

imply that δ_{2}^{2} has order 3^{n−4}. ■ 92

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For p ⩾ 5 it is much harder to calculate H^{∗}(BB̃; ℤ). Nevertheless we are able to prove the following theorem comparing the additive structure of H^{∗}(BB(n, ε); ℤ) for arbitrary p. THEOREM 4·13. For all m ⩾ 0, for all odd p, and all n ⩾ 6, the group H^{m}(BB(n, ε); ℤ) is independent of ε. Proof. First we consider the groups H^{m}(BB̃; ℤ). The E_{2}-page of the spectral sequence for B̃ as an extension of S^{1} by P_{2} has the property that E_{2}^{0,2j} is isomorphic to ℤ, whereas E_{2}^{i,j} is a finite p-group for (i, j) not of the form (0, 2j′). It follows that E_{∞}^{∗,∗} also has this property, and hence that H^{2m}(BB̃; ℤ) is an extension of a finite p-group by ℤ (which must of course be split) and H^{2m+1}(BB̃; ℤ) is a finite p-group. We know that B̃ has a subgroup isomorphic to P̃ of index p, and that the torsion in H^{∗}(BP̃; ℤ) in degree 2m (resp. in degree 2m + 1) has exponent p (resp. is trivial). Since Cor_{B} ; _{P}Res_{B} ; _{P} is multiplication by p it follows that in H^{2m}(BB̃; ℤ) the torsion has exponent at most p^{2}, and that H^{2m+1}(BB̃; ℤ) has exponent at most p (the odd degree case could also be deduced from lemma 3·12). Now fix for each m an isomorphism

H^{2m}(BB̃; ℤ) ≅ T^{2m} ⊕ ℤ,

where T^{2m} is the torsion subgroup of H^{2m}(BB̃; ℤ).

Now we consider the spectral sequence for BB(n, ε) as an S^{1}-bundle over BB̃. With notation as in lemma 4·9, we see that the differential d_{2} from E_{2}^{∗,1} to E_{2}^{∗,0} corresponds to multiplication by p^{n−4}δ′ − εβ from H^{∗}(BB̃; ℤ) to itself. During theorem 3·13 it was shown that H^{ev}(BB̃) maps on to H^{ev}(BB(n, ε)), or equivalently that E_{∞}^{od,1} is trivial. If we let φ stand for the map

φ : T^{2m} ⊕ ℤ ⟶ T^{2m+2} ⊕ ℤ

corresponding to multiplication by p^{n−4}δ′ − εβ from H^{2m}(BB̃; ℤ) to H^{2m+2}(BB̃; ℤ), then

E_{∞}^{2m+2,0} ≅ (T^{2m+2} ⊕ ℤ)/Im(φ) E_{∞}^{2m,1} ≅ Ker(φ)
E_{∞}^{2m+3,0} ≅ H^{2m+3}(BB̃; ℤ)/(p^{n−4}δ′ − εβ)H^{2m+1}(BB̃; ℤ).
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Since H^{od}(BB̃) has exponent p and n > 5, multiplication by p^{n−4}δ′ annihilates the group H^{2m+1}(BB̃; ℤ). Also −ε is a unit modulo p, so

E_{∞}^{2m+3,0} ≅ H^{2m+3}(BB̃; ℤ)/βH^{2m+1}(BB̃; ℤ).

Writing elements of H^{2m}(BB̃; ℤ) as ‘row vectors’ via the isomorphism with T^{2m} ⊕ ℤ, it may be checked that the map φ has matrix

φ = ( \begin{matrix}−εβ & 0 \\ p^{n−4}δ′ & p^{l} \end{matrix} ) : T^{2m} ⊕ ℤ ⟶ T^{2m+2} ⊕ ℤ,

where l is either n−4, n−3, or n−2, but depends only on m, not on ε. In any case, if p^{n−4} annihilates T^{2m}, which certainly happens for n ⩾ 6, then Ker(φ) and T^{2m+2} ⊕ ℤ/Im(φ) are independent of ε.

To complete the proof, it will suffice to show that H^{2m+3}(BB(n, ε); ℤ) is isomorphic to E_{∞}^{2m+2,1} ⊕ E_{∞}^{2m+3,0} for any choice of ε. The group E_{∞}^{2m+3,0} is a subgroup of H^{2m+3}(BB(n, ε); ℤ), and it follows from lemma 3·12 that this subgroup maps injectively to H^{2m+3}(BA; ℤ), where A is the abelian maximal subgroup of B(n, ε). Since A is isomorphic to C_{p^{n−2}} ⊕ C_{p}, the exponent of H^{2m+3}(BA; ℤ) is p, and hence if ξ is any element of H^{2m+3}(BB(n, ε); ℤ) such that pξ is an element of E_{∞}^{2m+3,0}, then pξ = 0. ■

94

5. The Integral Cohomology of the Held Group. In this section we shall examine integral cohomology of some finite groups by considering the restriction maps to the Sylow subgroups, and applying the results of section 2. Groups that we shall consider include extensions of (C_{p})^{2} by various subgroups of GL_{2}(p), and the Held group, a sporadic simple group of order 2^{10}.3^{3}.5^{2}.7^{3}.17 [He], [Co]. For the Held group we obtain explicitly the p-torsion subring of the integral cohomology for all odd primes p. C. B. Thomas has suggested [Th4] that the Atiyah-Hirzebruch-Rector spectral sequence [Re] might be a useful tool in studying the modular representation rings of groups. Since the E_{2} page of this spectral sequence for the p-modular ring consists of the cohomology of the group with coefficients of exponent prime to p our results for the Held group may have consequences concerning its 2-modular representation ring. Methods. If M is any p-local module for a finite group G (that is a module for which multiplication by any prime other than p is an isomorphism) then the restriction map from the cohomology of G with coefficients in M to that of its Sylow p-subgroup G_{p} is split injective, with splitting map a multiple of the corestriction. There is a characterisation of the image of this restriction due to Cartan-Eilenberg [CE] as the ‘stable elements’ of H^{∗}(BG_{p}; M)^{N(G_{p})}, that is the elements ξ satisfying

Res_{H∩gHg^{−1}}^{gHg^{−1}}c_{g}^{∗}(ξ) = Res_{H∩gHg^{−1}}^{H}(ξ).

Recent work of P. Webb [We] gives, for each of a wide variety of G-simplicial complexes, an alternative characterisation of the image of H^{∗}(BG; M) in H^{∗}(BG_{p}; M) as the kernel of a map from the direct sum of the cohomology of the vertex stabilisers to that of the edge stabilisers. If G never reverses an edge of the complex the maps involved are composed solely of restriction maps, so this characterisation can be simpler than the usual one. All the proofs given in this section use the stable element characterisation, but others were used in some of the author’s earlier proofs of these results. The author is indebted to

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J. C. Rickard for many discussions concerning Webb’s work.

If G_{p} (the Sylow subgroup of G) is abelian and we consider only trivial coefficients then a theorem of Swan [Sw] tells us that the stable elements are exactly the fixed points of H^{∗}(BG_{p}) under the action of its normaliser. (In [Sw] this theorem is stated for arbitrary coefficients, but this is incorrect. The author is indebted to D. J. Green for pointing this out to him and providing a counterexample.) The only cases of this theorem that we shall need are implied by the following lemma.

LEMMA 5·1. The corestriction from H^{i}(BC_{p}; ℤ) to H^{i}(BC_{p} ⊕ C_{p}; ℤ) is trivial for i > 0.

Proof. H^{i}(BC_{p} ⊕ C_{p}; ℤ) has exponent p for i > 0, and the restriction from C_{p} ⊕ C_{p} to C_{p} is onto. ■

COROLLARY 5·2. If a Sylow p-subgroup G_{p} of G is elementary abelian of rank two then H^{∗}(BG; ℤ) ≅ H^{∗}(BG_{p}; ℤ)^{N(G_{p})}. If G_{p} is isomorphic to P_{2} then the only non-trivial conditions for an element of H^{∗}(BG_{p}; ℤ)^{N(G_{p})} to be stable arise from intersections G_{p} ∩ G_{p}^{g} of order p^{2}. ■

We require the following lemma which is due to Dickson [Di].

LEMMA 5·3. Let GL_{2}(p) act in the usual way on a vector space spanned by x and x′. Then 𝔽_{p}[x, x′]^{SL_{2}(p)} is a polynomial algebra on generators a and b where

a = x^{p}x′ − x^{′p}x b = x^{p(p−1)} + x^{(p−1)^{2}}x^{′p−1} + … + x^{′p(p−1)},

and 𝔽_{p}[x, x′]^{GL_{2}(p)} is a polynomial algebra on a^{p−1} and b.

Proof. Consider the polynomials as functions from 𝔽_{p}^{2} to 𝔽_{p}. The action of SL_{2}(p) on 𝔽_{p}^{2} is transitive on non-zero elements, so if P is fixed it must represent either the trivial function, in which case it must divide by each of x′, x + ix′, and hence by a, or we may assume that it represents the function sending every non-zero point of 𝔽_{p}^{2} to 1. We shall describe such a polynomial as being identically one. We claim now that if P is identically

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one then the degree of P is divisible by p(p − 1). First, setting x = 0 and letting x′ vary we see that P is of the form xQ + x^{′n}, and that p − 1 divides n. Now

∑ λ_{i}x^{i}x^{′n−i} = P (x, x′) = P (x, x + x′) = ∑ λ_{i}x^{i}(x + x′)^{n−i},

where λ_{0} = 1. Balancing coefficients of xx^{′n−1} gives λ_{1} = λ_{1} + nλ_{0}, so p must divide n.

b = (∏_{i=0}^{p−1}(x + ix′))^{p−1} + x^{′p(p−1)}  (∗)

Equation (∗) makes apparent the fact that b is identically one, because the first term of the right hand side is one if x′ = 0 and zero otherwise. It is also apparent from (∗) that b is fixed by x′ ↦ x′, x ↦ x + x′, and there is a similar expression with x and x′ interchanged that illustrates the fact that b is fixed by x′ ↦ x′ + x, x ↦ x. These two elements generate SL_{2}(p), so b is invariant as claimed. Now given Q an arbitrary homogeneous polynomial fixed by SL_{2}(p), either Q is identically zero and we may divide Q by a, or Q is identically one and of degree divisible by p(p − 1), in which case for some n, Q − b^{n} is homogeneous of degree the same as Q and identically zero and fixed by SL_{2}(p).

The group GL_{2}(p) is generated by SL_{2}(p) and the map that fixes x and sends x′ to λx′, for λ a generator of 𝔽_{p}^{×}. Hence 𝔽_{p}[x, x′]^{GL_{2}(p)} is the fixed point subring of 𝔽_{p}[a, b] under the map which fixes b and sends a to λa. ■ COROLLARY 5·4. If H^{∗}(BC_{p} ⊕ C_{p}; ℤ) = ℤ[x, x′] ⊗ Λ[w] and GL_{2}(p) acts in the usual way on C_{p} ⊕ C_{p}, then

H^{∗}(BC_{p} ⊕ C_{p}; ℤ)^{SL_{2}(p)} = ℤ[a, b] ⊗ Λ[w] and
H^{∗}(BC_{p} ⊕ C_{p}; ℤ)^{GL_{2}(p)} = ℤ[a^{p−1}, b] ⊗ Λ[a^{p−2}w].

Proof. If λ ∈ GL_{2}(p), the action of λ sends w to det(λ)w. ■

Let G be an extension of (C_{p})^{2} by SL_{2}(p) or any subgroup of GL_{2}(p) containing SL_{2}(p). The metacyclic group P_{1} has a (C_{p})^{2} subgroup, but this contains a characteristic

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subgroup of order p (the subgroup of p-th powers). It follows that the Sylow subgroups of G are isomorphic to P_{2}, and so G must be the split extension. The groups (C_{p})^{2}: SL_{2}(p) and (C_{p})^{2}: GL_{2}(p) seem to arise frequently in cohomology calculations, so we determine the p-part of their cohomology rings below. THEOREM 5·5. Let H ≅ C_{p}⊕C_{p}, and let Q be a subgroup of GL_{2}(p) containing SL_{2}(p) and acting in the standard way on H. Also let C be a Sylow subgroup of Q, so C ≅ C_{p}, and H: C ≅ P_{2} is a Sylow subgroup of H: Q. Then the image of H^{∗}(BH: Q; ℤ) in H^{∗}(BH: C; ℤ) is the subring

{ξ ∈ H^{∗}(BH: C; ℤ)^{N(H:C)} | Res_{H}^{H:C}(ξ) ∈ H^{∗}(BH; ℤ)^{Q}}.

Note also that N_{H:Q}(H: C) = H: N_{Q}(C), that N_{Q}(C) is isomorphic to C_{p−1} ⊕ det(Q), and N_{Q}(C)/C_{Q}(C) is isomorphic to C_{(p−1)/2}. Proof. The remarks at the end of the statement are standard group theory, and were included to indicate how the main result can be applied. By considering the restriction maps from H: Q to H: C and H it may be seen that the image is no bigger than claimed. Conversely, corollary 5·4 tells us that given an element ξ of H^{∗}(BH: C; ℤ) fixed by the action of the normaliser we need only check that for each g ∉ N(H: C),

Cor_{H}^{H:C}Res_{H}^{g(H:C)g^{−1}}c_{g}^{∗}(ξ) = |C|ξ.

Since H is normal

Cor_{H}^{H:C}Res_{H}^{g(H:C)g^{−1}}c_{g}^{∗}(ξ) = Cor_{H}^{H:C}c_{g}^{∗}Res_{H}^{H:C}(ξ),

and this will be equal to |C|ξ if Res(ξ) is fixed by Q. ■ COROLLARY 5·6. If we take P_{2} ⩽ (C_{p})^{2}: SL_{2}(p) such that the subgroup ⟨B, C⟩ is the normal subgroup, then the image of H^{∗}(B(C_{p})^{2}: SL_{2}(p); ℤ) in H^{∗}(BP_{2}; ℤ) has module generators

ζ^{i}α^{j}β^{k}μ^{ε} for j > 0, ε = 0, 1 and i + 2j ≡ k (p − 1),
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ζ^{(p−1)i−j}χ_{j} for j < p − 1, ζ^{i(p−1)}(χ_{p−1} + β^{p−1}),

ζ^{i}α^{j}ν where i + 2j ≡ −3 (p − 1), and (ζβ)^{i}(ζ^{p−1} + β^{p(p−1)})^{j}μ^{ε}.

Proof. First we determine the fixed points of H^{∗}(BP_{2}) under the action of its normaliser, which is P_{2}: C_{p−1}. This group contains a chain of normal subgroups ⟨C⟩, ⟨B, C⟩, P_{2}. Let λ be a generator for 𝔽_{p}^{×}. The action by conjugation of a generator of C_{p−1} on ⟨B, C⟩ must be faithful, preserve ⟨C⟩, and have determinant one, so without loss of generality may be taken to be C ↦ C^{λ}, B ↦ B^{λ^{−1}}. Also this element acts on the quotient group P_{2}/⟨C⟩ = ⟨\overline{A}, \overline{B}⟩. The action on ⟨C⟩ is the determinant of the action on ⟨\overline{A}, \overline{B}⟩, so the element must send \overline{A} to \overline{A}^{λ^{2}}. The induced action on H^{∗}(BP_{2}) is

\begin{matrix}α & ↦ λ^{2}α \\ β & ↦ λ^{−1}β \end{matrix}  \begin{matrix}μ & ↦ μ \\ ν & ↦ λ^{3}ν \end{matrix}  \begin{matrix}ζ & ↦ λ^{p}ζ \\ χ_{i} & ↦ λ^{i}χ_{i} \end{matrix}

Since each monomial is an eigenvector, it is easy to identify the fixed subspace, which has the following module generators.

ζ^{i}α^{j}β^{k}μ^{ε} for j > 0, ε = 0, 1 and i + 2j ≡ k (p − 1),
ζ^{(p−1)i−j}χ_{j}, ζ^{i}α^{j}ν for i + 2j + 3 ≡ 0 (p − 1).

The kernel of the restriction to ⟨B, C⟩ is the ideal generated by α, χ_{2}, …, χ_{p−2}, χ_{p−1}+β^{p−1}, pζ, and ν, and this ideal together with the subring generated by ζ, β, and μ spans H^{∗}(BP_{2}) as a module. Identifying elements of this subring with their images under restriction to ⟨B, C⟩, it follows from corollary 5·4 that the fixed points of H^{∗}(B⟨B, C⟩) under the action of N(⟨B, C⟩) are generated by ζβ, ζ^{p−1} + β^{p(p−1)}, and μ. The result follows. ■

The Held Group. Now we shall consider the cohomology of the Held group He, a sporadic simple group of order 2^{10}.3^{3}.5^{2}.7^{3}.17, [He]. The Sylow 3- and 7-subgroups are isomorphic to P_{2}, so the results of section 2 will be relevant. By the end of this section we shall have determined H^{∗}(BHe; ℤ[\frac{1}{2}]) by considering separately each of the other primes that

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divide |He|. We list our results for the primes in order of increasing difficulty. In the sequel we shall assume any results concerning He and other groups that are stated explicitly in the Atlas [Co], but we shall prove everything else that we require in a series of lemmata, one for each p. Much of the content of these is in Held’s original paper [He].

The prime 17. If G has a cyclic Sylow subgroup C, it follows from Swan’s theorem (or directly from the stable element criterion if C has order p) that H^{∗}(BG; ℤ)_{p} is a polynomial algebra on a generator of degree 2|N_{G}(C) : C_{G}(C)|. More recently Thomas has shown that in this case the generator may be taken to be a Chern class of a representation (see [Th3], together with the observation that c.(−ρ)c.(ρ) = 1 so the Chern subring is generated by classes of actual representations). In He there are two classes of elements of order 17, so |N_{He}(C_{17}) : C_{He}(C_{17})| has order 8, and H^{∗}(BHe; ℤ)_{17} is a polynomial algebra on a generator of degree 16.

The prime 5.

LEMMA 5·7. The group He has a unique conjugacy class of elements of order 5, and its Sylow 5-subgroup is 5A^{2}, that is it is isomorphic to C_{5} ⊕ C_{5}. The centraliser of 5A^{2} is 5A^{2}, and N(5A^{2}) ≅ (C_{5})^{2}: C_{4}A_{4}. The group Aut((C_{5})^{2}) contains a unique class of C_{4}A_{4} subgroups.

Proof. The assertions concerning He are stated explicitly in the Atlas, except for the claim that 5A^{2} is its own centraliser. Note that C(5A) has order 300, and of course C(5A^{2}) ⊂ C(5A). He has only two classes of maximal subgroups of order divisible by 25, N(5A^{2}), and another isomorphic to S_{4}(4): C_{2}. The only maximal subgroups of S_{4}(4): C_{2} with order divisible by 25 are isomorphic to (A_{5} × A_{5}): (C_{2} × C_{2}). In this group the subgroup C_{5} × A_{5} has order 300 and centralises an element of order 5, so must be C(5A^{2}), but C_{5} × C_{5} is its own centraliser in C_{5} × A_{5}.

It remains to check the assertions concerning GL_{2}(5). A Sylow 2-subgroup of GL_{2}(5), for example the group of diagonal and ‘anti-diagonal’ matrices, is non-abelian of order

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32. It follows that any abelian subgroup of GL_{2}(5) of order 16 must contain the centre. P GL_{2}(5) ≅ S_{5}, so the result follows since S_{5} has a unique class of A_{4} subgroups. ■ THEOREM 5·8. H^{∗}(BHe; ℤ)_{5} is generated by elements α, β, γ and χ, where α has degree 16, β and γ have degree 24 and χ has degree 15, subject only to the relations that all generators have order 5 and γ^{2} = 3(β^{2} +γ^{3}). The cohomology operation δ_{p}P^{1}π_{∗} sends χ to a multiple of γ. Proof. It follows from corollary 5·2 that H^{∗}(BHe; ℤ)_{5} is isomorphic to the fixed points of H^{∗}(B5A^{2}; ℤ) under the action of N(5A^{2}). By lemma 5·7 we may take as generators for N(5A^{2})/5A^{2} considered as a subgroup of GL_{2}(p) the matrices

M_{1} = ( \begin{matrix}2 & 0 \\ 0 & 3 \end{matrix} ) M_{2} = ( \begin{matrix}2 & 0 \\ 0 & 2 \end{matrix} ) M_{3} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) M_{4} = ( \begin{matrix}1 & 1 \\ 2 & −2 \end{matrix} ) .

Note that these define a composition series for this group, in the sense that the subgroup generated by the first i of them is normal in the subgroup generated by the first i + 1 of them. Let H^{∗}(B5A^{2}; ℤ) be generated by elements δ and δ′ of degree 2 and ε of degree 3, such that the above matrices describe the action of C_{4}A_{4} on the basis δ, δ′ for H^{2}. Thus M_{1} sends δ to 2δ, δ′ to 3δ′, and fixes ε. The fixed point subring under the action of M_{1} is easily seen to be generated by δ^{4}, δ^{′4}, δδ′, and ε. Since M_{2} normalises M_{1}, M_{2} acts on this fixed point subring, and in fact it fixes δ^{4} and δ^{′4} and sends δδ′ to −δδ′ and ε to −ε. The fixed point subring is generated by δ^{4}, δ^{′4}, δ^{2}δ^{′2} and δδ′ε. The action of M_{3} fixes δ^{2}δ^{′2}, exchanges δ^{4} and δ^{′4} and sends δδ′ε to minus itself, so has fixed point subring generated by δ^{4} + δ^{′4}, δ^{2}δ^{′2}, and (δ^{5}δ′ − δ^{′5}δ)ε. This is a polynomial algebra on two generators of degree 8 tensored with an exterior algebra on one generator of degree 15.

The action of M_{4} clearly fixes the exterior generator, but has no fixed point in degree 8. To complete the proof it suffices to show that whenever C_{3} acts non-trivially on 𝔽_{5}[x, x′], then the fixed point subring is generated by elements a, b, c, of degrees two, three and three respectively (where x is of degree one), subject only to c^{2} = 3(b^{2} +a^{3}). For this, extend the action to one on 𝔽_{25}[x, x′], where it may be diagonalised. Note that if y is an eigenvector

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in degree one with eigenvalue ω, then ȳ (defined by conjugating the coefficients of y as an expression in x and x′) is an eigenvector of value ω̄. The fixed point subring over 𝔽_{25} is generated by yȳ, y^{3} and ȳ^{3}. An element of this ring is in 𝔽_{5}[x, x′] iff it is invariant under conjugation. Elements invariant under conjugation are the sum of a symmetric polynomial over 𝔽_{5} in y and ȳ, and \sqrt{3} times an antisymmetric polynomial over 𝔽_{5} in y and ȳ. The symmetric polynomials in y and ȳ expressible in terms of yȳ, y^{3} and ȳ^{3} form a polynomial ring on yȳ and y^{3} + ȳ^{3}, while the antisymmetric polynomials form a free module for the symmetric ones with basis y^{3} − ȳ^{3}. Our three generators are a = yȳ, b = y^{3} + ȳ^{3} and c = \sqrt{3}(y^{3} − ȳ^{3}). ■

The author wishes to thank Dr. D. J. Benson who found an error in an earlier statement of theorem 5·8. The prime 3. Lemma 5·9. Let P be a Sylow 3-subgroup of He. Then P ≅ P_{2}, and He contains two (self-inverse) conjugacy classes of elements of order 3, 3A and 3B. N(P) is isomorphic to P: D_{8} and D_{8} acts faithfully. The centre of P is 3A. There are two classes of C_{3} ⊕ C_{3} subgroups of He, 3A^{2} which is contained in a unique Sylow 3-subgroup, and 3A3B, which has

N(3A3B) ≅ C_{3}·(S_{3} × S_{4}) C(3A3B) ≅ C_{3}·(C_{3} × V_{4}).

There is a unique class of D_{8} subgroups of GL_{2}(3). Proof. The order of C(3B) is not divisible by 27, so Z(P) must be 3A. One of the maximal subgroups of He is N(3A) ≅ C_{3}·S_{7}, and we shall work largely within this group. For example, N(P) ⩽ N(3A), and we see that

N(P) = C_{3}·N_{S_{7}}(C_{3} × C_{3}) ≅ P: D_{8}.

If we write 3X for the conjugacy class in He represented by the 3-cycles in S_{7}, and 3Y for that represented by products of two disjoint 3-cycles, then

C(3A, 3X) ≅ C_{3}.(C_{3} × V_{4}) C(3A, 3Y) ≅ C_{3} × C_{3}

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We deduce that there are two classes of C_{3} ⊕ C_{3} subgroup in He, one of which must be 3A^{2}, and the other 3A3B containing a unique 3A subgroup. Now

|N(3A, 3B) : C(3A, 3B)| ⩽ 12 and |N(3A^{2}) : C(3A^{2})| ⩽ 48,

but N(3A, 3Y) ∩ N(3A) ≅ C_{3}·(S_{3} × S_{3}).C_{2}, which already has order 24 times that of C(3A, 3Y). Therefore 3Y = 3A, and N(3A^{2}) is either C_{3}·(S_{3}×S_{3}).C_{2} or a group containing this with index two. In either case N(3A^{2}) contains a unique Sylow subgroup. Also 3X = 3B, and N(3A, 3B) is as claimed.

Aut(𝔽_{9}, +) contains the subgroup generated by 𝔽_{9}^{×} and the Fröbenius automorphism, which is semidihedral of order 16, and contains a unique D_{8} subgroup. Since the Sylow 2subgroups of GL_{2}(3) have order 16 it follows that all D_{8} subgroups of GL_{2}(3) are conjugate.

■

THEOREM 5·10. H^{∗}(BHe; ℤ)_{3} is isomorphic to the subring of H^{∗}(BP_{2}; ℤ) generated by χ_{2}, α^{2} + β^{2}, α^{2}β^{2}, ζ(αν − βμ), and ζ^{2}. Proof. Identifying Aut(P_{2})/P_{2} with GL_{2}(p), it follows from lemma 5·9 that we may take N(P_{2}) as being generated by

M_{1} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )  M_{2} = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} )  M_{3} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

(Of course we only need M_{1} and M_{3}). The action of M_{1} fixes β, ν, and χ_{2}, and multiplies the other generators by −1. The action of M_{2} fixes α, μ, and χ_{2} and multiplies the other generators by −1. It follows that the fixed point subring under M_{1} and M_{2} is spanned by the elements

ζ^{2i}, ζ^{2i}χ_{2}, ζ^{2i+ε}α^{2j+ε}ν,
ζ^{i}α^{j}β^{k}ν^{ε} where ε + j ≡ i ≡ k (2), and j = 0 or k + ε ⩽ 3.

The action of M_{3} on the generators is given by

\begin{matrix}α & ↦ −β \\ β & ↦ α \end{matrix}  \begin{matrix}μ & ↦ ν \\ ν & ↦ −μ \end{matrix}  \begin{matrix}ζ & ↦ −ζ \\ χ_{2} & ↦ χ_{2} \end{matrix}
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and a simple calculation shows that the fixed points under the action of D_{8} are spanned by

ζ^{2i}χ_{2}, ζ^{2i+1}(α^{2j+1}ν − β^{2j+1}μ),
ζ^{2i}(α^{2j} + β^{2j}), ζ^{2i}α^{2j}β^{2},

and further that this is the subring generated by the elements of the statement. To complete the proof we must show that all these elements are stable. Without loss of generality we may take ⟨A, C⟩ and ⟨B, C⟩ to be the 3A3B subgroups of P_{2}, and by corollary 5·2 it suffices to check that for H = ⟨B, C⟩ and ξ any of the generators of the statement then

Cor_{H}^{P_{2}}c_{g}^{∗}Res_{H}^{P_{2}}(ξ) = 3ξ = Cor_{H}^{P_{2}}Res_{H}^{P_{2}}(ξ) for all g ∈ N_{He}(H).

If we take H^{∗}(BH; ℤ) ≅ ℤ[β′, γ] ⊗ Λ[δ], then the action of an element of N_{He}(H) sends β′ to λβ′, γ to λ′γ + λ′′β′ and δ to λλ′δ, where λ, λ′ ∈ 𝔽_{3}^{×} and λ′′ ∈ 𝔽_{3}. The restriction from P_{2} to H is given by

\begin{matrix}α & ↦ 0 \\ β & ↦ β′ \end{matrix}  \begin{matrix}μ & ↦ δ \\ ν & ↦ 0 \end{matrix}  \begin{matrix}ζ & ↦ γ^{3} − β^{2}γ \\ χ_{2} & ↦ −β^{2} \end{matrix} .

It can be checked that the images of the generators of the statement are fixed by N_{He}(H), so the generators satisfy the stability condition. ■

The prime 7.

LEMMA 5·11. Let P be a Sylow 7-subgroup of He. then P ≅ P_{2}, and He contains five conjugacy classes of element of order seven, 7A, 7B which are inverse to each other, 7C which is self-inverse, and 7D, 7E which are inverse to each other. The central elements are 7C, and in fact N(7C) = N(P) ≅ P: (S_{3} × C_{3}), with S_{3} × C_{3} acting faithfully on P. There is a unique class of such subgroups of GL_{2}(7). There are three conjugacy classes of C_{7} × C_{7} subgroups of He, with normalisers

N(7C, 7AB) ≅ P: (C_{3} × C_{3}) N(7C^{2}) ≅ (C_{7})^{2}: SL_{2}(7) N(7C, 7DE) ≅ P : (C_{2} × C_{3}). 104

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7C^{2} is its own centraliser. Some other subgroups are

(C_{7})^{2}: C_{6} ≅ N(7DE) ⩽ N(P) N(7AB) ≅ C_{7}: C_{3} × L_{2}(7),
(C_{7})^{2}: (C_{3} × C_{3}) ≅ N(7C, 7AB) ∩ N(7AB) ⩽ N(P).

Proof. The normaliser of a Sylow 3-subgroup of GL_{2}(7) is C_{6} ≀C_{2}, which contains a unique conjugacy class of C_{3} ≀ C_{2} ≅ S_{3} × C_{3} subgroup. The Atlas tells us that N(7C^{2}) is as claimed above, but L_{2}(7) is not a subgroup of GL_{2}(7), so the only possibility is that 7C^{2} is self-centralising. The Atlas tells us that N(7C) is P: (S_{3} × C_{3}) as claimed, and since no element of S_{3} × C_{3} can centralise a 7C^{2} subgroup of P we see that S_{3} × C_{3} acts faithfully on P. We already know that such an action is essentially unique, and so we fix generators for this group considered as acting on CĀ ; 7 ⊕ CB̄ ; 7 = P_{2}/⟨C⟩ as below.

( \begin{matrix}λ & 0 \\ 0 & λ′ \end{matrix} ) where λ^{3} = λ^{′3} = 1 ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Direct computation shows that there are four conjugacy classes of C_{7} subgroup in P under the action of N(P), which are tabulated in figure 5·1 together with their centralisers in N(P). For each of them except ⟨C⟩ the centraliser has index three in the normaliser.

C_{7} subgroup  Centraliser in N(P)
⟨C⟩  P.C_{3}
⟨AC^{i}⟩ ∼ ⟨BC^{i}⟩  (C_{7})^{2}: C_{3}
⟨AB^{2^{i}}C^{j}⟩  (C_{7})^{2}: C_{2}
⟨AB^{−2^{i}}C^{j}⟩  (C_{7})^{2}

5·1. Table of C_{7} subgroups of N(P).

He has only two classes of maximal subgroup whose order divides by 7^{3}, N(7C^{2}) and N(7C). We see that ⟨A, C⟩ cannot be 7C^{2} because its normaliser has order divisible by 9. Also its normaliser must be contained in N(7C), so is P.(C_{3} × C_{3}). ⟨A⟩ cannot be 7C by the above, but neither can it be 7DE since the order of C(7DE) is coprime to 3, hence

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⟨A⟩ must be 7AB. There are 7C^{2} subgroups, but ⟨C, AB^{2^{i}}⟩ cannot be 7C^{2} because it has elements centralised by an involution. It follows that ⟨AB^{2^{i}}⟩ must be 7DE and ⟨AB^{−2^{i}}⟩ is 7C. Comparing orders tells us that N(7DE) is contained in N(P) and is as claimed. The normaliser of 7AB is a maximal subgroup listed in the Atlas. The intersection of N(7C, 7AB) and N(7AB) has index seven in N(7C, 7AB), because this group permutes its 7AB subgroups transitively, so is as claimed. ■ THEOREM 5·12. With notation as in lemma 5·11, H^{∗}(BN(P); ℤ)_{7} is mapped via restriction to the subring of H^{∗}(BP; ℤ) generated by

α^{3} + β^{3}, α^{3}β^{3}, χ_{6}, α^{5}βμ − α^{2}β^{4}μ, ζαμ ζχ_{5}, ζ(α^{5}β^{2} − α^{2}β^{5}), ζ^{2}αβ,
ζ^{2}(α^{2}ν − β^{2}μ), ζ^{2}χ_{4}, ζ^{3}χ_{3}, ζ^{3}(α^{3} − β^{3}), ζ^{4}χ_{2}, ζ^{5}(α^{2}ν + β^{2}μ), ζ^{6}.

Proof. As in theorem 5·10 we take a basis for H^{∗}(BP; ℤ) ⊗ 𝔽_{7} consisting of the monomials

ζ^{i}, ζ^{i}χ_{j}, ζ^{i}α^{j}ν, ζ^{i}α^{j}β^{k}μ^{ε} where j = 0 or k + ε ⩽ 6.

All monomials are eigenvectors for the action of the C_{3} × C_{3} subgroup of N(P)/P, and it may be checked that the fixed monomials are

ζ^{3i}, ζ^{3i−j}χ_{j}, ζ^{3i+2}α^{3j+2}ν
ζ^{i}α^{j}β^{k}μ^{ε} where i + j + ε ≡ 0 and i + k ≡ ε modulo 3.

The action of the involution represented by the matrix in the statement of lemma 5·11 on the original generators is

\begin{matrix}α & ↦ β \\ β & ↦ α \end{matrix}  \begin{matrix}μ & ↦ −ν \\ ν & ↦ −μ \end{matrix}  \begin{matrix}ζ & ↦ −ζ \\ χ_{i} & ↦ (−1)^{i}χ_{i} \end{matrix}

and the fixed points under the whole of N(P)/P are seen to be spanned by

ζ^{6i}, ζ^{6i−j}χ_{j}, ζ^{6i+2}(α^{6j+2}ν − β^{6j+2}μ), ζ^{6i}(α^{3j} + β^{3j}),
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ζ^{6i−1}(α^{6j−1}ν + β^{6j−1}μ), ζ^{i}α^{j}β^{k}μ^{ε} where i ≡ 2j − ε ≡ j + k (6),

ζ^{i}(α^{j}β^{k}μ^{ε} + α^{j−3}β^{k+3}μ^{ε}) where i ≡ 2j − ε ≡ j + k + 3 (6),
ζ^{i}(α^{j}β^{k}μ^{ε} − α^{j−3}β^{k+3}μ^{ε}) where i ≡ 2j − ε + 3 ≡ j + k (6).

More calculation shows that the fifteen elements of the statement generate this subring.

■

In a sense this almost completes the calculation of H^{∗}(BHe; ℤ)_{7}, which may be described as the subring of stable elements of H^{∗}(BN(P); ℤ)_{7}. We know from corollary 5·2 that only conjugates P^{g} of P such that P ∩ P^{g} has order 49 can give any extra conditions, and by lemma 5·11 the only such intersections are of type 7C^{2}. It follows as in theorem 5·5 that ξ ∈ H^{∗}(BN(P); ℤ)_{7} is in the image of H^{∗}(BHe; ℤ) iff

Cor_{⟨AB^{−1},C⟩}^{P}φ^{∗}Res_{⟨AB^{−1},C⟩}^{P}(ξ) = pξ,  (∗)

where φ is some fixed element of Aut(⟨A, C⟩) of order 7 not fixing ⟨C⟩. (It is not obvious, but true, that all such φ will give the same condition.)

The description of H^{∗}(BHe; ℤ)_{7} given by condition (∗) is not very explicit, since the condition is difficult to check. For example, this description is not much use for calculating the Poincaré series of H^{∗}(BHe; ℤ)_{7} ⊗ 𝔽_{7}. The following lemma gives a more useful description of H^{∗}(BHe; ℤ)_{7}, which will allow us to completely determine H^{∗}(BHe; ℤ)_{7} in theorem 5·14.

LEMMA 5·13. Let K be the subgroup ⟨AB^{−1}, C⟩ of P, let Res = Res_{K}^{P}, and define ζ′, ε, δ, elements of H^{∗}(BK; ℤ) by

ζ′ = Res(ζ) ε = Res(α) = −Res(β) δ = Res(μ) = −Res(ν).

Now let S be the subring of H^{∗}(BK; ℤ) generated by ζ′ε, ζ^{′6}+ε^{42} and δ. Then H^{∗}(BHe; ℤ)_{7} is isomorphic to (Res)^{−1}(S) ∩ H^{∗}(BN(P); ℤ)_{7}.

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Proof. From corollary 5·2 it follows that an element ξ of H^{∗}(BN(P); ℤ)_{7} is in the image of the restriction from H^{∗}(BHe; ℤ)_{7} iff

c_{g}^{∗}Res_{g^{−1}P g∩P}^{P} (ξ) = Res_{P ∩gP g^{−1}}^{P}(ξ)

for all g such that P^{g} ∩ P has order p^{2}. This can only happen if P^{g} ∩ P is of type 7C^{2} (see lemma 5·11). Since N(P) acts transitively on such subgroups we may restrict to the case when P^{g} ∩ P is the subgroup K = ⟨AB^{−1}, C⟩ and g normalises K. Lemma 5·11 tells us that N(7C^{2}) ≡ (C_{7})^{2}: SL_{2}(7), and the ring S described in the statement is (by corollary 5·4) the fixed point subring of H^{∗}(BK; ℤ) under the action of SL_{2}(7). ■

THEOREM 5·14. With notation as in lemma 5·11, H^{∗}(BHe; ℤ)_{7} is mapped via restriction to the subring of H^{∗}(BP; ℤ) generated by the following elements.

α^{3} + β^{3}, χ_{6} − α^{3}β^{3}, ζαμ, ζχ_{5}, ζ^{2}αβ, ζ^{2}(α^{2}ν − β^{2}μ),
ζ^{2}χ_{4}, ζ^{3}χ_{3}, ζ^{3}(α^{3} − β^{3}), ζ^{4}χ_{2}, ζ^{5}(α^{2}ν + β^{2}μ), ζ^{6} − α^{39}β^{3}.

Proof. The twelve elements of the statement with α^{3}β^{3}, (α^{5}β − α^{2}β^{4})μ, and ζ(α^{5}β^{2} − α^{2}β^{5}) generate the subring of H^{∗}(BP) described in theorem 5·12. Also the images of the twelve elements under restriction from P to the subgroup K lie in the subring S (see lemma 5·13), so these elements are contained in H^{∗}(BHe; ℤ)_{7}. It remains to show that no other generators are required. It may be verified that the image of the subring of this statement under restriction to K is the subring S′ of S generated by ζ^{′6} + ε^{42}, ζ^{′2}ε^{2}, ζ^{′3}ε^{3}, ζ′εδ and ζ^{′2}ε^{2}δ. The elements α^{3}β^{3}, (α^{5}β − α^{2}β^{4})μ, and ζ(α^{5}β^{2} − α^{2}β^{5}) restrict to −ε^{6}, −2ε^{6}δ, and 2ζ′ε^{7} respectively. We claim now that the intersection of S and Res_{K}^{P}(H^{∗}(BN(P); ℤ)) is the subring S′ of S generated by the elements of the statement. Equivalently we claim that the intersection of S and the ring generated by S′, ε^{6}, ε^{6}δ and ζ′ε^{7} is S′. In even degrees S \ S′ is the set of elements of S divisible exactly once by ε, while elements of even degree in the ring generated by S′, ε^{6} and ζ′ε^{7} will be divisible by

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ε either at least twice or not at all. A similar argument works in odd degrees, since the odd degree part of S \ S′ is the set of elements not divisible by ε.

Now we have shown that the subring of H^{∗}(BP; ℤ) generated by the elements of the statement has the same image under restriction to K as H^{∗}(BHe; ℤ)_{7}, so we may assume that any remaining elements of H^{∗}(BHe; ℤ)_{7} are in the kernel of Res_{K}^{P}. It will suffice to show that the kernel of Res_{K}^{P} as a map from H^{∗}(BN(P); ℤ)_{7} is contained in the subring generated by the elements of the statement. We claim that this kernel is the ideal of H^{∗}(BN(P); ℤ)_{7} generated by α^{3} + β^{3}, ζ^{5}(α^{2}ν + β^{2}μ), ζ^{4}χ_{2}, ζ^{3}χ_{3}, ζ^{2}χ_{4}, ζχ_{5} and χ_{6} − α^{3}β^{3}. We easily reduce to the case of elements of the kernel of the form ζ^{i}P (α, β) or ζ^{i}(P (α, β) + λα^{r}ν). Notice that if n ⩾ 7 and

ζ^{j}(∑_{i=0}^{6} λ_{i}α^{n−i}β^{i} + λ_{7}β^{n}) ∈ H^{∗}(BN(P); ℤ)_{7}, then

ζ^{6k+j}(∑_{i=0}^{6} λ_{i}α^{6m+n−i}β^{i} + λ_{7}β^{6m+n}) ∈ H^{∗}(BN(P); ℤ)_{7},

so the remaining cases reduce to ζ^{i}P (α, β) and ζ^{i}(P (α, β)μ+λα^{r}ν)) where the degree of P is at most 13 and i ⩽ 5. It can be checked by lengthy calculation that all such elements are in the ideal of H^{∗}(BN(P); ℤ)_{7} generated by α^{3} + β^{3} and ζ^{5}(α^{2}ν + β^{2}μ). To check that the kernel is contained in the subring of the statement it suffices to find an expression for each product of one of the generators of the kernel (as an ideal) and one of α^{3}β^{3}, (α^{5}β −α^{2}β^{4})μ and ζ(α^{5}β^{2} − α^{2}β^{5}) in terms of the twelve elements of the statement. This too is a routine calculation. ■

109

6. Yagita’s Invariant. The following short section concerns an invariant related to free actions of groups on products of spheres. In [Ya] Yagita defined an integer invariant p∘(G) for a prime p and a finite group G which must divide n if G acts freely on a product (S^{n−1})^{k} with trivial action on H^{∗}((S^{n−1})^{k}; ℤ). The Invariant. Let u be a generator for H^{2}(BC_{p}; ℤ). To each inclusion i of C_{p} in G Yagita associates an integer m(i), which is the largest m such that i^{∗}(H^{∗}(BG; ℤ)) is contained in the subring of H^{2}(BC_{p}; ℤ) generated by u^{m}. Then p∘(G) is defined to be twice the lowest common multiple of all such m(i).

We recall that given a CW complex X with a free cellular action of G (such a space is referred to as a free G complex), there is a Cartan-Leray spectral sequence with E_{2} page H^{i}(BG; H^{j}(X)) converging to a filtration of H^{i+j}(X/G). Now let X be a finite free G complex, with cohomology ring isomorphic to an exterior algebra on k generators with trivial G action. An inclusion i : C_{p} ↣ G induces a map of Cartan-Leray spectral sequences, where the map on E_{2} pages is essentially the restriction, and the map on E_{∞} pages is induced by the projection from X/C_{p} to X/G. By considering this map of spectral sequences Yagita shows that m(i) must divide n.

While exploring the possibility of defining a similar invariant in terms of the images of Ch(G), the author discovered two errors in [Ya], and wrote the following letter to Yagita. At the time of writing there seemed to be no connection between the new invariant and actions on spheres, but the author has recently found a connection which is explained in lemma 6·1. The preprint referred to in the letter contained an incorrect proof of corollary 3·7, and a determination of the order of H^{n}(BG(a, 1); ℤ) which disagrees with Yagita’s theorem 2·4. We explain the method of calculation after the letter. Dear Prof. Yagita,

I am writing to you to report some work that I have done which was inspired by your paper ‘On the dimension of spheres whose product admits a free action by a non-abelian

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group’, Quart. J. Math. Oxford 36 (1985) 117–127. I have also found two mistakes in this paper. I find that theorem 2·4 is incorrect for odd primes p for the groups G_{2}(a, 1), but that your other results that rely on this theorem are correct. Secondly, your proof of lemma 1·7 is incorrect; in this case I have been unable to find a proof or a counterexample. Before explaining my criticisms, I shall define my adaptation, pc(G), of your invariant. My invariant is easier to calculate than yours, and gives an upper bound for yours, but does not seem to bear any direct relationship to actions on spheres. I shall consider only odd primes p.

You define, for each i : ℤ_{/p} ↣ G, m(i) to be the greatest integer m such that i^{∗}(R) ⊂ ℤ[u^{m}]/(pu^{m}), where R is H^{∗}(BG; ℤ). I define n(i) similarly, except that I take R to be the subring of H^{∗}(BG; ℤ) generated by Chern classes of representations of G. I then define

pc(G) = 2L.C.M.{n(i) | i : ℤ_{/p} ↣ G}.

(Incidentally, your stated definition of p∘(G) is L.C.M.{2m(i)}, which gives p∘(G) = 1 if p does not divide the order of G, and gives easy counterexamples to lemma 1·7. I shall use the definition 2L.C.M.{m(i)}.)

Various properties of pc(G), such as 1 to 4 below, may be proven using the proofs you use for p∘(G):

1. If G is abelian, then pc(G) = 2.

2. If N ↣ G ↠ Q, then pc(G) | L.C.M.{pc(Q), pc_{N}(G)}.

3. If H < G, then pc(H) | pc_{H}(G).

4. If H < G, then pc_{H}(G) | pc(G).

5. p∘(G) | pc(G), so my invariant gives an upper bound for yours. A lower bound would of course be more useful! It follows from 1 and 2 that

6. pc(G) = pc_{G′}(G), where G′ is the derived subgroup of G. Similarly, it can be shown

that

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6. ′ p∘(G) = p∘_{G′}(G).

If G is a minimal non-abelian p-group or an extra-special p-group, then G′ is isomorphic to ℤ_{/p}, so 6′ could be used to simplify your calculations for these groups.

Using 6 it may be shown that

7. If G is a minimal non-abelian p-group, then pc(G) = 2p, and hence

8. If G is a p-group, then pc(G) = 2 if and only if G is abelian. The regular representation of G restricts to a ℤ_{/p} subgroup as |G : ℤ_{/p}| copies of the

regular representation of ℤ_{/p}, so its total Chern class restricts to ℤ_{/p} as

(∏_{i=0}^{p−1}(1 − iu))^{|G:ℤ_{/p}|} = (1 − u^{p−1})^{|G:ℤ_{/p}|},

and so

9. pc(G) | 2(p − 1)p^{n−1}, where p^{n} is the p-part of |G|.

Using also representations of G induced up from irreducible representations of ℤ_{/p} I can show that pc(G) | 2|G|, but the calculation is longer.

I first noticed the flaw in lemma 1·7 while trying to prove the analogous result for pc(G). Let N ◁ G, and let x ∈ H^{∗}(BN). Then letting c_{t}^{∗} be the map H^{∗}(BN)⟶H^{∗}(BN) induced by n ↦ n^{t}, we have:

Res_{N}^{G}N_{N}^{G}(x) = ∏_{t∈G/N} c_{t}^{∗}(x)

(compare this with the formula for Res_{N}^{G}Cor_{N}^{G}), so the equation in your proof of lemma 1·7 is incorrect in general. I have been unable either to prove lemma 1·7 or to find a counterexample. However, the analogous statement for pc(G) is false:

Let the cyclic group of order p^{n} − 1 act on (ℤ_{/p})^{n} so as to permute the non-zero elements transitively (this can be done – it is the action by multiplication of the multiplicative group of the field of order p^{n} on its additive group). Now let N = (ℤ_{/p})^{n}, Q = ℤ_{/p^{n}−1}, and G the (unique) extension of N by Q corresponding to the above action. G has p^{n} − 1

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1-dimensional representations, which restrict trivially to any subgroup of order p, and one (p^{n} − 1)-dimensional representation, whose total Chern class restricts to any ℤ_{/p} subgroup as (1 − u^{p−1})^{p^{n−1}} = 1 − u^{(p−1)p^{n−1}}. Hence pc(G) = 2(p − 1)p^{n−1}, which does not divide (p^{n} − 1)pc(N).

To calculate p∘(G), I followed a suggestion of the late Prof. J. F. Adams: Firstly extend the action of Q on H^{∗}(BN; ℤ) = ℤ_{/p}[x_{1}, …, x_{n}] to an action of Q on 𝔽_{p^{n}}[x_{1}, …, x_{n}], calculate the fixed subring under this action, then calculate the subring of this fixed by the Galois group of 𝔽_{p^{n}} over ℤ_{/p}. Using this method it may be shown that such groups G do not give counterexamples to p∘(G) | (|Q|, p∘(N)).

Your theorem 2·4, in the case G_{1} (the metacyclic groups) is a special case of a result of C. T. C. Wall from ‘Resolutions for extensions of groups’, Proc. Camb. Phil. Soc. 57 (1961) 251–5. The case G_{2}(1, 1) is studied in G. Lewis ‘The integral cohomology rings of groups of order p^{3}’ Trans. Amer. Math. Soc. 132 (1968) 501–29. In this spectral sequence d_{2} is not trivial; for example d_{2} : E_{2}^{1,4} → E_{2}^{3,3} is not trivial. In my preprint, which I have enclosed, I give sufficient information (during the proof of theorem 3·6) to calculate the order of H^{n}(BG_{2}(a, 1); ℤ) for a > 1 (I use the name M(a + 2) for your G_{2}(a, 1)). I find that the order of H^{n}(BG_{2}(a, 1); ℤ) is 0, p^{a+1}, p^{2}, p^{a+3} for n = 1, 2, 3, 4 respectively. For i + j ⩽ 4, I calculate that the addititve structure of E_{2} for the spectral sequence you consider is

\begin{matrix}ℤ_{/p^{a}} ⊕ ℤ_{/p} \\ ℤ_{/p} & ℤ_{/p} \\ ℤ_{/p^{a}} & ℤ_{/p} & ℤ_{/p} \\ − & − & − & − \\ ℤ & − & ℤ_{/p} & − & ℤ_{/p} \end{matrix}

I believe, therefore, that in the spectral sequence for ⟨A, C⟩ ↣ G_{2}(a, 1) ↠ ℤ_{/p}, where a > 1, there is a non-trivial d_{n} : E_{n}^{i,4−i} → E_{n}^{i+n,5−i−n} for some i and n. I think that the

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values of i and n are 1 and 2 respectively.

I have no plans to publish this work, but I shall include it in my Ph. D. thesis. Remarks. Since receiving the above letter, Yagita has retracted the non-metacyclic cases of his theorem 2·4, and has sent the author a proof of lemma 1·7 under the strong condition that the action of G on H^{∗}(BN) is nilpotent. The groups G(a, 1) considered by Yagita may be presented as follows.

⟨A, B, C | A^{p^{a}} = B^{p} = C^{p} = [A, C] = [B, C] = 1 [A, B] = C⟩

Yagita’s theorem 2·4 claimed that the spectral sequence with integer coefficients for G(a, 1) expressed as an extension of ⟨A, C⟩ by C_{p} collapses. We may consider G(a, 1) as a central extension of ⟨A^{p}⟩ by P_{2}. Since G(a, 1) is nilpotent of class two the corresponding circle group is isomorphic to S^{1} × P_{2}, which has cohomology given by the Künneth theorem.

H^{∗}(BS^{1} × P_{2}; ℤ) ≅ ℤ[τ] ⊗ H^{∗}(BP_{2}; ℤ)

The class τp^{a−1} + β in H^{2}(BS^{1} × P_{2}; ℤ) is the first Chern class of a bundle over B(S^{1} × P_{2}) with total space BG(a, 1). We could of course choose any other non-zero element of H^{2}(BP_{2}; ℤ) instead of β. It is easy now to calculate the order of H^{n}(BG(a, 1); ℤ) using the spectral sequence for this S^{1} bundle.

A linear action of G on S^{2n−1} is an action induced by an n dimensional complex representation of G. Linear actions on products of spheres are products of linear actions. Linear actions induce the trivial action on the cohomology of the spheres because GL_{n}(ℂ) is connected. An element g of G will act fixed point freely in the linear action given by a product of representations ρ_{1} × … × ρ_{k} if and only if there is a j such that the restriction of ρ_{j} to ⟨g⟩ does not contain the trivial representation of ⟨g⟩. We obtain the following lemma. LEMMA 6·1. If G has a linear action on a product of 2n − 1 dimensional spheres such that every element of G of order p acts fixed point freely, then pc(G) divides 2n.

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Proof. It suffices to check that for each i : C_{p} ↣ G then n(i) (as defined in the above letter) divides n. Let the action be specified by maps ρ_{1}, …, ρ_{k} from G to GL_{n}(ℂ), and let C be the image of C_{p} in G. If C acts fixed point freely, then there is a ρ = ρ_{j} such that Res_{C}^{G}(ρ) does not contain the trivial representation. Then

c.(Res_{C}^{G}(ρ)) = ∏_{l=1}^{n}(1 + λ_{l}u)

where u is a generator for H^{2}(BC; ℤ) and λ_{l} is non-zero. But then c_{n}(Res_{C}^{G}(ρ)) is a nonzero multiple of u^{n}.

■

115

7. The Davis Complex. The following chapter, which is largely expository although we hope to obtain new results in this area eventually, concerns a construction due to M. Davis [Da], which assigns to a Coxeter group G a simplicial complex D(G), which is contractible and has a simplicial G-action with finite stabilisers. We give a simplified account of a special case of Davis’ construction, and an elementary account of an example due to Bestvina [Be] of a group whose cohomological dimension over 𝔽_{2} is greater than its cohomological dimension over ℚ. None of these results are original, although our proofs differ from those of [Da] and [Be]. Finally we suggest an application to the construction of a 4-manifold with interesting homological properties.

DEFINITION 7·1. A group G is said to be a graph product if it is generated by a collection G_{1}, …, G_{l} of finite subgroups, such that for each i ≠ j the subgroup generated by G_{i} and G_{j} is either the free product G_{i} ∗ G_{j} or the direct product G_{i} × G_{j}. In the special case when each G_{i} is a group of order two, G is a right angled Coxeter group. A subgroup of G generated by a subset of the G_{i} shall be called a special subgroup. Note that the trivial subgroup is defined to be special (generated by the empty set of G_{i}’s).

DEFINITION 7·2. If G is a graph product, then its Davis complex D(G) is the simplicial complex associated to the poset whose elements are the left cosets of the finite special subgroups of G, ordered by inclusion.

Write G(S) for the special subgroup of G generated by the set of G_{i} for i an element of S. We may represent any n-simplex of D(G) in the form (g, S_{0}, S_{1}, …, S_{n}), where g is an element of G and S_{0}, …, S_{n} is a strictly increasing chain of subsets of {1, …, l} such that G(S_{n}) is finite. Another such symbol (h, T_{0}, …, T_{n}) represents the same simplex if and only if T_{i} = S_{i} for all i and gG(S_{0}) = hG(S_{0}). The stabiliser of the simplex (g, S_{0}, …, S_{n}) is the subgroup gG(S_{0})g^{−1} of G, which is finite.

We shall not prove that the Davis complex is contractible, but we shall prove the 116

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following easier result. PROPOSITION 7·3 (Davis). The Davis complex of a graph product is acyclic. Proof. If for all pairs i, j in {1, …, l} the subgroup of G generated by G_{i} and G_{j} is G_{i}×G_{j}, then G is itself the direct product of all of the G_{i}’s, and hence finite, and so D(G) is a cone with vertex the unique left coset of G itself. If not then without loss of generality we may assume that G_{1} and G_{l} generate G_{1} ∗ G_{l}. Define subgroups H, K and L of G by

H = G({1, …, l − 1}), K = G({2, …, l}), L = G({2, …, l − 1}).

Then H ∩ K = L, and G = H ∗_{L} K. Consider now the subcomplex X of D(G) with simplices those (g, S_{0}, …, S_{n}) such that l ∉ S_{n}. By fixing a transversal T to H and expressing g = th for some t ∈ T and h ∈ H, we see that X is isomorphic to a disjoint union of copies of D(H), indexed by T. Note however that the action of H on the D(H) corresponding to t is twisted by conjugation by t. Since inductively D(H) may be assumed to be acyclic we see that X has no homology except in degree zero, and that H_{0}(X) is isomorphic as a ℤG-module to ℤG/H (the permutation module with basis the left cosets of H). Similarly, if we define Y and Z by

Y = {(g, S_{0}, …, S_{n})|1 ∉ S_{n}}, Z = {(g, S_{0}, …, S_{n})|1, l ∉ S_{n}},

then these subcomplexes have only degree zero homology, and as ℤG-modules

H_{0}(Y) ≅ ℤG/K, H_{0}(Z) ≅ ℤG/L.

Any simplex (g, S_{0}, …, S_{n}) of D(G) must be contained in either X or Y since S_{n} cannot contain both 1 and l, and the intersection of X and Y is Z. The Mayer-Vietoris sequence for D(G) implies that H_{i}(D(G)) = 0 for i > 1, and gives

0⟶H_{1}(D(G))⟶ℤG/L⟶^{η}ℤG/H ⊕ ℤG/K⟶H_{0}(D(G))⟶0.

The map η is often studied in the proof of the Mayer-Vietoris theorem for group cohomology (see [Br]), and it is known that η is injective and has cokernel ℤ. ■

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COROLLARY 7·4. If H is a torsion-free subgroup of a graph product G (for example the kernel of the map from G to the direct product of the G_{i}’s), then the chain complex for D(G) is an H-free resolution for ℤ. The chain complex for D(G) with rational coefficients is a ℚG-projective resolution for ℚ.

Proof. The chain complex for D(G) is a resolution for the trivial module by Proposition 7·3. The modules that occur in this resolution are permutation modules with basis G/K for various finite subgroups K. If H is torsion-free, then it cannot meet such a K, so acts freely. The module RG/K is projective provided that the order of K is invertible in R.

■

Before stating any further theorems it is useful to introduce another simplicial complex K(G) associated to the graph product G. K(G) has as simplices the subsets S of {1, …, l} such that G(S) is finite. Equivalently, K(G) has vertices {1, …, l}, edges the pairs i and j such that G_{i} and G_{j} commute, and is the full simplicial complex on its 1-skeleton. (Recall that a simplicial complex is said to be full if whenever its 1-skeleton contains a complete graph on n vertices then the complex contains an (n − 1)-simplex whose boundary is that graph.) The barycentric subdivision of any simplicial complex is full. Given a (finite) full simplicial complex K, and a finite group G_{v} for each vertex v of K, we may define a graph product G with K(G) = K, by taking the free product of all the G_{v}’s and adding the relations that G_{v} and G_{w} commute if (v, w) is an edge of K.

PROPOSITION 7·5. The Davis complex D(G) may be obtained by taking a free G-orbit of copies of the cone on the barycentric subdivision of K(G), and identifying together parts of their boundaries as described below.

Proof. By the boundary of a cone, we mean the simplices not containing the cone point. Simplices of K(G) correspond to non-empty subsets S of {1, …, l} such that G(S) is finite, so simplices of its barycentric subdivision correspond to ascending chains (S_{0}, …, S_{n}) of non-empty subsets such that G(S_{n}) is finite. We may add a cone point to this by al-

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lowing our chains to contain the empty set. Now we see that simplices of G × CK(G)′ are representable uniquely by symbols (g, S_{0}, …, S_{n}), where S_{0}, …, S_{n} is a chain of subsets with G(S_{n}) finite. Putting the equivalence relation on these symbols described after Definition 7·2 is equivalent to identifying parts of the boundaries of the cones. ■

From now onwards, we shall only consider the case when G is a right-angled Coxeter group

THEOREM 7·6 (Bestvina). Let X be the topological space obtained by attaching a disc D^{2} to a circle S^{1} using the map z ↦ z^{n} from S^{1} to S^{1}. Let G be any right-angled Coxeter group such that K(G) ≅ X, and let H be a torsion-free subgroup of G of finite index. Then H has cohomological dimension three, but H^{3}(H; M) has exponent dividing n for all modules M. Also H^{3}(H; ℤH) is cyclic of order n.

Proof. By Corollary 7·4, the Davis complex provides a finite free resolution of ℤ over ℤH of length three. Using Proposition 7·3 we shall study this resolution. The rest of this proof is very similar to a ‘bare hands’ proof that H^{2}(X) is cyclic of order n. Firstly, orient C(K(G)′) (the cone on the barycentric subdivision of K(G)) by coherently orienting the cone on a disc having the same 2-simplices as K(G)′, and then identifying the sides of the cone. The third stage (C_{3}) of the resolution consists of a direct sum of copies of ℤG, one for each 2-simplex of K(G)′. The second stage (C_{2}) of the resolution consists of a direct sum of three sorts of pieces. Firstly, copies of ℤG corresponding to interior 2-simplices of C(K(G)′) containing the cone point. The map from C_{3} to each of these components will be non-zero on exactly two of C_{3}’s summands, and will be the identity on one and minus the identity on the other. Secondly, copies of ℤG corresponding to 2-simplices of C(K(G)′) containing the cone point and an edge of the disc. The map from C_{3} to each of these components will be the identity map on n of C_{3}’s summands, and zero on the rest. Finally, for each 2-simplex σ, a copy of ℤG/L, where L is the minimal vertex of σ (necessarily a subgroup of order two). The map from C_{3} to this component will be

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projection on one of C_{3}’s summands, and zero on the rest.

Now apply Hom_{ℤH}( , M) to this resolution, and consider the ‘adjoint map’ from C^{2} to C^{3}. C^{3} consists of a direct sum of copies of N = Hom_{ℤH}(ℤG, M), and C^{2} consists of a direct sum of three types of module, the first and second types being isomorphic to N. The map from a summand of the first type to C^{3} is of the form

m ↦ (0, …, 0, m, 0, …, 0, −m, 0, …, 0),

and since the interior of the disc is connected the images of the first type summands suffice to identify all the copies of N that comprise C^{3}. The map from a second type summand to this quotient N is multiplication by n, and we see that H^{3}(H; M) has exponent dividing n.

Now specialise to the case when M = ℤH, so that N ≅ ℤG. Already by quotienting C^{3} by the image of first and second type summands of C^{2} we have shown that H^{3} is a quotient of RG, where R = ℤ/(n). A summand of C^{3} of the third type has the form ℤG/⟨g_{i}⟩, where the g_{i} are the Coxeter generators for G, and the effect of quotienting by these is that

H^{3}(H; ℤH) ≅ RG/(∑_{i}(1 + g_{i})RG).

Since G is presented by the g_{i} subject only to relations involving even powers of each g_{i} we see that the subgroup G′ of the group of units of RG generated by −g_{1}, …, −g_{l} is isomorphic to G and RG′ = RG. Now H^{3} is just RG′ modulo its augmentation ideal, or R. ■

Next we shall look for conditions for D(G) to be a combinatorial manifold.

LEMMA 7·7. The link of a simplex of D(G) is isomorphic to a suspension of the link of a simplex of K(G)′, except that the 0-simplex (g, ∅) has link isomorphic to K(G)′.

Proof. The link of the simplex (g, S_{0}, S_{1}, …, S_{n}) is the join of various pieces, which we shall examine separately. Firstly there is the complex corresponding to the poset of special

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cosets strictly contained in gG(S_{0}). This is the barycentric subdivision of the Coxeter complex of G(S_{0}), that is the subdivision of the m dimensional analogue of the octahedron (where S_{0} contains m + 1 elements), so is homeomorphic to S^{m} except when S_{0} = ∅, when it is empty. Secondly, there is the complex corresponding to the poset of special cosets lying strictly between gG(S_{i}) and gG(S_{i+1}). If S_{i+1} \ S_{i} contains m elements then this complex is the barycentric subdivision of the surface of an (m − 1)-simplex (or empty if m = 1)—this is because faces of an (m−1)-simplex correspond to non-empty subsets of its vertices. Thirdly, there is the complex corresponding to the poset of special cosets strictly containing gG(S_{n}). This is isomorphic to the link in K(G)′ of any |S_{n}|−1-simplex with S_{n} as its minimal vertex, except when S_{n} = ∅, in which case it is the whole of K(G)′. ■ COROLLARY 7·8. D(G) is a combinatorial (n + 1)-manifold if and only if K(G) is homeomorphic to S^{n}. Proof. Keep track of the dimensions of the spheres that arise in the proof of the preceeding lemma. ■

I. M. Chiswell has used the Davis complex to provide a formula for the Euler characteristic of a graph product [Ch], which we state here only in the case when G is a right-angled Coxeter group. THEOREM 7·9 (Chiswell). If G is a right-angled Coxeter group, then

χ(G) = 1 − \frac{1}{2} ∑_{i⩾0} \frac{n_{i}}{2^{i}} ,

where n_{i} is the number of i-simplices in K(G). ■

P. H. Kropholler has suggested combining Corollary 7·8 and Theorem 7·9 to try to exhibit a closed aspherical 4-manifold of negative Euler characteristic as follows. Given any full triangulation of the three sphere, form the corresponding right-angled Coxeter group G, and let H be a torsion-free subgroup of finite index. H acts freely and cocompactly on D(G), so D(G)/H is a closed manifold. The strong version of Davis’ theorem (namely

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that D(G) is contractible) implies that D(G)/H is aspherical, and its Euler characteristic is |G : H| times that of G. Unfortunately this seems not to work. For example: PROPOSITION 7·10. If K is any triangulation of S^{3} having m_{i} i-simplices, and G is the Coxeter group corresponding to the barycentric subdivision of K, then χ(G) is positive. Proof. Find n_{i} (the number of i-simplices of K′ in terms of the m_{j}, and substitute in to the formula of Theorem 7·9. It is helpful to note that the n_{i} (and the m_{i}) satisfy the following relations:

n_{0} − n_{1} + n_{2} − n_{3} = 0 2n_{3} = n_{2}.

(The first of these comes from χ(S^{3}) = 0, and the second from the fact that each 2-simplex in a 3-manifold bounds exactly two 3-simplices. ■

In the case of triangulations of S^{3} arising from convex polytopes, the theory of toric varieties may be brought to bear on this question. G. K. Sankaran has pointed out that given a triangulation of S^{3}, the signature of the cup product as a quadratic form on H^{4} of the corresponding toric variety is a positive multiple of the expression occuring in Theorem 7·9.

122

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