Some examples in the integral and Brown-Peterson cohomology of p-groups.

I. J. Leary

School of Mathematical Sciences, Queen Mary and Westfield College, London. E1 4NS N. Yagita Musashi Institute of Technology, Tamazatumi, Setagaya, Tokyo.

Introduction.

For a finite group G, we define the Chern ring, Ch(G), to be the subring of H^{even}(G; ℤ) generated by Chern classes of representations of G. We say that G has p-rank n if n is maximal such that G contains a subgroup isomorphic to (C_{p})^{n}. In [3] 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 (AHSS) gives rise to a filtration of the representation ring of G. Atiyah 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. (It is clear that Ch(G) consists of universal cycles because the AHSS for BU(n) collapses.) Weiss discovered that the alternating group A_{4} gives a counterexample to this conjecture [16], and Thomas has exhibited many counterexamples, all of which have order divisible by more than one prime [14].

Thomas showed that the split metacyclic p-groups and various other p-groups of prank two have the property that the Chern subring is the whole of the even degree integral cohomology, and conjectured that this property would hold for all p-groups of p-rank two [12], [13], [15]. The group A_{4} shows that the conjecture cannot be extended to groups of non-prime power order. AlZubaidy claimed to have verified this conjecture, but some of his proofs are flawed [1], [2]. Recently Huebschmann and Tezuka-Yagita have shown that Ch(G) = H^{even}(G; ℤ) for any metacyclic p-group G [6], [11]. For p ≥ 5 Blackburn’s classification [4] implies that the only p-groups of p-rank two not covered by the above theorems are the groups G(n, ϵ) defined below. We show that Ch(G(n, ϵ)) is strictly contained in H^{even}(G(n, ϵ); ℤ) for each such group. These groups also afford p-group counterexamples to the conjecture of Atiyah described above.

Similar calculations may be made in the Brown-Peterson cohomology rings of these groups. These enable us to give a negative answer to a question of Landweber [7], who asked if Chern classes generate the Brown-Peterson cohomology of every p-group.

The examples. The groups which we shall consider may be presented as

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

where p is a prime not equal to 2 or 3, n ≥ 4, and for fixed p and n there are two isomorphism classes of such groups, depending whether ϵ is either 1 or a quadratic nonresidue modulo p. The group G(n, ϵ) has order p^{n}. In the sequel we shall refer to G instead of G(n, ϵ) unless the values of n and ϵ are important. The subgroup M generated by B and C is maximal (and hence normal) and is isomorphic to C_{p} ⊕ C_{p^{n−2}}. We define one dimensional representations θ and ϕ of the group M by

\begin{matrix}θ : B^{j}C^{k} & ↦ exp(2πij/p) \\ ϕ : B^{j}C^{k} & ↦ exp(2πik/p^{n−2}). \end{matrix}

The action of the quotient group G/M on the representation ring of M is that conjugation by A sends θ to θ ⊗ ϕ^{⊗p^{n−3}} and sends ϕ to ϕ ⊗ θ^{⊗ϵ}. Later we shall define elements of H^{2}(M; ℤ) and BP^{2}(BM) as Chern classes, and the action of G/M on these elements will be determined by its action on the representations θ and ϕ.

The group G has only 1- and p-dimensional irreducible representations because it has an abelian subgroup (M in fact) of index p. A one dimensional representation of G must restrict trivially to ⟨B⟩, and a p-dimensional representation of G restricts to ⟨B⟩ as either p copies of the same representation of ⟨B⟩, or as the sum of one copy of each of the onedimensional representations of ⟨B⟩. The examples Ind_{M}^{G}(θ) and Ind_{M}^{G}(ϕ) show that both these alternatives do occur.

Now define generators β, γ for H^{2}(M; ℤ) by

β = c_{1}(θ) γ = c_{1}(ϕ),
so that H^{even}(M; ℤ) ≅ ℤ[β, γ]/(pβ, p^{n−2}γ),

and let β′ be the restriction to ⟨B⟩ of β, so that

H^{∗}(⟨B⟩; ℤ) ≅ ℤ[β′]/(pβ′).

Lemma 1. With notation as above, the image of Ch(G) under restriction to ⟨B⟩ is the subring of H^{∗}(⟨B⟩; ℤ) generated by β^{′p−1} and β^{′p}. For all m ≥ 0,

β^{′m+p−1} = −Res_{⟨B⟩}^{G}Cor_{M}^{G}(γ^{p−1}β^{m}).

Proof. If ρ is a 1-dimensional representation of G then its Chern class restricts trivially to ⟨B⟩. If ρ is a p-dimensional representation of G, then either ρ restricts to ⟨B⟩ as p-copies of the same representation, in which case

Res(c.(ρ)) = (1 + iβ′)^{p} = 1 + iβ^{′p},

or as one copy of each representation, in which case

Res(c.(ρ)) = ∏_{i=0}^{p−1}(1 + iβ′) = 1 − β^{′p−1}.

By applying the double coset formula we see that

Res_{⟨B⟩}^{G}Cor_{M}^{G}(γ^{p−1}β^{m}) = Res_{⟨B⟩}^{M}(∑_{i=0}^{p−1} c_{A^{i}}^{∗}(γ^{p−1}β^{m})

= Res_{⟨B⟩}^{M}(∑_{i=0}^{p−1}(γ + iϵβ)^{p−1}(β + ip^{n−3}γ)^{m})

= −β^{′m+p−1}  ■

Remarks. The image of Res_{⟨B⟩}^{G} is precisely the subring of H^{∗}(⟨B⟩; ℤ) generated by β^{′p−1}, β^{′p}, β^{′p+1}, …, β^{′2p−3}. One way to show this is by considering the subgroup N of G generated by A and B. This subgroup is normal in G, and is the non-abelian group of order p^{3} and exponent p. Using Lewis’ calculation of H^{∗}(N; ℤ) [9], it may be shown that the image of H^{∗}(N; ℤ)^{G/N} under restriction to ⟨B⟩ does not contain β^{′i} for i < p − 1.

Corollary 2. Ch(G) is strictly contained in H^{even}(G; ℤ). Moreover, Ch(G) does not map onto H^{∗}(G; ℤ) modulo its nilradical.

Proof. We know that β^{′p+1} is in Res_{⟨B⟩}^{G}(H^{even}(G; ℤ)), but not in Res_{⟨B⟩}^{G}(Ch(G)). ■

Corollary 3. In the AHSS for G, write B_{∞}(G) for the universal boundaries, and Z_{∞}(G) for the universal cycles. Then Ch(G) + B_{∞}(G) is strictly contained in Z_{∞}(G).

Proof. The AHSS for ⟨B⟩ collapses, so Res_{⟨B⟩}^{G}(B_{∞}(G)) is trivial. Corestrictions of Chern classes are universal cycles, so β^{′p+1} ∈ Res_{⟨B⟩}^{G}(Z_{∞}(G)), but β^{′p+1} ∉ Res_{⟨B⟩}^{G}(Ch(G) + B_{∞}(G)). ■

For any generalised cohomology theory H and any group K, we may define Ch_{H}(K) to be the subring of H^{∗}(BK) generated by ρ^{∗}(H^{∗}(BU)) for all representations ρ of K in a unitary group U. We now give a result analogous to Corollary 2 for Brown-Peterson cohomology.

Lemma 4. Ch_{BP} (G) is strictly contained in BP^{∗}(BG).

Proof. As in the integral cohomology case, define elements β and γ in BP^{2}(M) by β = c_{1}(θ), γ = c_{1}(ϕ), and also define β′ = Res_{⟨B⟩}^{M}(β), so that

BP^{∗}(BM) ≅ BP_{∗}[[β, γ]]/([p]β, [p^{n−2}]γ), BP^{∗}(B⟨B⟩) ≅ BP_{∗}[[β′]]/([p]β′),

where [r]x stands for the BP formal group sum of r copies of x. Let ‘≡’ stand for congruence modulo the ideal of BP^{∗}(B⟨B⟩) generated by p, v_{1}, v_{2}, . . .. As in Lemma 1, if ρ is a p-dimensional representation of G, then either

Res_{⟨B⟩}^{G}(c.(ρ)) = (1 + [i]β′)^{p} ≡ 1 + iβ^{′p}
or Res_{⟨B⟩}^{G}(c.(ρ)) = ∏_{i=0}^{p−1}(1 + [i]β′) ≡ 1 − β^{′p−1}.

Also, we have that

Res_{⟨B⟩}^{G}Cor_{M}^{G}(γ^{p−1}β^{2}) = Res_{⟨B⟩}^{M}(∑_{i=0}^{p−1}(γ +_{BP} [iϵ]β)^{p−1}(β +_{BP} [ip^{n−3}]γ)^{2}
≡ −β^{′p+1}.  ■

Our original proofs of these results involved calculation with BP^{∗}(BN), which has been determined by Tezuka-Yagita [10], and with the integral cohomology of the nonabelian maximal subgroups of G, determined by Leary [8]. Using these methods we obtain more information concerning BP^{∗}(BG) and H^{∗}(G; ℤ), which we intend to publish later.

We thank Dr. C. B. Thomas for suggesting investigations of the Chern subring and for his encouragement. The second author also thanks M. Tezuka and M. Kaneda for many suggestions. The work of the first author has been supported by the SERC.

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