Contemporary Mathematics Volume 00, 0000

Applications of the Minami-Webb Formula to Splitting Classifying Spaces

JOHN MARTINO AND STEWART PRIDDY

We obtain a simplification of the general theory of stable splittings of classifying spaces by showing that the summands occurring in the Minami-Webb formula are determined by an eigenvalue of a certain matrix. Applications include new splittings and a rederivation of several known results as well as groups whose cohomologies are isomorphic as rings and as modules over the Steenrod algebra but whose cohomologies are not isomorphic as algebras over the Steenrod algebra.

1. Introduction.

Let G be a finite group. It is now well understood that the indecomposable stable summands of the classifying space BG, localized at a prime p, correspond to certain simple modules for F_{p}Out(Q) where Q ranges over subgroups of a Sylow p-subgroup of G. As one would expect the usefulness of this theory [BF, MP1, MP2] depends on the complexity of G. The purpose of this note is to provide some new techniques which simplify the task of decomposing BG. One by-product is a strikingly simple example showing how the stable and unstable homotopy types of classifying spaces can subtly differ: groups G, H are given such that (1) H^{∗}(BG) and H^{∗}(BH) are isomorphic as algebras, (2) BG and BH are stably equivalent and so H^{∗}(BG) and H^{∗}(BH) are isomorphic as modules over the Steenrod algebra, (3) the cohomologies are not isomorphic as algebras over the Steenrod algebra and so BG and BH have different unstable homotopy types, completed at p. As another application we give a complete stable decomposition of the classifying space of the Borel subgroup of SL_{3}(F_{5}) at p = 5. Various known decompositions are rederived as well.

1991 Mathematics Subject Classification. Primary 55R35; Secondary 20J06, 55P42.

Our approach uses the Minami-Webb formula. If G acts on a geometry Δ (e.g., a simplicial complex) and the geometry is locally contractible (i.e., for every p-subgroup P of G, Δ^{P} is contractible), then the geometry can be used to study the cohomology of G. For example, see the work of D. Quillen [Q] where he uses the poset A_{p}(G) of non-trivial p-subgroups. By using smaller geometries homotopy equivalent to A_{p}(G) one is led to reduction formulas for group cohomology. One such found by P. Webb [Wb] is

H^{∗}(BG; F_{p}) = ∑_{H∈C_{p}(G)} \frac{f(H)}{[G : H]}H^{∗}(BH; F_{p}),

where the sum is in a Grothendieck group completion with addition the direct sum,

C_{p}(G) = {H ∈ G | H/O_{p}(H) is a cyclic p′-group}

(the elements of C_{p}(G) are referred to as cyclic mod-p groups), and f is a Möbius function defined by ∑_{K≥H} f(K) = 1 for all H ∈ C_{p}(G). The Möbius function f only takes on non-zero values on maximal cyclic mod-p subgroups and intersections thereof. In [Mi] N. Minami showed that the sum actually occurs in the Grothendieck completion of p-local topological spaces with addition the wedge product,

BG_{(p)} = ⋁_{H∈C_{p}(G)} \frac{f(H)}{[G : H]}BH_{(p)}.

We will refer to this formula as the Minami-Webb formula. Suitably interpreted the Minami-Webb formula is a unique expression of BG_{(p)} (see [MP2]). If the sums are taken over conjugacy classes in C_{p}(G) then the index [G : H] is replaced by [N_{G}(H) : H], but the Möbius function f is still defined using the poset of all subgroups in C_{p}(G).

In this paper we give some theorems and propositions which facilitate the use of the Minami-Webb formula. To do this we show how the general theory simplifies when applied to the terms of the formula. In fact an eigenvalue of a certain matrix may be used to compute the stable homotopy type of BH_{(p)} where H is a cyclic mod-p group (Theorem 3.1). We use the Möbius function f to give a topological proof of a result on Swan groups originally proved using group theoretic methods. We use the Minami-Webb formula to derive some interesting facts about the 2-localization of BΣ_{6}. In the process we give the example mentioned above of two finite groups whose cohomologies are isomorphic as rings and as modules over the Steenrod algebra, but whose cohomologies are not isomorphic as algebras over the Steenrod algebra.

The paper is organized into the following sections. In section 2 we collect a few preliminaries. In section 3 we show how eigentheory can be used to analyze cyclic mod-p groups. Section 4 is devoted to using the coefficients of the Minami-Webb formula to show the metacyclic group M_{3}(p) is a Swan group. In section 5 we use the Minami-Webb formula and eigentheory to analyze BΣ_{6} localized at 2. Section 6 contains two examples illustrating how the eigentheory and the Minami-Webb formula may be used to compute stable summands.

In the following sections all groups are finite and all spaces and spectra are localized at a prime p.

2. Preliminaries.

For the convenience of the reader we recall some background material on the general theory of stable splittings.

We refer the reader to [MP2] for a discussion of the correspondence between the original summands of BP and the simple modules of F_{p}Out(P). An original summand (a.k.a. dominant summand) of BP is a summand that does not occur as the summand of the classifying space of any proper subgroup of P. Recall {BP, BP } is the ring of stable self-maps of BP and J(P) is the ideal of all stable maps that factor through the classifying spaces of groups of order less than |P |. The quotient

{BP, BP }/J(P) ⊗ F_{p} ≈ F_{p}Out(P),

where Out(P) is the outer automorphism group of P.

A related concept to an original summand is that of a linked summand. A summand X is linked in BQ if, for the corresponding idempotent e ∈ {BQ, BQ}, e ∉ \overline{J}_{τ} (Q), where \overline{J}_{τ} (Q) ⊆ \{BQ, BQ\} is the ideal generated by maps of the form BQ ⟶^{tr} BR ⟶^{Bh} BQ where R < Q. For ẽ ∈ F_{p}Out(Q), if e is a summand of ẽ then we say that X is linked to the corresponding original summand in BQ. As with original summands each linked summand is in one-to-one correspondence with a simple module of the ring R(Q) = (\{BQ, BQ\} ⊗ F_{p})/\overline{J}_{τ} (Q). Note F_{p}Out(Q) ⊆ R(Q) and there exists an ideal J ⊆ R(Q) such that F_{p}Out(Q) ≈ R(Q)/J. Therefore, the simple modules of F_{p}Out(Q) are simple modules of R(Q) where J acts as zero.

The situation is particularly nice when P is an abelian p-group. This case was extensively analyzed by J. Harris and N. Kuhn [HK]. The transfer map plays no role in splitting BP. All the summands either originate in or are linked in BP. Furthermore, a simple R(P) module restricts to a simple F_{p}Aut(P) module. Thus for a finite group G with an abelian Sylow p-subgroup, the multiplicity of a summand Y linked to an original summand X is exactly the same as the multiplicity of X. For example, L(2) is linked to L(3) in B(Z/2)^{3}, thus if a group G with Sylow 2-subgroup isomorphic to (Z/2)^{3} has two copies of L(3) in BG then there are exactly two copies of L(2) is BG. Since the transfer plays no role, the normalizer of a Sylow p-subgroup determines the classifying space BG. This gives a topological explanation of the fact, first observed by R. Swan [Sw], that the normalizer controls fusion in G.

We recall that a p-group P is called a Swan group if H^{∗}(BG) ≈ H^{∗}(N_{G}(P)) for every finite group G with P as Sylow p-subgroup. Hence, abelian groups are Swan groups. The reader should refer to [P] for more explanation of Swan groups.

In general, the p-local structure of a group may be used to determine its classifying space.

Theorem 2.1 [MP2]. Let G be a finite group, X a summand originating in BQ, Q a p-subgroup of G. Then the number of summands stably homotopy equivalent to X in BG is

∑_{i} dim_{k} \overline{W}_{i}M_{i},

where i ranges over the conjugacy classes Q_{i} of p-subgroups in which X either originates in or is linked in BQ_{i}, M_{i} is the corresponding simple module in R(Q_{i}), k is the splitting field for M_{i} (independent of Q_{i}), and \overline{W}_{i} = ∑ w, w ∈ N_{G}(Q_{i})/Q_{i}

Remark 1. If Y is linked in BP and originates in BQ then Q is a retract of P [MP2].

Remark 2. The element \overline{W}_{i} ∈ F_{p}Out(Q_{i}) equals BQ_{i} ⟶^{Bincl} BG ⟶^{tr} BQ_{i} mod J(Q_{i}).

3. Eigenvalue One.

In this section we describe a simple way to analyze the classifying space of a cyclic mod-p group.

Theorem 3.1. Let H = P ⋊ C be a cyclic mod-p group, X an original summand of BP, and M the corresponding simple F_{p}Out(P) module. If x ∈ C, a generator of C, acts as the matrix A on the module M, then

(1) X is a summand of BH if and only if 1 is an eigenvalue of A and

(2) the multiplicity of X in BH is the dimension of the eigenspace for 1 over End(M).

Proof. Without loss of generality we may assume that C acts faithfully on P (i.e., C ≤ Aut(P)). Let n = |C|. Since A^{n} = I, the eigenvalues of A are nth roots of unity in \overline{F}_{p}, the algebraic closure of F_{p}. The matrix A is diagonalizable, since if there are 1s off the diagonal in the Jordan canonical form of A then A^{n} has nλ^{n−1} off the diagonal for some nth root of unity λ and n ≢ 0 mod p. Hence the element \overline{C} = 1 + x + x^{2} + ⋯ + x^{n−1} acts as

I + A + A^{2} + ⋯ + A^{n−1} = [ \begin{matrix}I_{d} & 0 \\ 0 & 0 \end{matrix} ]

where d is the dimension of the eigenspace for 1. This follows from the fact that if λ is a nontrivial nth root of unity then

1 + λ + λ^{2} + ⋯ + λ^{n−1} = 0.

Therefore, dim \overline{C}M = d completing the proof. □

4. M_{3}(p) Is a Swan Group.

For a p-group P, a transfer summand X is a summand of BP where the idempotent BP ⟶ X ⟶ BP involves a transfer map (i.e., the idempotent is in \overline{J}_{τ} (P)). Transfer summands are summands that either originate or are linked in proper subgroups of P. In [MP3] we showed that for a p-group P if the classifying space has no transfer summands, then P is a Swan group.

Let M_{3}(p) denote the metacyclic group of order p^{3} and exponent p^{2}.

M_{3}(p) = ⟨x, y|x^{p^{2}} = y^{p} = 1, yxy^{−1} = x^{p+1}⟩.

For p odd, G. Glauberman has shown M_{3}(p) is a Swan group [MP3 (sic)]. His argument is group theoretic and is rather ad hoc. Our purpose is to derive this fact from a general result in the theory of stable splittings of classifying spaces. The group M_{3}(p) is an example of a p-group with transfer summands, which is nevertheless a Swan group.

Let P be a fixed p-group and write BP ≅ X ∨ Y where Y is the wedge sum of all transfer summands of BP. Thus X is the wedge sum of all summands either originating in or linked in BP.

Proposition 4.1. If Y is a summand of BG for every group G with P as a Sylow p-subgroup then P is a Swan group.

Proof. Since BG is a summand of BN_{G}(P) it suffices to show they have the same stable type. We use the Minami-Webb formula [Mi]: let C_{p}(G) be the set of all cyclic mod-p subgroups of G. Then

BG = ⋁_{(H)} \frac{f(H)}{[N_{G}(H) : H]}BH

stably, where H runs over the conjugacy classes of C_{p}(G) and f : C_{p}(G) → Z is the Möbius function given by

∑_{J≤K} f(K) = 1

for J, K ∈ C_{p}(G). This expression is unique [MP2]. The following lemma exhibits certain restrictions on f.

Lemma 4.2. Let Q ≤ G be p-subgroup and set S = ∑ \frac{f(H)}{[N_{G}(H):H]} where H runs over the conjugacy classes of C_{p}(G) with Sylow p-subgroup isomorphic to Q. Then

(1) S = 1 if Q ≈ P.

(2) S = 0 if Q is not a proper retract of a subgroup conjugate to P.

Remark. J. Thévenaz has shown [Th] the hypothesis that Q is not a subretract is unnecessary.

Proof. The principal original summand P S(Q) of BQ is a summand of multiplicity one in BK for every group K with Sylow p-subgroup isomorphic to Q. Thus if Q ≈ P, (1) follows immediately. Under the hypothesis in (2) that Q is not a subretract the principal summand P S(Q) is not a summand of any classifying space with Sylow p-subgroup of order greater than |Q| [MP1; Cor.4.4]. Hence S = 0 also follows immediately. □

Write

BG = ⋁_{i} α_{i}BH_{i} ∨ ⋁_{j} β_{j}BH′_{j},

where the Sylow p-subgroups of each H_{i} are isomorphic to P and the Sylow psubgroups of each H′_{j} are isomorphic to a p-group of order less than |P |. The normalizer determines whether an indecomposable summand of X is in either BG or BN_{G}(P). They only differ in transfer summands.

By Lemma 4.2, ∑_{i} α_{i} = 1, and by the hypothesis, Y is a summand of BG and ⋁_{i} α_{i}BH_{i}. Hence ⋁_{j} β_{j}BH′_{j} = 0 which implies by the uniqueness (independence) [MP2] that each β_{j} = 0. Thus all the maximal cyclic mod-p subgroups of G are contained in N_{G}(P), so they have the same lattice of maximal cyclic mod-p subgroups and intersections of maximal subgroups. Therefore,

BG ≅ BN_{G}(P).

Hence P is a Swan group. □

Turning now to the proof that P = M_{3}(p) is a Swan group for p odd, we first recall from [D]

BP ≅ ⋁_{i=0}^{p−2} X_{i} ∨ ⋁_{j=0}^{p−2} L(2, j) ∨ ⋁_{k=0}^{p−2} L(1, 1, k)

where the X_{i} are original summands of BP. Each L(2, j) is a transfer summand of BM_{3}(p). By arguments of [DMP], L(2, j) is a summand of BG. The fate of the summands L(1, 1, k) are controlled by linkage either to an X_{i} or an L(2, j). So the transfer summands consist of the L(2, j)s and the L(1, 1, k)s linked to them. Therefore, the sum of all the transfer summands is a summand of the classifying space of every group with a Sylow p-subgroup isomorphic to M_{3}(p). Hence M_{3}(p) is a Swan group by Proposition 4.1.

Proposition 4.3. For a finite group G with P a Sylow p-subgroup, BG ≅ BN_{G}(P) if and only if f(K) = 0 for all K ∈ C_{p}(G) with a Sylow p-subgroup not isomorphic to P.

Proof. If BG ≅ BN_{G}(P) then it follows by the uniqueness of the Minami-Webb formula that each f(K) = 0. On the other hand, if each f(K) = 0 then all the maximal cyclic mod-p subgroups contain P, and G and N_{G}(P) have the same lattice of maximal cyclic mod-p subgroups of intersections thereof. □

5. On the Cohomology and Stable Type of BΣ_{6}.

In this section we use the Minami-Webb formula to study BΣ_{6}. All spaces and spectra are localized at the prime p = 2.

Let std_{n} : Σ_{n} → O(n) be the standard orthogonal representation of the symmetric group Σ_{n}. We recall that H^{∗}(BΣ_{4}; F_{2}) ≈ F_{2}[a, b, c]/ac where

a = w_{3}(std_{4}), b = w_{2}(std_{4}), c = w_{1}(std_{4}).

The 2-local stable type is given by

BΣ_{4} ≅ BP SL_{2}(F) ∨ L(2) ∨ BZ/2

where F = F_{q} is a finite field with q odd such that 4 = ν_{2}(q^{2} − 1). In [MtP] this result is only stated for q = 7 but the result holds since this condition guarantees that the dihedral group D_{8} is a Sylow 2-subgroup of P SL_{2}(F).

Proposition 5.1. Let G be a finite group with Minami-Webb formula

BG ≅ ⋁ α_{i}BH_{i}.

If P is a finite p-group, then G × P has Minami-Webb formula

B(G × P) ≅ ⋁ α_{i}B(H_{i} × P).

Proof. The Möbius function f in the Minami-Webb formula is only nonzero on maximal and intersections of maximal cyclic mod-p subgroup of G × P. Clearly, every maximal cyclic mod-p subgroup is of the form H ×P. Hence, every intersection of maximal cyclic mod-p subgroups of that form. Therefore, G and G×P have the same lattice of maximal cyclic mod-p subgroups and intersections thereof. The corresponding values of the Möbius functions and indices for G and G × P are thus the same. □

Theorem 5.2. Stably localized at 2,

(1) BΣ_{6} ≅ B(P SL_{2}(F) × Z/2).

(2) BΣ_{6} ≅ B(D_{8} × Z/2) + B(A_{4} × Z/2) − B(Z/2)^{3}.

(3) A complete stable decomposition is given by

BΣ_{6} ≅ P S(D_{8} × Z/2) ∨ BP SL_{2}(F) ∨ 2L(3) ∨ 2L(2) ∨ B(Z/2)

where P S(D_{8} × Z/2) is the principal original summand of B(D_{8} × Z/2).

Theorem 5.3.

(1) The cohomology of BΣ_{6} is

H^{∗}(BΣ_{6}; F_{2}) ≈ F_{2}[w_{1}, w_{2}, w_{3}, u_{3}]/(u_{3}^{2} + u_{3}(w_{3} + w_{2}w_{1}))

where w_{i} = w_{i}(std_{6}) and i^{∗}(u_{3}) = a under the restriction i^{∗} : H^{∗}(BΣ_{6}; F_{2}) → H^{∗}(B(Σ_{4} × Σ_{2}); F_{2}).

(2) As algebras over F_{2} (but not as algebras over the Steenrod algebra A)

H^{∗}(BΣ_{6}; F_{2}) ≈ H^{∗}(B(P SL_{2}(F) × Z/2); F_{2}).

Remark. Σ_{6} and P SL_{2}(F)×Z/2 are examples of groups whose cohomologies are isomorphic as algebras over F_{2}, but not as algebras over A. However, their A module structures with higher Bocksteins agree since their classifying spaces are equivalent at 2. The cohomology ring of Σ_{6} was computed by A. Adem, J. Maginnis, and J. Milgram [AMM]. We give another derivation suitable for our purposes.

Proof of Theorem 5.2. Part (2) of Theorem 5.2 is just the Minami-Webb formula for Σ_{6} [MP2;Wb, 7.5].

For part (1) we apply Proposition 5.1 to

BP SL_{2}(F) = BD_{8} + BA_{4} − B(Z/2)^{2},

and observe that BΣ_{6} and B(P SL_{2}(F) × Z/2) have identical Minami-Webb formulas.

For part (3), complete stable splittings for B(Z/2)^{3} and B(D_{8} × Z/2) are found in [Mt] and [MP1] respectively.

B(Z/2)^{3} ≅ B[(Z/2)^{3} ⋊ (Z/7 ⋊ Z/3)] ∨ 3X_{31} ∨ 3BZ/2 ∨ 3X_{32} ∨ 3BA_{4} ∨ 8L(3) ∨ 8L(2). B(D_{8} × Z/2) ≅ P S(D_{8} × Z/2) ∨ 2X_{31} ∨ 3BZ/2 ∨ 2X_{32} ∨ 2BA_{4} ∨ BP SL_{2}(F)

∨ 8L(3) ∨ 8L(2).

For the remaining group A_{4} × Z/2 in the Minami-Webb formula in part (2), a stable decomposition can be obtained from [CK; Thm.7.1]. Alternatively we derive this splitting from Theorem 3.1 using the fact that A_{4} × Z/2 = [(Z/2)^{2} ⋊ Z/3]×Z/2, is a cyclic mod-2 group. A Sylow 2-subgroup of A_{4}×Z/2 is isomorphic to (Z/2)^{3}, and up to isomorphism there are four simple modules of F_{2}GL_{3}(F_{2}). These are the trivial module 1, the three-dimensional standard module V, its dual V^{#}, and the eight-dimensional Steinberg module St. The corresponding original summands are B[(Z/2)^{3} ⋊(Z/7⋊Z/3)], X_{32}, X_{31}, and L(3) respectively.

Let x ∈ A_{4} represent a generator of Z/3. The element x acts as the matrix

[ \begin{matrix}0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] ,

which has a one-dimensional eigenspace for the value 1. Hence X_{32} and by duality X_{31} have multiplicity one in B(A_{4} × Z/2). The Steinberg module St is the quotient of H^{4}(B(Z/2)^{3}; F_{2}) by image of the Steenrod operations. The element x acts as the matrix

A = [ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \end{matrix} ] .

Hence,

A − I = [ \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] .

The first six columns of A − I are clearly linearly independent, while the last two columns are each the sum of the third and fifth columns. So, the eigenspace for 1 is two-dimensional, and L(3) has multiplicity two in B(A_{4} × Z/2). The multiplicities of the non-original summands are accounted for by linkage. The summand BZ/2 is linked to X_{31}, BA_{4} is linked to X_{32}, and L(2) is linked to L(3). Hence,

B(A_{4}×Z/2) ≅ B[(Z/2)^{3}⋊(Z/7⋊Z/3)]∨X_{31}∨BZ/2∨X_{32}∨BA_{4}∨2L(3)∨2L(2).

Part (3) now follows from the Minami-Webb formula in part (2). □

Proof of Theorem 5.3.

(1) We begin by showing the existence of a unique element u_{3} which restricts in the required manner. Since [Σ_{6} : Σ_{4} × Σ_{2}] is odd, the restriction i^{∗} is injective. Also H^{∗}(BΣ_{4}; F_{2}) is detected by the elementary abelian 2-subgroups Δ_{2} = ⟨(13)(24), (12)(34)⟩ and T = ⟨(12), (34)⟩. More explicitly, if i_{Δ} : Δ_{2} → Σ_{4} and i_{T} : T → Σ_{4} are the inclusions then it is well known that

i_{Δ}^{∗}(a) = x^{2}y + xy^{2}, i_{Δ}^{∗}(b) = x^{2} + xy + y^{2}, i_{Δ}^{∗}(c) = 0

and

i_{T}^{∗}(a) = 0, i_{T}^{∗}(b) = xy, i_{T}^{∗}(c) = x + y.

These facts follow since i_{Δ} is the regular representation while std_{4}∘i_{T} is conjugate to the inclusion O(1) × O(1) → O(2) → O(4). Now we define u_{3} = tr^{∗}(x^{2}y) where tr^{∗} : H^{∗}(BE; F_{2}) → H^{∗}(BΣ_{6}; F_{2}) is the transfer and E = Δ_{2} × Σ_{2}. Since W = N_{Σ_{6}}(E)/E = GL_{2}(F_{2}) we have

(i_{Δ} × Σ_{2})^{∗}tr^{∗} = \overline{W}

where \overline{W} = Σc_{w}, w ∈ W. Similarly,

(i_{T} × Σ_{2})^{∗}tr^{∗} = 0

([MtP; Prop.1.4]). By an easy computation \overline{W} x^{2}y = x^{2}y + xy^{2} and a ∈ H^{3}(B(Σ_{4}×Σ_{2}); F_{2}) is the only element which restricts to x^{2}y+xy^{2} in H^{3}(BE; F_{2}) and to 0 in H^{3}(B(T × Σ_{2}); F_{2}). Thus i^{∗}(u_{3}) = a.

By the Whitney formula for Stiefel-Whitney classes we have

i^{∗}(w_{3}) = a + bz, i^{∗}(w_{2}) = b + cz, i^{∗}(w_{1}) = c + z.

Computing we find i^{∗}(u_{3}^{2}+u_{3}(w_{3}+w_{2}·w_{1})) = 0 which gives the desired relation. Furthermore, it is well known from Nakaoka’s computations or more explicitly by Webb that the Poincaré series of H^{∗}(BΣ_{6}; F_{2}) is (1+t^{3})/(1−t)(1−t^{2})(1−t^{3}). Thus this is the only relation. For use below we note Sq^{2}(w_{3}) = w_{3}w_{2} + u_{3}w_{1}^{2}.

(2) We recall from [MtP; Thm.2.6] the cohomology algebra H^{∗}(BP SL_{2}(F); F_{2}) ≈ F_{2}[v_{2}, v_{3}, \overline{v}_{3}]/(\overline{v}_{3}^{2} + v_{3}\overline{v}_{3}) with Steenrod operations

Sq^{1}(v_{2}) = v_{3}, Sq^{2}(v_{3}) = v_{2}v_{3}, Sq^{1}(\overline{v}_{3}) = 0, Sq^{2}(\overline{v}_{3}) = v_{2}\overline{v}_{3}.

An isomorphism of algebras f : H^{∗}(B(P SL_{2}(F) × Z/2); F_{2}) → H^{∗}(BΣ_{6}; F_{2}) may be specified by

f(\overline{v}_{3}) = u_{3}, f(v_{3}) = w_{3} + w_{2} · w_{1}, f(v_{2}) = w_{2}, f(z) = w_{1}.

It is easily checked that f preserves the required relation.

Any other isomorphism g must satisfy g(z) = w_{1}. Thus by the addition of z^{2} we may assume the generator v_{2} is chosen so that g(v_{2}) = w_{2} and Sq^{1}(v_{2}) = v_{3} as before. If g preserves Steenrod operations then g(v_{3}) = w_{3} + w_{2}w_{1} by the Wu formula. However g(v_{2}v_{3}) = g(Sq^{2}(v_{3})) = Sq^{2}g(v_{3}) = Sq^{2}(w_{3}+w_{2}w_{1}) = w_{3}w_{2}+ u_{3}w_{1}^{2} + w_{2}^{2}w_{1} + w_{3}w_{1}^{2} + w_{2}w_{1}^{3} while g(v_{2}v_{3}) = g(v_{2})g(v_{3}) = w_{2}(w_{3} + w_{2}w_{1}). Thus an algebra isomorphism can not preserve Steenrod operations.

6. Further Examples.

In this section we give two examples showing how Theorem 3.1 and the Minami-Webb formula may be used in conjunction. All spaces and spectra are localized at the prime p.

Let p = 5 and consider the Borel subgroup B_{3} of SL_{3}(F_{5}) consisting of upper triangular matrices over F_{5}. Then B_{3} = U_{3} ⋊ D where U_{3} is the subgroup of upper triangular matrices with ones down the diagonal and D = Z/4 × Z/4 is the group of matrices generated by

h_{1} = [ \begin{matrix}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{matrix} ] , h_{2} = [ \begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix} ]

Inspection shows there are six maximal mod 5 cyclic subgroups H_{1}, ..., H_{6} each having the form U_{3}⋊Z/4 where the Z/4 is generated by h_{1}, h_{2}, h_{1}h_{2}, h_{1}^{2}h_{2}, h_{1}h_{2}^{2}, h_{1}h_{2}^{3} respectively. These intersect in three subgroups K_{1}, K_{2}, K_{3} of the form U_{3} ⋊Z/2 where Z/2 is generated by h_{1}^{2}, h_{2}^{2}, h_{1}^{2}h_{2}^{2} respectively and lastly U_{3} itself. The resulting Minami-Webb formula is

(1)  BB_{3} = ∑_{i=1}^{6} \frac{1}{4}BH_{i} − ∑_{j=1}^{3} \frac{1}{8}BK_{j} − \frac{1}{8}BU_{3}

From this one can obtain a complete decomposition using Theorem 3.1. We restrict ourselves to sketching the derivation of the multiplicity of the original summands, i.e., those originating in U_{3}. These summands, denoted by X_{q,k}, 0 ≤ q ≤ 4, 0 ≤ k ≤ 3, are described in [DP]. They correspond to the simple F_{5}Out(U_{3}) modules. These modules in turn correspond to simple F_{5}GL_{2}(F_{5}) modules M_{q,k} under the map induced by projection π : U_{3} → U_{3}/Z where Z ≈ Z/3 is the center ([DP; Cor. 1.5]). If V is the standard module for GL_{2}(F_{5}) then M_{q,k} = S^{q}(V) ⊗ (det_{2})^{k} where S^{q}(V) is the q-fold symmetric product and det_{2} is the determinant representation. Then

π(h_{1}) = [ \begin{matrix}−1 & 0 \\ 0 & 3 \end{matrix} ] , π(h_{2}) = [ \begin{matrix}3 & 0 \\ 0 & −1 \end{matrix} ]

Applying Theorem 1.3 and (1) one obtains the following multiplicities for the X_{q,k} in BB_{3}.

Summand  Multiplicity
X_{0,k}  1
X_{1,k}  0
X_{2,k}  0
X_{3,k}  0
X_{4,k}  2

As a second example let p = 2 and define V = (Z/2)^{3} and G_{168} = (Z/2)^{3} ⋊ (Z/7 ⋊ Z/3), where Z/7 acts as F_{8}^{∗} under the identification V ≈ F_{8} and Z/3 is generated by the Frobenius automorphism x ↦ x^{2}. By the work of S. Mitchell [Mt] we have the stable splitting

B(Z/2)^{3} ≅ BG_{168} ∨ 3X_{31} ∨ 3BZ/2 ∨ 3X_{32} ∨ 3BA_{4} ∨ 8L(3) ∨ 8L(2).

into indecomposable summands.

Applying the Minami-Webb formula we get

BG_{168} = B(V ⋊ Z/3) + \frac{1}{3}B(V ⋊ Z/7) − \frac{1}{3}BV.

Decompositions of B(V ⋊ Z/3) and B(V ⋊ Z/7) were given by Carlisle and Kuhn ([CK; Th. 7.1]). We shall rederive these results using Theorem 3.1. By the primitive normal basis theorem (see [Da]) it is possible to choose an element ω ∈ F_{8} so that ω generates F_{8}^{∗} and {ω, ω^{2}, ω^{4}} forms a basis for F_{8} over F_{2}. Furthermore, ω, which generates Z/7, acts on the basis {1, ω, ω^{2}} as the matrix

A = [ \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ] .

and the Frobenius automorphism, which generates Z/3, acts on the basis {ω, ω^{2}, ω^{4}} as the matrix

B = [ \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ] .

The value 1 is not an eigenvalue of A, but it is an eigenvalue of B with a one-dimensional eigenspace. Hence, by duality and linkage we have

B(V ⋊ Z/7) ≅ BG_{168} ∨ aL(3) ∨ aL(2).
B(V ⋊ Z/3) ≅ BG_{168} ∨ X_{31} ∨ BZ/2 ∨ X_{32} ∨ BA_{4} ∨ bL(3) ∨ bL(2).

Theorem 3.1 can now be used on the Steinberg module St to determine the integer scalars a and b. But suppose that the Steinberg module is not readily available. We know that BG_{168} is indecomposable so the Minami-Webb formula tells us

b + \frac{1}{3}a − \frac{8}{3} = 0.

The only non-negative integer solutions to the equation are

a = 8 b = 0
a = 5 b = 1
a = 2 b = 2.

Let U_{3} be the subgroup of upper triangular matrices in GL_{3}(F_{2}) with ones on the diagonal, and for H ≤ GL_{3}(F_{2}) by \overline{H} we mean the formal sum of elements in H. The Steinberg idempotent is e = \overline{U}_{3} · \overline{Σ}_{3} and St = eF_{2}GL_{3}(F_{2}). Let B be the cyclic subgroup generated by B. Thus e and e · \overline{U}_{3} · \overline{B} are both elements of \overline{B}St. Since e · \overline{Σ_{3}/B} = 0 and e · \overline{U}_{3} · \overline{B} · \overline{Σ_{3}/B} = e^{2} = e, the two elements are not equal. Thus by Theorem 2.1 b is neither 0 nor 1. Hence a = 2 and b = 2. Therefore, we have complete decompositions of B(V ⋊ Z/7) and B(V ⋊ Z/3).

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Department of Mathematics and Statistics,

E-mail address: martino@math-stat.wmich.edu

Department of Mathematics, Northwestern University, Evanston, IL 60208 E-mail address: priddy@math.nwu.edu