LECTURES ON THE STABLE HOMOTOPY OF BG

STEWART PRIDDY

notes by Michael Hill

Let G be a finite group. Our goal is to study the stable homotopy of the classifying space BG completed at some prime p. For ease of notation, we shall always assume that any space in question has been p-completed. Our fundamental approach is to decompose the stable type of BG into its various summands. This is useful in addressing many questions in homotopy theory especially when the summands can be identified with simplier or at least better known spaces or spectra. It turns out that the summands of BG appear at various levels related to the subgroup lattice of G. Moreover since we are working at a prime, the modular representation theory of automorphism groups of p-subgroups of G plays a key role. These automorphisms arise from the normalizers of these subgroups exactly as they do in p-local group theory. The end result is that a complete stable decompositon of BG into indecomposable summands can be described (Theorem 6) and its stable homotopy type can be characterized algebraically (Theorem 7) in terms of simple modules of automorphism groups.

This paper is a slightly expanded version of lectures given at the International School of the Hanoi Conference on Algebraic Topology, August 2004. The author is extremely grateful to Mike Hill for taking notes and producing a TeX document which formed the basis of the present work. Additional comments and references have been added to make the exposition reasonably self-contained. Many of the results of this paper were obtained jointly with my longtime collaborator, John Martino.

1. Preliminaries

1.1. What do we mean by “stable homotopy”. Given a pointed space (X, ∗), let ΣX = X × I/X × {0, 1} ∪ {∗} × I denoted the reduced suspension. We can represent this pictorially as:

One can also quickly check that ΣS^{n} = S^{n+1}, where S^{n} is the n-sphere.

Now, H̃_{∗}(ΣX) = H̃_{∗−1}(X), and for any space Y, [ΣX, Y] is a group. The composition is as defined for homotopy groups: if we have two homotopy classes [f], [g], then we let [f] · [g] denote the composite [∇ ∘ (f, g) ∘ π′], where π′: ΣX → ΣX ∨ ΣX is the “pinch” map defined by

Date: May 24, 2006.

1991 Mathematics Subject Classification. Primary: 55P42; Secondary: 55R35, 20C20.

Key words and phrases. classifying space, stable splittings, transfer.

[(x, t)] ↦ \begin{cases}([(x, 2t)], ∗) & 0 < t < 1/2 \\ (∗, [(x, 2t − 1)]) & 1/2 ≤ t ≤ 1 \end{cases}

and ∇: ΣX ∨ ΣX → ΣX is the “fold” map. The fact that we can add maps gives the theory a very different, more algebraic flavor, than that of unstable homotopy theory. Moreover, the same proof as for ordinary homotopy groups shows that [Σ^{2}X, Y] is an abelian group.

For us, “stable” just means that we can suspend any number of times, even an infinite number, as needed. More precisely let QY = colimΩ^{n}Σ^{n}Y. Then we define the stable homotopy classes of maps {X, Y } = [X, QY]. If X is a finite complex {X, Y } = colim[Σ^{n}X, Σ^{n}Y]. For a discussion of spectra, see [1].

1.2. Classifying Spaces. Let G be a finite group. Define EG to be a free, contractible G-space, and let BG denote the quotient EG/G. The contractibility of EG shows us that BG is a space with a single non-trivial homotopy group: π_{1}(BG) = G. We give two explicit constructions of EG and then give an application.

1.2.1. Milnor’s Definition. We define

EG = ⋃_{n} \underbrace{G ∗ ⋯ ∗ G}_{n},

where ∗ denotes the join of two spaces, which we take to be the suspension of the smash product. Since the join includes a suspension, the greater the number of copies of G being joined, the higher the connectivity, and so EG is contractible.

As an example, we take G = ℤ/2. In this case,

E ℤ/2 = ⋃_{n} \underbrace{ℤ/2 ∗ … ℤ/2} = ⋃_{n} S^{n−1} = S^{∞},

and the ℤ/2 action is the usual diagonal reflection action, and in this case, the quotient B ℤ/2 is just ℝP^{∞}.

1.2.2. Simplicial Model. We can think of a group as a category with a single object and whose morphisms are the elements of the group. We can now pull in the categorical construction of the nerve, and this will give us a model for BG.

First we recall briefly the definition of a simplicial set. A more complete reference is [13]. Let Δ be the category whose objects are the sets 0, …, n for all n and whose morphisms are non-decreasing maps. A simplicial set is a contravariant functor from Δ to the category of sets. We can think of a simplicial set as a collection of sets indexed by the natural numbers together with a large family of structure maps called faces and degeneracies which satisfy certain properties, modeled dually on the inclusion of faces in the standard simplices in ℝ^{n}.

To any simplicial set S, we can associate a topological space, the geometric realization, |S|. Loosely speaking, this is defined by putting a copy of the standard n−simplex in for every element of S_{n} and gluing them all together via the face and degeneracy maps.

To any category C, we can associate a simplicial set, the nerve, NC_{∗}. The k−simplices of NC are the k−tuples of composable morphisms in C. The face maps are induced by the various ways to compose adjacent maps (or to forget the ends), and the degeneracies comes from inserting the identity map in various places. We define the classifying space BC to be the geometric realization of the nerve. With a little work, one can quickly show that this construction is functorial.

In the case C = G, a finite group, NC_{k} = G^{k}, since all morphisms are composable. We then get a model of BG by taking the geometric realization.

This construction has some very nice advantages over the previous one, and to show this, we need a small proposition.

Proposition 1. If F_{0}, F_{1} : C → C′, and H is a natural transformation from F_{0} to F_{1}, then BF_{0} ≃ BF_{1} as maps BC → BC′, and the homotopy is given by BH on B(C × {0 → 1}) = BC × [0, 1].

This immediately gives us an important result about conjugation.

Corollary 1. Let x ∈ G, and let C_{x}(g) = x^{−1}gx denote conjugation by x. We then have BC_{x} ≃ Id_{BG}.

Proof. There is a natural transformation between C_{x}, viewed as an endo-functor of G, and the identity functor given by “multiplication by x”:

\xymatrix{e \ar[r]^{C_{x}(g)} \ar[d]_{x} & e \ar[d]^{x} \\ e \ar[r]_{g} & e},

where e denotes the single object in the category. In other words, the morphism x : e → e is a natural transformation between C_{x} and the identity, and the result follows. □

It is this simple corollary which gives us the basic connection between group theory and the homotopy theory of classifying spaces.

1.3. Group Cohomology. The space EG allows us to define group homology and cohomology. The singular chains C_{∗}(EG) is a ℤ[G]−free resolution of ℤ.

Definition. For any G−module M, let H_{∗}(G; M) = H(C_{∗}(EG) ⊗_{ℤ[G]} M), and let H^{∗}(G; M) = H(Hom_{ℤ[G]}(C_{∗}(EG), M).

Note in particular that if M = ℤ, the trivial G−module, then H_{∗}(G; M) = H_{∗}(BG) and similarly for cohomology. If M is not trivial, then H_{∗}(G; M) can be similarly related to H_{∗}(BG) but with twisted coefficients.

In what follows all cohomolgy is taken with simple coefficients in 𝔽_{p}.

2. Stable Splittings

Suppose that ΣBG = X_{1} ∨ ⋯ ∨ X_{N}. If we can do this, then for any generalized cohomology theory E,

E^{∗}(BG) = E^{∗+1}(ΣBG) = ⨁ E^{∗+1}X_{i}.

In general, this is a simpler object to study. We want now to find ways to relate the X_{i} to G itself.

2.1. Summands via Idempotent Self-maps. Let e: X → X for some pointed space X. If e^{2} ≃ e, we call e a homotopy idempotent. We now form the mapping telescope eX = T el(X, e) which is the homotopy colimit of the diagram X ⟶^{e} X. More explicitly, we start with the disjoint union

∐_{n≥0}(X × [2n, 2n + 1])

and identify (x, 2n + 1) with (e(x), 2n + 2) and pinch (∗, t) to a point.

In this case,

π_{∗}(eX) = colim π_{∗}(X) = e_{∗}π_{∗}(X),

where the structure maps in the limit are e_{∗}. A similar statement holds for homology.

If X is a suspension, then we can add and subtract maps, and in particular, we can form a map X → eX ∨ (1 − e)X whenever e is a homotopy idempotent. From the above comments, this is an equivalence. Our next task is then to find idempotents in [X, X]. In the case of X = BG or ΣBG, we shall get the first layer of these from algebra, using the Aut(G) action on [BG, BG].

Rather than looking at homotopy classes of maps, we’ll look at stable homotopy classes of self maps {BG, BG}. Under composition, this has the structure of a ring. The group of stable homotopy self-maps carries an action of Aut(G) via the map which sends α ∈ Aut(G) to the stable class of Bα. This therefore extends to a map of rings from ℤ[Aut(G)] → {BG, BG}. If we can find idempotents in ℤ[Aut(G)], then we can push them forward to stable homotopy idempotents. Since we are interested in working one prime at a time and since idempotent theory is easier for completed rings, we shall assume BG is completed at p and consider the induced map ℤ_{p}[Aut(G)] → {BG, BG}. If G is a p-group then BG is already p-complete.

We start by reducing mod p, since any idempotent e ∈ 𝔽_{p}[Aut(G)] lifts to an idempotent in ℤ_{p}[Aut(G)]. Moreover, if we have a primitive orthogonal idempotent decomposition 1 = e_{1} + ⋯ + e_{n} in 𝔽_{p}[Aut(G)], where e_{i}e_{j} = 0 for i ≠ j, e_{i}^{2} = e_{i} then this lifts to a decomposition of the same form in ℤ_{p}[Aut(G)].

Example 1. If G = ℤ/p, then Aut(G) = ℤ/(p − 1). If p = 3, then we can readily find two idempotents in 𝔽_{3}[ℤ/2], namely −1−e and −1+e, where e is the non-trivial element in ℤ/2.

In general, there are p − 1 primitives with idempotents given by

e_{i} = ∏_{j≠i} \frac{ξ − a^{i}}{a^{i} − a^{j}} , i = 0, .., p − 2

where ξ is the generator of ℤ/p − 1 and a is the element in 𝔽_{p} = ℤ/p by which ξ acts.

Proposition 2. Stably and p-completed,

B ℤ/p ≃ X_{0} ∨ ⋯ ∨ X_{p−2},

where X_{i} = e_{i}B ℤ/p.

We will say more about X_{0} below. One can also try to use the full ring of endomorphisms for an abelian p-group P. This approach has been thoroughly studied by Harris and Kuhn [6].

3. Transfer Maps

Let H ⊂ G be a subgroup of index [G : H] = n. If we take the quotient of EG by H, then we get BH, since EG is contractible and being G free forces it to be H free. We can further quotient by all of G to get a map BH → BG and the fiber of this map is G/H. In other words, we have an n-sheeted cover BH → BG. The map BH → BG is also easily seen to be equivalent to B of the inclusion H → G.

The transfer is a stable map which goes from BG back to BH. In cohomology, we can easily define it. Let π: X → Y be an n-sheeted cover. For each simplex Δ ∈ C_{∗}(Y), we can find n simplices in C_{∗}(X) lying over it. The transfer is the map in homology induced by

Δ ↦ ∑_{Δ′∈π^{−1}(Δ)} Δ′.

If we compose now with the projection map, then it is clear that the composite is simply multiplication by n.

Actually getting a stable map requires a little more work. Write G = ∐ τ_{i}H. Given an element τ_{i}, left multiplication by g ∈ G sends it to τ_{σ(g)(i)}h_{i,g}. This gives us a permutation representation σ: G → Σ_{n}, and a homomorphism

G → H^{n} ⋊ Σ_{n} = Σ_{n} ≀ H
g ↦ (h_{1,g}, …, h_{n,g}, σ(g)).

We define the transfer to be the map adjoint to the composite

\xymatrix{BG \ar[r] \ar@{-->}[drrr] & B(Σ_{n} ≀ H) \ar[r]^{=} & BH^{n} ×_{Σ_{n}} BΣ_{n} \ar[r] & (QBH)^{n} ×_{Σ_{n}} BΣ_{n} \ar[d]_{Θ} \\ & & & QBH} ,

where Θ is the Dyer-Lashof map arising from the infinite loop structure of QBH. It is not difficult to see that in homology this map agrees with the previous definition for the covering BH → BG. Actually the map we have defined is sometimes referred to as the reduced transfer. Let BG^{+} denote BG with an added disjoint basepoint so that BG^{+} ≃ BG ∨ S^{0}. Then it is easy to extend this definition to a stable map tr : BG^{+} → BH^{+} which is multiplication by [G, H] on the bottom cell. For a detailed exposition see [7]; another approach is given by Adams [1].

3.1. Properties of the Transfer and Corollaries. We have already seen homologically that the composite BG ⟶^{tr} BH ⟶^{Bi} BG is multiplication by the index [G : H].

Corollary 2. If H ⊂ G, and [G : H] is prime to p, then BG is a stable summand of BH when completed at p.

Proof. Since [G : H] is prime to p, it is a unit in ℤ_{p}, and multiplication by it is an equivalence. Thus the transfer and inclusion give the splitting. □

Corollary 3. Stably and completed at p, BΣ_{p} is a stable summand of B ℤ/p.

Proposition 3 (Properties of the Transfer). We will write tr_{H}^{G} for the transfer with H considered as a subgroup of G.

(1) If H = G, then tr_{H}^{G} = Id.

(2) If K ⊂ H ⊂ G, then tr_{K}^{G} = tr_{H}^{G} ∘ tr_{K}^{H}.

(3) The transfer is natural with respect to maps of coverings.

(4) The “Double Coset Formula” holds: If H, K ⊂ G, write G = ∐ KxH for some collection of x ∈ G. Let tr_{x} denote the transfer BK → B(K∩x^{−1}Hx), and let i_{x} denote the inclusion xKx^{−1} ∩ H → H. Then if i_{K} is the map BK → BG, we have

tr_{H}^{G} ∘ i_{K} = ∑_{x} i_{x} ∘ C_{x^{−1}} ∘ tr_{x}.

Lemma 1. If G is an elementary abelian p-group, and H ⊊ G, then the transfer induces the zero map in mod p cohomology.

Proof. The map i_{H}^{∗} is surjective, since H sits inside G as a summand. Since

tr_{H}^{G}^{∗}i_{H}^{∗}(x) = [G : H]x = p^{n}x = 0,

we conclude that tr^{∗} = 0. □

Corollary 4. If V ⊂ G is elementary abelian, then

i_{V}^{∗} ∘ tr_{V}^{G}^{∗} = ∑_{w∈N(V)/V} C_{w}^{∗}.

Proof. Let K = V in the double coset formula and let x vary over coset representatives. □

It follows from Corollary 2 that if G_{p} is a Sylow p-subgroup, then BG is a stable summand of BG_{p} after p-completion. We now specialize to the case that G_{p} = V is elementary abelian.

Theorem 1. If V ⊂ G is an elementary abelian Sylow p-subgroup, then (1) H^{∗}(G) ≅ H^{∗}(V)^{W} , where W = N_{G}(V)/V is the Weil group.

(2) BN_{G}(V) → BG is an H ℤ/p-equivalence, even unstably.

Proof. The second result follows immediately from the first.

For the first part, note that we always have H^{∗}(G) ⊂ H^{∗}(V). Since conjugation acts as the identity on H^{∗}(G), we must have

H^{∗}(G) ⊂ H^{∗}(V)^{N_{G}(V)}.

Since conjugation by V is trivial in cohomology, H^{∗}(V)^{N_{G}(V)} = H^{∗}(V)^{W} . From the previous corollary, we know that the composite

H^{∗}(V) ⟶^{tr^{∗}} H^{∗}(G) ⟶^{i^{∗}} H^{∗}(V)^{W}

is just ∑_{w∈W} C_{w}. Since |W | is prime to p, it is invertible in ℤ_{p}, and e = ∑ C_{w}/|W | is an idempotent invariant under the action of W. Conversely, all invariants arise in this way, since on the subalgebra of W -invariants, the composite is just multiplication by |W | and is therefore invertible. This in particular shows that i^{∗} is surjective, and the result follows. □

Example 2. For G = Σ_{p}, the Sylow p-subgroups are ℤ/p, and N_{Σ_{p}}(C_{p})/C_{p} = ℤ/(p − 1). Now H^{∗}(Σ_{p}) = H^{∗}(ℤ/p)^{W}. W acts on H^{∗}(ℤ/p) = E(x_{1}) ⊗ 𝔽_{p}[y_{2}] as multiplication by a generator of 𝔽_{p}^{×} on x_{1} and y_{2} = β(x_{1}). The fixed point algebra is then generated by x_{1}y_{2}^{p−2} and y_{2}^{p−1} as an unstable algebra over the Steenrod algebra. This shows that

H^{∗}(Σ_{p}; 𝔽_{p}) = \begin{cases}ℤ/p & ∗ = 0, −1 mod p \\ 0 & otherwise \end{cases} .

These dimensions explain why the Steenrod operations occur where they do, just as a similar computation for the map B(ℤ/p×ℤ/p) → B(ℤ/p≀ℤ/p) → BΣ_{p^{2}} yields the Adem relations.

With more work one can show a generalization of the previous theorem.

Theorem 2 (Harris-Kuhn[6]). If P ⊂ G is a Sylow p−subgroup, and P is an abelian p−group, then

(1) H^{∗}(G) ≅ H^{∗}(P)^{W} = H^{∗}(N_{G}(P))

(2) BN_{G}(P) → BG is an H 𝔽_{p}-equivalence.

3.2. Modular Representation Theory. If p divides the order of the automorphism group, then the representation theory of Aut(G) over 𝔽_{p} lies in the realm of modular representation theory and hence becomes more complicated. We demonstrate this with some basic examples of increasing trickiness.

Example 3. Let G = V_{2} = ℤ/2 × ℤ/2, and take p = 2. Now

Aut(V_{2}) = GL_{2}(𝔽_{2}) = Σ_{3}

has order divisible by 2, and we can find simple generators:

σ = [\begin{matrix}0 & 1 \\ 1 & 1 \end{matrix}] , τ = [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] .

In R = 𝔽_{2}[Aut(V_{2})] the element

f_{1} = 1 + σ + σ^{3}

is an idempotent, since f_{1}^{2} = 3f_{1}. Since τστ = σ^{2}, f_{1} and f_{2} = 1 − f_{1} are central idempotents. With a small bit of work, one can show

Proposition 4. R ≅ Rf_{1} × Rf_{2}, and Rf_{1} = E(γ), where γ = ∑_{g∈GL} g and Rf_{2} = M_{2}(𝔽_{2}).

We can lift f_{1} and f_{2} to idempotents e_{1} and e_{2} in {BV_{2}, BV_{2}}, so we conclude that

BV_{2} = e_{1}BV_{2} ∨ e_{2}BV_{2}.

The first summand we can identify, as it is clearly the same as B of the semi-direct product

(ℤ/2 × ℤ/2) ⋊ ℤ/3 = A_{4}.

Now f_{2} can be written as the sum of two idempotents F_{1} and F_{2}, where

F_{1} = (1 + τσ)(1 + τ), F_{2} = (1 + τ)(1 + τσ).

These can also be lifted to idempotents in ℤ_{2}[Aut(V)], so BV_{2} splits further as

(1)  BV_{2} = BA_{4} ∨ ẽ_{1}BV_{2} ∨ ẽ_{2}BV_{2}.

Finally, ẽ_{1}BV_{2} ≃ ẽ_{2}BV_{2}, since we have a sequence

F_{1}R ⟶^{1+τ} F_{2}R ⟶^{1+τσ} F_{1}R ⟶^{1+τ} F_{2}R

in which the composite of any two successive arrows is the identity. It is known that

ẽ_{1}BV_{2} = L(2) ∨ B ℤ/2,

where L(2) is a spectrum that is related to Steenrod operations of length two. We therefore have the following result of Mitchell [8]:

BV_{2} = BA_{4} ∨ (L(2) ∨ ℝP^{∞}) ∨ (L(2) ∨ ℝP^{∞}).

in which the summands are indecomposable.

Example 4. Let G = D_{8}, the dihedral group of order 8. One can show that Aut(D_{8}) = D_{8}, so this is a two group.

Lemma 2. If G is a p-group, then 𝔽_{p}[G] has only one simple module, the trivial one.

The lemma follows from the fact that the augmentation ideal is nilpotent in this case

In the case G = D_{8}, the lemma shows that we have only one idempotent in 𝔽_{2}[Aut(G)], namely the element 1. Nevertheless, we have the following splitting, [8]:

BD_{8} = BP SL_{2}(𝔽_{7}) ∨ (L(2) ∨ ℝP^{∞}) ∨ (L(2) ∨ ℝP^{∞}).

Using ring theory, we can get a more direct relationship between the structure of R = 𝔽_{p}[Aut(G)] and idempotents. Let J be the Jacobson radical of R, namely the elements annihilating all simple R−modules or equivalently the intersection of all maximal ideals. Ring theory tells us that R/J is semisimple and therefore splits as as a product of matrix rings over division algebras:

R/J = M_{n_{1}}(D_{1}) × ⋯ × M_{n_{k}}(D_{k}).

The simple R-modules are just the columns of the various matrix rings. Lifting the idempotents from this decomposition, we obtain in R a primitive orthogonal decomposition

1 = ∑_{j} e_{j} = n_{1}e_{1} + ⋯ + n_{k}e_{k}.

For the earlier example of V_{2}, the Jacobson radical is the ideal generated by the element γ, and so

𝔽_{2}[Aut(V_{2})]/J = 𝔽_{2} ×M_{2}(𝔽_{2}),

giving us the correct number of factors for the decomposition of BV_{2} in (1). To obtain the complete decomposition of (1) one must use the full endomorphism group End(V_{2}).

Example 5. Next, we look at a more complicated example, V_{3} = (ℤ/2)^{3}. In this case, R = 𝔽_{2}[GL_{3}(𝔽_{2})] has 4 simple modules:

Module 𝔽_{2} V_{3} V_{3}^{∗} St_{3}
Dimension 1 3 3 8

where V_{3}^{∗} is the contra-gredient module and St_{3} is the Steinberg module (described explicitly in Example 6 below). This means that we have a splitting

BV_{3} = (ẽ_{1}BV_{3}) ∨ 3(ẽ_{2}BV_{3}) ∨ 3(ẽ_{3}BV_{3}) ∨ 8(ẽ_{4}BV_{3}).

As before, we can identify the first summand. In this case, it is the same as B of the group 𝔽_{8}^{×} ⋊Gal(𝔽_{8} / 𝔽_{2}) obtained by taking the semidirect product of V_{3} = 𝔽_{8} with ℤ/7 ⋊ ℤ/3 = Gal(𝔽_{8} / 𝔽_{2}).

Generalizing the example of the dihedral group D_{8}, we consider P = D_{2^{n}}, the dihedral group of order 2^{n}. We have the following splitting

Theorem 3 (Mitchell-Priddy). BD_{2^{n}} = BP SL_{2}(𝔽_{q}) ∨ 2L(2) ∨ 2B ℤ/2, where q = p^{k} for p odd and D_{2^{n}} is a Sylow 2-subgroup of P SL_{2}(𝔽_{q}).

This condition translates to saying that n = ν_{2}(\frac{q^{2}−1}{2}) is the order of 2 in (q^{2} − 1)/2.

We can also explain the existence of the summands 2L(2) ∨ 2B ℤ/2. There are two non-conjugate copies of ℤ/2 × ℤ/2 sitting inside D_{2^{n}}, and the summands in question appear via transfers from BD_{2^{n}} to B of these subgroups, followed by projection onto the summands.

4. The Segal Conjecture and its Consequences

While studying the idempotents in the ring {BG, BG} via Aut(G) provides a good bit of information about the splittings of BG, if G = P is a p-group then the Segal conjecture completely determines the ring {BG, BG}, so it is to this that we turn.

Let A(P, P) be the Grothendieck ring of P × P sets which are free on the right. The sum is given by disjoint union, and the product is the product over P. If P_{0} ⊂ P is a subgroup, and ρ: P_{0} → P is a homomorphism, then we can define elements of A(P, P) by

P ×_{ρ} P = P × P/ ∼

where (xp_{0}, y) ∼ (x, ρ(p_{0})y), p_{0} ∈ P_{0}. As a group, A(P, P) is a free abelian group on these transitive sets. There is a homomorphism

α : A(P, P) → {BP, BP }

defined by

α(P ×_{ρ} P) = (BP_{+} ⟶^{tr_{P_{0}}^{P}} BP_{0}_{+} ⟶^{Bρ} BP_{+}).

Upon completion this map is essentially an isomorphism. More precisely A(P, P) contains an ideal Ã(P, P) which is a free abelian group on the classes P ×_{ρ} P − (P/P_{0} × P) and we have

Theorem 4 (G. Carlsson [3]; Lewis, May, McClure [5]). The map α induces a ring isomorphism

α̃ : Ã(P, P) ⊗ ℤ_{p} → {BP, BP }.

Corollary 5. (1) {BP, BP } is a finitely generated, free ℤ_{p}-module.

(2) BP splits as a finite wedge X_{1} ∨ … X_{n}, where the X_{i} are indecomposable p-complete spectra, unique up to order and equivalence.

Proof. The first is immediate. For the second, after tensoring with 𝔽_{p} we have a finite dimensional 𝔽_{p}-algebra. This means that 1 = ∑ e_{i} is a decomposition into primitive idempotents unique up to order and conjugation. □

Corollary 6. Given a finite p-group P, there exist finitely many stable homotopy types of BG with P a Sylow p-subgroup of G.

Proof. We have already seen that if P is a Sylow p-subgroup of G, then we have a splitting BP ≃ BG ∨ Rest. The finiteness result of the previous corollary gives this one. □

Corollary 7. Each summand of BP is also an infinite complex.

Proof. By the first part of the first corollary, we know that {BP, BP } is torsion free. If X were both a finite complex and a summand of BP, then BP → X → BP, the projection followed by the inclusion, would be a torsion free map. However, if X is a finite complex, then the identity map of X has torsion, since X is p-complete, so the torsion free composite must as well. □

Theorem 5. (Nishida [14]) Given G, G′ finite groups with BG ≃ BG′ stably at p, then the Sylow p-subgroups of G and G′ are isomorphic.

We shall derive this from a more general result in Section 6.

4.1. Analysis of Indecomposable Summands. We will now assume that X is an indecomposable summand of BP for P a fixed p-group.

Definition. X originates in BP if it does not occur as a summand of BQ for any Q ⊊ P. X is a dominant summand of BP if it originates in BP.

The notion of dominant summand is due to Nishida. As an example, for BV_{2} the dominant summands are BA_{4} and L(2).

It is also clear that every X must originate in some subgroup Q of P.

Now let J(P) be the ideal in {BP, BP } generated by the maps which factor through BQ for some proper subgroup Q ⊊ P. In other words, these are the maps which arise from transitive sets for which P_{0} is a proper subgroup or if P_{0} = P from proper (i.e. non surjective) endomorphisms ρ : P_{0} → P. Since every summand X = eBP for some idempotent e ∈ {BP, BP }, X is dominant iff e ∉ J(P). Furthermore we have an isomorphism of rings

ℤ_{p}[Out(P)] ⟶^{i} {BP, BP } ⟶^{π} {BP, BP }/J(P) = ℤ_{p}[Out(P)].

This follows by remembering that {BP, BP } is generated by transfers followed by homomorphisms. If the subgroup for the transfer is proper, or if the homomorphism is not surjective then this map is in J(P), so all that we have left over are the automorphisms of P. This gives us the following equivalence:

{Homotopy types of dominant summands of BP } →

{Isomorphism classes of simple 𝔽_{p}[Out(P)] -modules}

given by eBP ↦ e_{0} 𝔽_{p}[Out(Q)], where e_{0} is determined as follows: since e is primitive we can find a primitive idempotent ẽ ∈ ℤ_{p}[Out(P)] such that i(ẽ) = e mod J. Then e_{0} is the mod p reduction of ẽ.

Let S be a simple 𝔽_{p}[Out(Q)] module and let X_{Q,S} be its corresponding dominant summand of BQ.

Theorem 6. ([9], [2]) There is a complete stable decomposition unique up to order and equivalence of factors

BP = ⋁_{Q,S}n_{Q,S}X_{Q,S}

where Q runs over the subgroups of P, S runs over the simple 𝔽_{p}[Out(Q)] modules, and n_{Q,S} is the multiplicity of X_{Q,S} in BP.

4.2. Principal Dominant Summand. Among all dominant summands, there is a distinguished one corresponding to the trivial module. If we decompose 1 ∈ 𝔽_{p}[Aut(P)] into primitive orthogonal idempotents, then we can consider the image of them under the augmentation ring map ϵ : 𝔽_{p}[Aut(P)] → F_{p} defined by sending all g ∈ Aut(P) to 1 ∈ 𝔽_{p}. Since e_{i}^{2} = e_{i}, these must map to either 0 or 1 under the augmentation map. Additionally, exactly one must map to 1, since the augmentation sends 1 to 1, and e_{i}e_{j} = 0 for i ≠ j. We denote by e_{0} the idempotent that maps to 1 and say that it is the principal idempotent. The corresponding summand will be denoted X_{0} and called the principal dominant summand.

Proposition 5 (Nishida [14]). X_{0} is a summand of BG for all G with P a Sylow p-subgroup.

For two examples, for B ℤ/p, the principal dominant summand is BΣ_{p}, and for BV_{2}, it is BA_{4}.

4.3. Ring of Universally Stable Elements. For a fixed p-group P, we define the ring of universally stable elements as

I(P) = ⋂_{G⊃P as a Sylow p-subgroup} Im(H^{∗}(G) → H^{∗}(P)).

Theorem 7 (Evens-Priddy, [4]). H^{∗}(P) is a finite module over I(P). This implies that H^{∗}(P) is a finitely generated algebra over I(P) of the same Krull dimension, which in turn implies I(P) is a finitely generated 𝔽_{p}-algebra.

This follows from Quillen’s theorem [16].

Proposition 6. [4] If E is an elementary abelian p-group of rank n, then I(E) = H^{∗}(E)^{GL(E)}, except when p = n = 2, where it is H^{∗}(BA_{4}).

In general, I(P) is not realizable as the cohomology of a spectrum. In certain familiar cases, however, it is not only realizable but also connected to the cohomology of the principal dominant summand.

Proposition 7. [4] We have the following rings of universally stable elements.

\begin{matrix}I(D_{2^{n}}) = H^{∗}(P SL_{2}(𝔽_{q})) & n = ν_{2} (\frac{q^{2}−1}{2}) \\ I(Q_{2^{n+1}}) = H^{∗}(SL_{2}(𝔽_{q})) & n = ν_{2} (\frac{q^{2}−1}{2}) \\ I(SD_{2^{n}}) = H^{∗}(SL_{3}(𝔽_{q})) & n = ν_{2}((q^{2} − 1)(q + 1)), q = 3 mod 4 \end{matrix}

For all of these groups, X_{0} is B of the group shown.

5. Summands of Supergroups

Recall that if X is an indecomposable summand of BQ, then we say it is dominant if it is not a summand of BQ′ for any subgroup Q′ ⊊ Q. If P is a p-group, and Q ⊂ P is a subgroup with X a dominant summand of BQ, we can ask when X is a summand of BP.

Example 6. We recall the definition of the Steinberg module. Let e_{St_{n}} ∈ R = 𝔽_{p}[GL_{n}(𝔽_{p})] be defined by

e_{St_{n}} = \frac{1}{[GL_{n}(𝔽_{p}) : U_{n}]} ( ∑_{b∈B_{n},σ∈Σ_{n}}(−1)^{σ}bσ) ,

where B_{n} is the Borel subgroup of upper triangular matrices and U_{n} is the unipotent subgroup thereof. This element is idempotent and primitive, and if we let St_{n} = e_{st_{n}}R, then St_{n} is a simple, projective module of dimension p^{\binom{n}{2}}.

If V is an elementary abelian group of dimension n, then

e_{st_{n}}BV = L(n) ∨ L(n − 1),

where L(n) = Σ^{−n}(Sp^{p^{n}}(S^{0})/Sp^{p^{n−1}}(S^{0})) and Sp^{n}(X) = X^{n}/Σ_{n} is the symmetric product. [8]

Proposition 8. [15] If P is a p-subgroup of rank n (i.e. n is largest number such that (ℤ/p)^{n} is a subgroup of P), then L(n) is a summand of BP iff P contains a self-centralizing elementary abelian subgroup of rank n.

Example 7. For p = 2 and low values of n, we have the following examples. n = 2 , L(2) is a summand of BD_{2^{n}}. n = 1 , L(1) is actually ℝP^{∞}, and we have seen already instances when this occurs

as a summand. n = 1 , As a “non-example”, L(1) is not a summand of BQ_{8}, since the ℤ/2 ⊂ Q_{8}

is central.

Now recall the principal summand X_{0}. Let P S(X_{0}, t) be its Poincaré series in mod-p cohomology. By the dimension of X_{0} we mean the order of the pole of P S(X_{0}, t) at t = 0.

Proposition 9. [11] X_{0} has dimension n in BP if the rank of P is n.

5.1. Generalized Dickson Invariants. For the Dickson invariants, we normally start with the algebra H^{∗}(E)^{GL(E)}. If E is an elementary abelian p-group of rank n, then we can form the composite E ⟶^{Reg} Σ_{E} → U(p^{n}), where Reg is the regular representation of E acting on itself. If we look in cohomology, then Chern classes c_{p^{n}−p^{i}} map to the Dickson invariants c_{i}. These Dickson invariants carve out a polynomial invariant subalgebra of H^{∗}(BE)^{GL(E)} of dimension n.

For a general p-group P, we can formally mirror the above construction. Let ρ denote the composite complex representation

ρ : P ⟶^{Reg} Σ_{|P |} → U(|P |),

and define the (generalized) Dickson invariants of P to be the image under ρ^{∗} of

𝔽_{p}[c_{p^{s}(p^{n}−p^{i})} : i = 0, …, n − 1; s = |P |/p^{n}] ⊂ H^{∗}(U(|P |)

where s = |P |/p^{n}. This obviously forms a subalgebra of H^{∗}(BP) which we will denote D(P).

Proposition 10. If P is a p-group of rank n, then (1) D(P) is a polynomial ring of dimension n.

(2) D(P) ⊂ H^{∗}(BP)^{Out(P)}.

Proof. This is easy to see. For the first part, we use the fact that the composite of the inclusion of E, an elementary abelian subgroup of rank n, into P followed by the regular representation map to Σ_{|P |} is the same as p^{s} times the regular representation map of E. By the naturality of cohomology, the classes c_{p^{s}(p^{n}−p^{i})} pull back to the classes (c_{p^{n}−p^{i}})^{p^{s}}.

For the second part, given an automorphism f of P, we can form a commutative square

\xymatrix{P \ar[d]_{f} \ar[r] & Σ_{P} \ar[d]^{C_{f}} \\ P \ar[r] & Σ_{P}} .

As before conjugation induces the identity in the cohomology of the classifying space; this shows that the pullback of any classes coming from H^{∗}(Σ_{P}) lies in the invariants of H^{∗}(P) under Out(P). □

6. Stable Classifications of BG at p

The following result gives a classification of the stable type of BG in terms of its p-subgroups Q and associated Out(Q) modules. Let Rep(Q, G) = Hom(Q, G)/G and Inj(Q, G) ⊂ Rep(Q, G)be the classes of injections. Let Cen(Q, G) ⊂ Inj(Q, G) be represented by monomorphisms α : Q → G such that C_{G}(Imα)/Z(Imα) is a p′-group.

Theorem 8 (Martino-Priddy, [10]). Let G and G′ be finite groups. The following are equivalent:

(1) BG ≃ BG′ at p.

(2) For every finite p-group Q, there is an isomorphism of Out(Q)-modules

𝔽_{p}[Cen(Q, G)] ≅ 𝔽_{p}[Cen(Q, G′)].

Note: The proof of a related classification result of [10] contains an error, see [12] for a correction.

A stable equivalence BG ≃ BG′ at p induces an isomorphism

𝔽_{p}[Inj(Q, G)] ≅ 𝔽_{p}[Inj(Q, G′)]

of Out(Q) modules. From this we can easily derive Nishida’s result, Theorem 5:

Corollary 8. If BG ≃ BG′ stably at p, then P ≅ P′ where P, P′ are respective Sylow p-subgroups.

Proof. We have

0 ≠ 𝔽_{p}[Inj(P, G)] ≅ 𝔽_{p}[Inj(P, G′)].

This implies that P ⊂ P′. Reversing P and P′ gives the desired desired conclusion P ≅ P ” □

Corollary 9. If BG ≃ BG′ stably at p, then G and G′ have the same number of conjugacy classes of p-subgroups of order |Q| for all Q.

Proof. It is easy to see that

𝔽_{p}[Inj(Q, G)] = ⨁_{Q_{1}⊂G^{(Q_{1}), Q_{1}≅Q}} 𝔽_{p}[Out(Q)] ⊗_{𝔽_{p}[W_{G}(Q_{1})]} 𝔽_{p},

where

W_{G}(Q_{1}) = N_{G}(Q_{1})/C_{G}(Q_{1}) · Q_{1}

is the “Weyl group” of Q_{1}. From the third part of the theorem, we have an isomorphism

⨁_{Q_{1}⊂G^{(Q_{1}), Q_{1}≅Q}} 𝔽_{p}[Out(Q)] ⊗_{𝔽_{p}[W_{G}(Q_{1})]} 𝔽_{p} ≅ ⨁_{Q_{1}⊂G′^{(Q_{1}), Q_{1}≅Q}} 𝔽_{p}[Out(Q)] ⊗_{𝔽_{p}[W_{G}(Q_{1})]} 𝔽_{p},

and if we apply to this the functor 𝔽_{p} ⊗_{𝔽_{p}[Out(Q)]}(·), then we conclude that

⨁_{Q_{1}≅Q^{(Q_{1}), Q_{1}⊂G}} 𝔽_{p} ≅ ⨁_{Q_{1}≅Q^{(Q_{1}), Q_{1}⊂G′}} 𝔽_{p} .

□

Definition. Let H, K be subgroups of G. We say that H and K are pointwise conjugate in G if there is a bijection of sets H ⟶^{ϕ} K such that ϕ(h) = g(h)hg(h)^{−1} for some g(h) ∈ G depending on h. This is equivalent to the statement that

|H ∩ (g)| = |K ∩ (g)|

for all g ∈ G.

Corollary 10. Assume that G and G′ have normal Sylow p-subgroups P and P′ respectively. Then BG ≃ BG′ stably at p if and only if there is an isomorphism P ⟶^{ϕ} P′ such that W_{G}(P) is pointwise conjugate to ϕ^{−1}W_{G}(P′)ϕ.

References

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Department of Mathematics, Northwestern University, Evanston, IL 60208, USA

E-mail address: priddy@math.northwestern.edu