A NOTE ON THE HOMOTOPY TYPE OF BSL_{3}(ℤ)_{2}^{∧}

RAN LEVI

Abstract. It is known that for p-perfect groups G of finite virtual cohomological dimension and finite type mod-p cohomology, the pcompleted classifying space BG_{p}^{∧} has the property that ΩBG_{p}^{∧} is a retract of the loop space on a simply-connected, 𝔽_{p}-finite, p-complete space. In this note we consider a particular example where this theorem applies, namely we study the homotopy type of BSL_{3}(ℤ)_{2}^{∧}. It particular we analyze ΩBSt_{3}(ℤ)_{2}^{∧}, a double cover of ΩBSL_{3}(ℤ)_{2}^{∧}, and obtain a splitting theorem for it in terms of 2-primary Moore spaces and fibres of degree 2^{r} maps on spheres. We also give a formula for the Poincaré series of H_{∗}(ΩBΓ_{p}^{∧}; 𝔽_{p}) for a general group Γ, as above, in terms of possibly simpler components. This formula is used to calculate the mod-2 homology of ΩBΓ_{2}^{∧} for Γ = SL_{3}(ℤ) or St_{3}(ℤ) as modules over a certain tensor subalgebra.

0. Introduction

Let p be a prime number and let Γ be a p-perfect group of finite virtual cohomological dimension and mod-p cohomology of finite type. Then it shown in [4] that in this case there exists a simply-connected, 𝔽_{p}-finite, p-complete space S and a map f : S ^{}BΓ_{p}^{∧} such that Ωf has a right homotopy inverse. If one makes the additional hypothesis that the p-adic cohomology of BΓ is of finite type then it follows from [1] that S can be thought of as the p-completion of a simply connected finite complex. Thus one can expect a rich interplay between the the homotopy theory associated to classifying spaces of groups Γ, as above, and the homotopy theory of finite complexes. In this note we investigate this relationship for the first non-trivial example among a particularly interesting family of groups, namely the special linear groups over the integers SL_{n}(ℤ). The case we study is n = 3, where the cohomology was computed by Soulé [8] in 1978. For n ≥ 1 let St_{n}(ℤ) denote the Steinberg group of rank n over the integers. For any n the group St_{n}(ℤ) is a central extension of SL_{n}(ℤ) by ℤ/2ℤ.

Throughout this paper we denote the groups SL_{3}(ℤ) and St_{3}(ℤ) by Sl and St respectively. We show that the spaces Ω(BSl)_{2}^{∧} and Ω(BSt)_{2}^{∧} are retracts of the loop spaces on certain 4-dimensional torsion complexes. Moreover, these complexes are shown to be given by wedges of mod-2^{r} Moore spaces. We then proceed with a calculation of the mod-2 homology of Ω(BSt)_{2}^{∧} and Ω(BSl)_{2}^{∧} as finitely generated free modules over a certain subalgebra. This calculation is an easy consequence of a general formula for the Poincaré series of H_{∗}(ΩBΓ_{p}^{∧}; 𝔽_{p}), which we derive, expressing it in terms, which for our examples are easily computable.

1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55Q52.

.

Our calculations are used to obtain a splitting result for Ω(BSt)_{2}^{∧} in terms of an infinite product, where one of the factors is the fibre of the degree 16 map on the sphere S^{3} and the other factors are given in terms of mod-2^{r} Moore spaces. This splitting gives a considerably large family of non-trivial elements in the 2-primary unstable algebraic K groups K_{3,∗}(ℤ) = π_{∗}(BSL_{3}(ℤ)_{2}^{∧}).

The point of this calculation is to show how homotopy theoretic information can be derived in those examples out of some naive representation theory associated to the groups under consideration and cohomological information. For n > 3 there isn’t nearly as much known about the cohomology of SL_{n}(ℤ) as is the case for n = 3. However, the reader will notice that we do not use any high dimensional cohomological information but rather only need to know the cohomology of St_{3}(ℤ) up to dimension 4. Thus one may hope that our techniques might generalize to higher values of n.

The paper is organized as follows. Section 1 contains a preliminary discussion of the main claims of [4], which are a prerequisite for our study here. A summary of results is in section 2. The rest of the paper contains proofs of our statements.

It is a pleasure to thank Brayton Gray and Stewart Priddy for useful discussions. We also thank the referee for some useful comments.

1. Preliminaries

Recall that a group Γ is p-perfect if its first mod-p cohomology vanishes. Let Γ be a p-perfect group of finite virtual cohomological dimension and mod-p cohomology of finite type. It is shown in [4] that for some p-compact group X, there exists a map f : BΓ_{p}^{∧} ⟶ BX, such that the homotopy fibre X_{Γ} of f is 𝔽_{p}-finite, i.e. its mod-p cohomology is a finite dimensional mod-p vector space. One may always take BX to be the p-completion of the classifying space of a suitable compact Lie group. A map f as above is said to be a faithful homotopy representation of Γ in X. A general reference for p-compact groups is [3].

Given a faithful homotopy representation of a group Γ as above in a connected p-compact group X, one gets, upon p-completion, a fibre sequence

X ⟶^{ι} (X_{Γ})_{p}^{∧} ⟶^{j} BΓ_{p}^{∧} ⟶^{f} BX. (1)

A note on the homotopy type of ΩBSL_{3}(ℤ)_{2}^{∧} 3

Let S_{f}(Γ) denote the p-completion of the mapping cone of ι. Notice that S_{f}(Γ) is 𝔽_{p}-finite. The universal property of cofibrations in conjunction with the fact that BΓ_{p}^{∧} is p-complete gives a map ρ : S_{f}(Γ) ⟶ BΓ_{p}^{∧} and the main result of [4] is that Ωρ has a right homotopy inverse. In general one has the formula

(2) ΩS_{f}(Γ) ≃ ΩBΓ_{p}^{∧} × ΩΣ(X ∧ ΩBΓ_{p}^{∧})_{p}^{∧}.

Notice that this theorem implies that, given any explicit CW structure on BΓ and letting B_{n}Γ denote n-skeleton (for instance one may take the n-th stage of Milnor’s classifying space construction for Γ), then for some n > 0 the p-completion of the inclusion B_{n}Γ ^{}BΓ admits a right homotopy inverse after looping. The minimal n for which this holds is called the p-essential dimension ed_{p}(Γ) of the group Γ. It is of course independent of the choice of a CW structure. In [4] the author gives an upper bound on ed_{p}(Γ). For groups of positive vcd, the bound is given by the sum of the minimal dimension of a faithful homotopy representation of Γ and its vcd.

2. statement of results

For any n ≥ 2 and any prime power p^{r}, let P^{n}(p^{r}) denote the mod-p^{r} Moore space, i.e. the cofibre of the degree p^{r} map on the sphere S^{n−1}.

Theorem 2.1. The 2-essential dimension of Sl and St is precisely 4. Moreover, let S = P^{3}(2)∨P^{3}(2)∨P^{4}(4)∨P^{4}(8) and let S̃ = P^{3}(2)∨P^{4}(4)∨ P^{4}(16). Then there are maps α : S ⟶ (BSl)_{2}^{∧} and α̃ : S̃ ⟶ (BSt)_{2}^{∧} such that Ωα and Ωα̃ have right homotopy inverses.

As we shall see, formula (2) gives that there exist homotopy equivalences

ΩS ≃ Ω(BSl)_{2}^{∧} × ΩΣ(SO(3) ∧ Ω(BSl)_{2}^{∧})

and

ΩS̃ ≃ Ω(BSt)_{2}^{∧} × ΩΣ^{4}Ω(BSt)_{2}^{∧}.

One has to be a bit careful here in general, as a statement as above will be true only after an extra completion on the right hand side. However the factors on the right hand side of both equations are purely 2-torsion spaces and thus are 2-complete to begin with.

These equivalences imply that 8 and 16 annihilate the reduced homology of Ω(BSl)_{2}^{∧} and Ω(BSt)_{2}^{∧} respectively and, furthermore, that this order of torsion is actually present in the respective homology groups. The second bound is a direct consequence of the more explicit description of Ω(BSt)_{2}^{∧} given below. However the existence of torsion of order 8 in the homology of Ω(BSl)_{2}^{∧} is mildly curious as the homology of (BSl)_{2}^{∧} has exponent 4. At any rate the fact that loop space homology has an exponent at all is rather unusual. The reader is referred to [5] for a general theorem in this fashion for finite groups. It is worth pointing out though, that for higher values of n, the groups SL_{n}(ℤ) have rational homology, so an exponent result should not be expected.

We also remark that the upper bound for the 2-essential dimension of these groups, as follows from [4], is 6. Thus the approximation there is not best possible in these examples.

Since S and S̃ in Theorem 2.1 are finite suspensions, their loop space homology algebras are finitely generated free associative algebras and so an immediate corollary is the following

Corollary 2.2. The algebras H_{∗}(Ω(BSl)_{2}^{∧}; 𝔽_{2}) and H_{∗}(Ω(BSt)_{2}^{∧}; 𝔽_{2}) are finitely generated.

Next we get a more explicit hold on the homology of Ω(BSl)_{2}^{∧} and Ω(BSt)_{2}^{∧}. To do that we first obtain a general formula for the Poincaré series of ΩBΓ_{p}^{∧} for a general group Γ, satisfying our hypotheses. For a space Y let P_{Y} denote the Poincaré series for H_{∗}(Y; 𝔽_{p}).

Theorem 2.3. Let Γ be a group of finite vcd and finite type mod-p cohomology. Let a faithful homotopy representation f of Γ in some p-compact group X be given and let S = S_{f}(Γ) be defined as above. Then

P_{ΩBΓ_{p}^{∧}} = \frac{P_{ΩS}P_{X}}{1 − P_{ΩS} + P_{ΩS}P_{X}}

One could hope that it would be easier to calculate P_{ΩS} than P_{ΩBΓ_{p}^{∧}}, since S is 𝔽_{p}-finite, where BΓ_{p}^{∧} generally has infinite dimensional mod-p cohomology. Indeed in our examples the respective spaces S and S̃ of Theorem 2.1 are suspensions, a fact which makes the calculation very easy. More generally, one has the following

Corollary 2.4. With the notation of Theorem 2.3, suppose that S is a co-H space. Then

P_{ΩBΓ_{p}^{∧}}(t) = \frac{tP_{X}}{1 − P_{S} + tP_{X}}

It should be pointed out that finite or finite vcd groups Γ with the property that ΩBΓ_{p}^{∧} is a retract of the loop space on a finite co-H space appear to be rather rare.

Having computed the Poincaré series of ΩBΓ_{2}^{∧} for our examples, we are now able to get most of the product structure in the respective homology algebras.

Theorem 2.5. Let T denote the free associative algebra over 𝔽_{2} generated by elements a_{1}, b_{2}, c_{2}, x_{2} and d_{3}, where degrees are given by subscripts. Let H denote H_{∗}(Ω(BSl)_{2}^{∧}; 𝔽_{2}) and let H̃ denote H_{∗}(Ω(BSt)_{2}^{∧}; 𝔽_{2}). Then H and H̃ are finitely generated free T -modules. In fact there are isomorphisms of T -modules

H ≅ T ⊗ H_{∗}(SO(3); 𝔽_{2}) and H̃ ≅ T ⊗ H_{∗}(S^{3}; 𝔽_{2}).

Finally we show that, as a space, Ω(BSt)_{2}^{∧} can be expressed in terms of simpler building blocks.

Theorem 2.6. Let Y denote the space P^{3}(2) ∨ P^{4}(4) and let P denote ΩS^{3} ∧ ΩY. Then there is a homotopy equivalence

Ω(BSt)_{2}^{∧} ≃ ΩΣP × S^{3}{16} × ΩY. (1)

As we observe below, Ω(BSt)_{2}^{∧} is a double cover of Ω(BSl)_{2}^{∧}. Thus (BSl)_{2}^{∧} and (BSt)_{2}^{∧} have the same 2-connected cover. Notice that Theorem 2.6 gives the homotopy type of ΩBSt_{3}(ℤ)_{2}^{∧} in terms of very familiar spaces. Elementary homotopy theory can be used to show that ΣP splits as an infinite wedge, where most of the summands are mod-2 and 4 Moore spaces, while the others are given by iterated smash products of mod-2 Moore spaces. The factor ΩY also admits a splitting with similar summands. Thus the theorem yields a large family of non-zero 2-primary classes in the unstable algebraic K-theory of the integers K_{3,∗}(ℤ)_{2}^{∧} = π_{∗}(BSL_{3}(ℤ)_{2}^{∧}).

3. Finite complexes associated to SL_{n}(ℤ)

For any Euclidean domain E, the Steinberg group St_{n}(E) is a central extension of SL_{n}(E). Moreover, the extension is natural with respect to ring homomorphisms. Thus, consider the faithful representation of SL_{n}(ℤ) in SL_{n}(ℝ), given by the natural inclusion. For E = ℤ or ℝ the kernel of the extension above is given by ℤ/2ℤ and one gets the corresponding commutative diagram of central extensions.

Applying the classifying space functor followed by R-completion to this diagram, where R ⊆ ℚ or R = 𝔽_{p}, and replacing Lie groups by their maximal compact subgroups, we get a homotopy commutative diagram of fibrations:

[figure: Z(n); =; Z(n); (Bℤ/2ℤ)_{R}^{∧}; BSt_{n}(ℤ)_{R}^{∧}; BSL_{n}(ℤ)_{R}^{∧}; (Bℤ/2ℤ)_{R}^{∧}; =; BSpin(n)_{R}^{∧}; BSO(n)_{R}^{∧}] (2)

The theory presented in section 1 gives that the cofibres S(n) and S̃(n) of the maps from SO(n)_{R}^{∧} and Spin(n)_{R}^{∧} to Z(n), respectively, have the property that

ΩS(n)_{R}^{∧} ≃ ΩBSL_{n}(ℤ)_{R}^{∧} × ΩΣ(SO(n)_{R}^{∧} ∧ ΩBSL_{n}(ℤ)_{R}^{∧})_{R}^{∧}

and

ΩS̃(n)_{R}^{∧} ≃ ΩBSt_{n}(ℤ)_{R}^{∧} × ΩΣ(Spin(n)_{R}^{∧} ∧ ΩBSt_{n}(ℤ)_{R}^{∧})_{R}^{∧}.

Notice that S(n) is the R-completion of the cofibre of the map from SL_{n}(ℝ) to Z′(n), where Z′(n) denotes the orbit space SL_{n}(ℝ)/SL_{n}(ℤ). The corresponding description can be given for S̃(n).

By [4], for R = 𝔽_{p}, the mod-p cohomological dimension of both S(n) and S̃(n) is bounded above by the sum of the cohomological dimension of SO(n) and the vcd of SL_{n}(ℤ). The vcd of SL_{n}(ℤ) was computed by Borel and Serre [2] to be \frac{n(n−1)}{2}, which apparently is the same as the topological dimension of SO(n). Thus n(n − 1) is an upper bound for the cohomological dimension of S(n). This is also an upper bound for the p-essential dimension of these groups for every prime p although an improvement, which we don’t discuss here, can be obtained for odd primes. Notice that a bound for the 2-essential dimension of Sl and St, given by this discussion is 6, whereas the actual 2-essential dimension is 4, as we show below.

4. The cohomology of Sl and St and the related homotopy types

In [8] Soulé presents a cohomology calculation for Sl and St with coefficients in the 2-local integers ℤ_{(2)}. His methods though are applicable to any ring of coefficients, a fact which will be of use in the sequel. The actual calculation will be of limited benefit to us, but for the sake of completeness we start by recording it.

Theorem 4.1 (Soulé). There are isomorphisms

H^{∗}(BSl; ℤ_{(2)}) ≅ P [x_{3}, y_{3}, a_{4}, b_{4}, t_{5}, u_{6}, v_{6}]/I

and

H^{∗}(BSt; ℤ_{(2)}) ≅ P [α_{3}, η_{4}, ν_{4}]/J,

where degrees are given by subscripts. The ideal I is given by

2x = 2y = 4a = 4b = 2t = 2u = 2v = 0

vx = vb = vt = vu = yx = yb = yt = yu = 0
v^{2} + vy^{2} = ab + xt = au + ax^{2} = au + t^{2} = 0
xu + bt = ab^{2} + u^{2} = tu + tx^{2} = 0.

The ideal J is given by 2α = 4η = 16ν = α^{2} = αη = ην = 0

Notice that the cohomology of St is dramatically simpler than that of Sl. In fact the cohomology of St is periodic, as one can see from Theorem 4.1 or observe as follows. Recall that Spin(3) ≃ S^{3}. Thus, by pulling back the central column of Diagram, 2 we get a fibration

(S^{3})_{2}^{∧} ^{}Z ^{}(BSt)_{2}^{∧},

where Z = Z(3). Since Z is of finite mod-2 cohomological dimension and since the fibration is principal, it follows that there is a non-trivial transgression in the associated Serre spectral sequence, which grants periodicity in positive dimensions. It takes an easy exercise in undetermined coefficients to see that the periodicity class, given by the image of transgression, is aη + bν, where a and b are the classes of some odd integers modulo 4 and 16 respectively. The following is a straight forward consequence of the Serre spectral sequence.

Lemma 4.2. There is an isomorphism of abelian groups

H^{∗}(Z; R) ≅ H^{∗}(P^{3}(2) ∨ P^{4}(4) ∨ S^{3}; R),

For R = ℤ_{(2)} or 𝔽_{2}.

Consider the map (S^{3})_{2}^{∧} ^{}Z. The Serre spectral sequence calculation easily implies that the induced map is degree 16 on integral cohomology. On the other hand notice that there is a commutative diagram of cofibrations

[figure: S^{3}; Z; S̃; SO(3); π; Z; =; S] (3)

and that π is degree 2 on the top class. Here of course S and S̃ are S(3) and S̃(3) of the preceding section respectively. It is now easy to conclude the following

Lemma 4.3. There are isomorphisms of abelian groups

H^{∗}(S; R) ≅ H^{∗}(P^{3}(2) ∨ P^{3}(2) ∨ P^{4}(4) ∨ P^{4}(8); R),

and

H^{∗}(S̃; R) ≅ H^{∗}(P^{3}(2) ∨ P^{4}(4) ∨ P^{4}(16); R),

For R = ℤ_{(2)} or 𝔽_{2}.

Notice that the map ρ̃: S̃ ^{}(BSt)_{2}^{∧} can be thought of as the inclusion of the 4-skeleton, but that S actually has torsion of order 8 whereas the integral cohomology of BSl is annihilated by 4. Furthermore, by the discussion in the preceding section, Ωρ and Ωρ̃ have right homotopy inverses. We proceed by analyzing the homotopy structure of S and S̃.

Throughout the end of this section we omit the 2-completion symbol at our convenience. All statement thus hold up to 2-completion. The homotopy type of S̃ can be determined rather easily if one knows its cohomology with ℤ/16ℤ coefficients. Indeed consider the additive cohomological structure, given in Lemma 4.3. Clearly the 3-skeleton S̃^{(3)} is given by P^{3}(2) ∨ S^{3} ∨ S^{3}. Analyzing π_{3} of this space is elementary; there is a free summand of rank 2 given by π_{3}(S^{3} ∨ S^{3}) with generators λ_{1}, λ_{2} and a class of order 4 given by the generator of π_{3}(P^{3}(2)). The last is the class of the composite S^{3} ^{η}^{}S^{2} ^{π}^{}P^{3}(2), where η is the Hopf map.

Thus each of the two attaching maps for 4-cells in S̃ is given as a sum aλ_{1}+bλ_{2}+cπη, where a and b are the appropriate powers of 2 and c ∈ ℤ/4ℤ. Clearly S̃ is a wedge of Moore spaces if and only if c = 0 for each one of the attaching maps. Notice that if c is odd, the corresponding cohomology class will be the cup square of the bottom class. Thus such a summand can be detected by mod-2^{r} cohomology for r ≥ 1. On the other hand 2πη can be detected in mod-2^{r} cohomology for r ≥ 2. Thus, in particular, by computing the cup product structure in H^{∗}(S̃; ℤ/16ℤ), we can complete the analysis of its homotopy type.

Proposition 4.4. All cup products in H^{∗}(S̃; ℤ/16ℤ) are trivial. Thus there is a homotopy equivalence

S̃ ≃ P^{3}(2) ∨ P^{4}(4) ∨ P^{4}(16).

Proof. The map ρ̃: S̃ ^{}(BSt)_{2}^{∧} produced above induces an isomorphism on mod-2^{r} cohomology for r ≥ 1 through dimension 4. Thus our claim follows by using Soulé’s method to calculate the mod-16 cohomology of St in low dimensions.

Indeed Soulé shows that the 2-completed homotopy types of BSl and BSt can be obtained as the 2-completions of certain homotopy colimits, which we describe below (compare to [6]). Thus the respective cohomology can be computed in terms of the spectral sequence associates to a homotopy colimit. In both cases the spectral sequence degenerates into a long exact, Mayer-Vietoris type, long exact sequence which allows an easy calculation of the cohomology. We proceed by describing this long exact sequence for St.

Consider the diagram

Σ_{4}  Σ_{4}  Σ_{4}
[figure: j_{2}; j_{1}; i_{1}; i_{2}]
ℤ/2ℤ  D_{8},

where the morphisms j_{1} factors through the alternating group A_{4} and j_{2} does not; the morphisms i_{k}, k = 1, 2, are specified by the requirement that they are not conjugate in D_{8}.

Applying the classifying space functor B(−) to this diagram one obtains a map ϕ from the homotopy colimit of the resulting diagram of spaces to the classifying space BSl, which induces a mod-2 cohomology isomorphism and thus a homotopy equivalence after 2-completion. In particular ϕ induces a cohomology isomorphism with respect to any 2-local ring of coefficients.

Now, consider the group Σ̃_{4} given as the unique central extension of Σ_{4} by ℤ/2ℤ which covers the projection Q_{16} ^{}D_{8}, the target being the Sylow 2-subgroup of Σ_{4}. Then the diagram above can be pulled back along the projection of Σ̃_{4} onto Σ_{4} to obtain a new diagram

Σ̃_{4}  Σ̃_{4}  Σ̃_{4}
[figure: j̃_{2}; j̃_{1}; ĩ_{1}; ĩ_{2}]
ℤ/4ℤ  Q_{16}.

The process described above for Sl applies here to give a homology decomposition of BSt at 2, namely, applying B(−) and taking the homotopy colimit, one obtains a map ϕ̃ to BSt, which induces a cohomology isomorphism with respect to any 2-local ring of coefficients.

Notice that the nature of these diagrams implies that in the associated spectral sequence for the cohomology of the homotopy colimit, the nontrivial terms appear on the 0 and 1 vertical lines. In other word the functor lim^{i}(H^{∗}(−)) applied to these diagrams obtains trivial values for i > 1. Thus for the diagram corresponding to the Steinberg group one obtains each cohomology groups as an extension

0 ^{}lim_{D}^{1} H^{n−1}(−) ^{}H^{n}(BSt) ^{}lim_{D} H^{n}(−) ^{}0,

where D is the diagram for St. Splicing these sequences dimensionwise with the exact sequence defining the inverse limit and its first derived functor and canceling redundant terms, one gets, as in [8], a long exact sequence with respect to any 2-local ring of coefficients:

⋯ ^{}H^{i}(St) ^{α}^{}H^{i}(Σ̃_{4})^{×3} ^{d}^{}H^{i}(Q_{16}) × H^{i}(ℤ/4ℤ) ^{}⋯ (4)

where the map d is given by the formula

d(x, y, z) = (ĩ_{2}^{∗}(x) − ĩ_{1}^{∗}(y), j̃_{1}^{∗}(y) − j̃_{2}^{∗}(z))

and α is the ring homomorphism induced by the natural map from the 3-fold free product of Σ̃_{4} into St.

The cohomology of all groups involved is well known. In particular an explicit description of the cohomology of Σ̃_{4} is in [8]. We write an additive set of generators for these groups in low dimensions. Throughout this section mod-16 coefficients are assumed. For 1 ≤ k ≤ 4 the groups H^{k}(Σ̃_{4}) are generated by elements σ_{1}, η_{2}, ϵ_{3} and τ_{4}, where dimensions are given by subscripts and orders are 2, 2, 16 and 16 respectively. Similarly, the groups H^{k}(Q_{16}) are generated by α_{1}, β_{1}, γ_{2}, δ_{2}, ϵ′_{3} and τ′_{4} of orders 2, 2, 2, 2, 16 and 16 respectively. Finally the groups H^{k}(ℤ/4ℤ) are generated by a_{1}, b_{2}, c_{3} and d_{4} all of order 4. Using Soulé’s integral cohomology calculation and the universal coefficients theorem, it is an elementary exercise to compute the following table, describing the effect of the maps induced on cohomology by the homomorphisms in the diagram above.

σ  η ϵ τ
j̃_{1}^{∗} 0  0 c d
j̃_{2}^{∗} 2a  2b c d
ĩ_{1}^{∗} α  γ ϵ′ τ′
ĩ_{2}^{∗} α + β γ + δ ϵ′ τ′

From this table one easily calculates the homomorphism d in (4) up to i = 4. In particular one has from the data above that the map α is injective in dimension 4 and zero in dimension 2. By naturality of the cup product one concludes that the cup square of the generator of H^{2}(St) vanishes. This completes the proof.

The homotopy type of S is calculated similarly. However, a direct calculation of the cohomology of Sl is not necessary.

Proposition 4.5. There is a homotopy equivalence

S ≃ P^{3}(2) ∨ P^{3}(2) ∨ P^{4}(4) ∨ P^{4}(8).

Proof. Consider Diagram (3) above. It follows immediately that the map

H^{4}(S; ℤ/16ℤ) ^{}H^{4}(S̃; ℤ/16ℤ)

is injective. By naturality of the cup product, it follows at once that all cup products of elements in H^{2}(S; ℤ/16ℤ) ≅ ℤ/2ℤ ⊕ ℤ/2ℤ vanish. Thus a simple argument as above, involving the possible attaching maps for 4-cells in S, gives that the two 4-cells are attached only to the respective 3-cells, as detected by cohomology, and the result follows.

This completes the proof of Theorem 2.1 except for the claim that the 2-essential dimension of the groups under consideration is precisely 4. It is obviously bounded above by 4. Furthermore, a simply-connected 2dimensional space of finite type must be equivalent to a wedge of 2-spheres. Thus for any group Γ whose homology contains p-torsion, the p-essential dimension is larger than 2. Thus it remains to show that ed_{2}(Γ) > 3 in our examples.

Indeed, independently of the precise value of the 2-essential dimension, we compute the homology of Ω(BSl)_{2}^{∧} and Ω(BSt)_{2}^{∧}. It follows immediately from the calculation that in both cases the loop space homology has indecomposable elements in dimension 3. On the other hand, every simply-connected 3-dimensional space T is a suspension and thus its loop space homology is a free associative algebra on generators in dimensions 1 and 2. It follows at once that if Γ is any group of p-essential dimension 3 then the homology of ΩBΓ_{p}^{∧} is generated by elements in degrees 1 and 2. Since this is not the case in our examples, the proof is complete.

5. A formula for the Poincaré series

Let P_{X} denote the Poincaré series for the mod-p homology of a space X. Let Γ be a p-perfect finite vcd group satisfying the extra hypothesis of

having a finite type mod-p cohomology module. Let a faithful homotopy representation f of Γ in some p-compact group X be given and construct the space S = S_{f}(Γ), as explained in section 1 above. Then there is a homotopy equivalence

ΩS_{f}(Γ) ≃ ΩΣ(X ∧ ΩBΓ_{p}^{∧}) × ΩBΓ_{p}^{∧}.

Now since H_{∗}(ΩΣA) ≅ T [H̃_{∗}(X)], where T denotes the tensor algebra, one has

P_{ΩΣ(X∧ΩBΓ_{p}^{∧})} = \frac{1}{1 − P̃_{X} P̃_{ΩBΓ_{p}^{∧}}} = \frac{1}{1 − (P_{X} − 1)(P_{ΩBΓ_{p}^{∧}} − 1)},

where P̃ denotes the Poincaré series for the reduced homology. Thus we get the formula

P_{ΩS} = \frac{P_{ΩBΓ_{p}^{∧}}}{1 − (P_{X} − 1)(P_{ΩBΓ_{p}^{∧}} − 1)}

It is now easy to solve for P_{ΩBΓ_{p}^{∧}} and get the formula of Theorem 2.3.

Having established this, Corollary 2.4 follows from the observation that if S is a co-H space then

P_{ΩS}(t) = \frac{t}{1 + t − P_{S}} .

6. the loop space homology

Let H and H̃ denote the homology algebras of Ω(BSl)_{2}^{∧} and Ω(BSt)_{2}^{∧} respectively. Let P_{H} and PH̃ denote the respective Poincaré series.

Proposition 6.1. With the notation above

P_{H}(t) = \frac{1 + t^{3}}{1 − t − 3t^{2} − t^{3}}

and

PH̃ = \frac{1 + t + t^{2} + t^{3}}{1 − t − 3t^{2} − t^{3}}

Proof. Since both S and S̃ are suspensions, we can apply Corollary 2.4, which gives

P_{H}(t) = \frac{tP_{SO(3)}}{1 − P_{S} + tP_{SO(3)}} and PH̃ = \frac{tP_{S^{3}}}{1 − PS̃ + tP_{S^{3}}} .

Using the information obtained above about S and S̃, the reader immediately verifies the proposition.

Recall that we have fibrations

ΩZ ^{}Ω(BSl)_{2}^{∧} ^{}(S^{3})_{2}^{∧}

and

ΩZ ^{}Ω(BSt)_{2}^{∧} ^{}SO(3)_{2}^{∧}.

But notice that Z is a suspension with the Poincaré series for the mod-2 homology of ΩZ given by \frac{1}{1−t−3t^{2}−t^{3}} . On the other hand the numerators of the fractions above give the Poincaré polynomials of SO(3) and S^{3} respectively. Thus the Serre spectral sequences for the fibrations above both collapse at the E^{2} page and we get the statement of Theorem 2.5.

7. splitting in terms of Moore spaces

Let Z be, as before, the fibre of the map (BSt)_{2}^{∧} ^{}B(S^{3})_{2}^{∧}. Then the restriction map H^{∗}((BSt)_{2}^{∧}) ^{}H^{∗}(Z) is an epimorphism with respect to any 2-local ring of coefficients. Thus the same argument as made for S̃ in Proposition 4.4 gives that Z ≃ Y ∨ S^{3}, where Y = P^{3}(2) ∨ P^{4}(4). Thus there is a diagram of fibrations

[figure: F̃; S^{3}; S^{3} ⋊ ΩY; Ω(BSt)_{2}^{∧}; S^{3}; Z; ΩY; q; ∗; Y; p] (5)

Notice that q has a right homotopy inverse, since p does. Hence, Ω(BSt)_{2}^{∧} being a loop space, splits as a product

Ω(BSt)_{2}^{∧} ≃ F̃ × ΩY.

The following two diagrams give a hold on F̃. Let P denote ΩS^{3} ∧ ΩY.

[figure: ΩΣP; ∗; ΣP; F̃; q̃; S^{3}{16}; ΣP; F̃; S^{3}; S^{3} ⋊ ΩY; F̃; =; S^{3}; S^{3} ⋊ ΩY; S^{3}{16}; q̃; S^{3}; =; [16]; S^{3}; p_{1}; ∗; S^{3}; [16]; =; S^{3}; p_{1}] (6)

The left diagram implies that q̃ has a right homotopy inverse, since p_{1} has one and since the 3-skeleton of S^{3} ⋊ΩY is given by S^{3}. The right diagram, on the other hand, gives that the fibration

ΩΣP ^{}F̃ ^{}S^{3}{16}

is principal and hence that there is a homotopy equivalence

F̃ ≃ ΩΣP × S^{3}{16}.

Thus we get the statement of Theorem 2.6,

Ω(BSt)_{2}^{∧} ≃ ΩΣ(ΩS^{3} ∧ ΩY) × S^{3}{16} × ΩY.

To end this section, we point out that Σ(ΩS^{3}∧ΩY) splits into an infinite wedge, where the factors are mod-2 and 4 Moore spaces and iterated smash products of mod-2 Moore spaces. These splittings are elementary but we describe them in some detail for the benefit of the reader who might not be a homotopy theorist. Good references are [9] and [7].

First, if X is a space then the James splitting theorem gives that

ΣΩΣX ≃ Σ ⋁_{k≥1} X^{(k)},

where X^{(k)} denotes the k-fold smash product of X. It is also easy to see that for r > s ≥ 1, there is the identity

P^{n}(2^{r}) ∧ P^{k}(2^{s}) ≃ P^{n+k−1}(2^{s}) ∨ P^{n+k}(2^{s}).

However the cohomology of iterated smash products of mod-2 Moore spaces contains Steenrod squares of excess larger than 1. Thus these smash products do not split as a wedge.

In view of this one has

Σ(ΩS^{3} ∧ ΩY) ≃ ⋁_{i=0}^{∞} S^{3+2i} ∧ ΩY ≃ ⋁_{i=0}^{∞} Σ^{3+2i}ΩΣ(P^{2}(2) ∨ P^{3}(4)).

For each factor Σ^{3+2i}ΩΣ(P^{2}(2) ∨ P^{3}(4)) there is a decomposition

Σ^{3+2i}ΩΣ(P^{2}(2) ∨ P^{3}(4)) ≃ Σ^{3+2i} ⋁_{j=1}^{∞}(P^{2}(2) ∨ P^{3}(4))^{(j)}.

The claim follows.

References

1. V. Belfi and C. Wilkerson, Some examples in the theory of p-completions, Ind. Univ. Math. Jour. 25 (1976), no. 6, 565–576.

2. A. Borel and J-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1974), 244–297.

3. W. Dwyer and C. Wilkerson, Homotopy fixed point methods for Lie groups and finite loop spaces, Ann. Math. 139 (1994), no. 2, 395–442.

4. R. Levi, On p-completed classifying spaces of discrete groups and finite complexes , to appear in Journal of the London Math. Soc.

5. , On finite groups and homotopy theory, Mem. AMS 567 (1995).

6. Norihiko Minami, On the classifying spaces of SL_{3}(ℤ), St_{3}(ℤ) and finite Chevalley groups, Contemporary Mathematics, vol. 158, 1994, pp. 175–185.

7. J. Neisendorfer, Primary Homotopy Theory, Mem. AMS 25 (1980), no. 232.

8. C. Soulé, The cohomology of SL_{3}(ℤ), Topology 17 (1978), 1–22.

9. G. W. Whitehead, Element of Homotopy Theory, Grad. Texts in Math., vol. 61, Springer-Verlag, 1978.

Dept. of mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208, U.S.A.

  E-mail address: ran@@math.nwu.edu