The mod 2 cohomology of the linear groups over the ring of integers

Dominique Arlettaz, Mamoru Mimura, Koji Nakahata, Nobuaki Yagita

Abstract. This paper completely determines the ring structure, with an explicit description of the generators, of the mod 2 cohomology of the linear groups GL(ℤ) , SL(ℤ) and St(ℤ) and its module structure over the Steenrod algebra.

1. Introduction

Recently, C. Weibel deduced from V. Voevodsky’s proof [V] of the Milnor conjecture the complete calculation of the 2-torsion of the algebraic K-theory of the ring of integers ℤ (see [W]). Of course, this has immediate consequences on the mod 2 cohomology of the infinite general linear group GL(ℤ) and more generally on the understanding of the space BGL(ℤ)^{+} . If p is a prime ≡ 3 or 5 mod 8 , M. Bökstedt introduced in [Bök] (see also [M] and Section 4 of [DF]) a space J(p) which is defined by the pull-back diagram

J(p) ⟶^{h′} BO
↓ f′_{p}  ↓ c
F Ψ^{p} ⟶^{b} BU ,

where F Ψ^{p} is the fiber of (Ψ^{p} − 1) : BU → BU (recall that F Ψ^{p} ≃ BGL(𝔽_{p})^{+} by Theorem 7 of [Q2]), b the Brauer lifting and c the complexification. The fibers of the horizontal maps are homotopy equivalent to the unitary group U . In fact, Bökstedt was more precisely interested in the covering space JK(ℤ, p) of J(p) corresponding to the cyclic subgroup of order 2 of π_{1}J(p) ≅ ℤ ⊕ ℤ/2 . After completion at the prime 2 , he constructed a map

φ̃ : (BGL(ℤ)^{+})_{ˆ2} ⟶ JK(ℤ, p)_{ˆ2}

which induces a split surjection on all homotopy groups. Recall that the localization exact sequence in K-theory implies that

(BGL(ℤ[\frac{1}{2}])^{+})_{ˆ2} ≃ (BGL(ℤ)^{+})_{ˆ2} × (S^{1})_{ˆ2} .

Therefore, φ̃ provides a map

φ : (BGL(ℤ[\frac{1}{2}])^{+})_{ˆ2} ⟶ J(p)_{ˆ2}

We would like to thank Christian Ausoni for his helpful comments on Bökstedt’s work [Bök]. The third author thanks the Swiss National Science Foundation for financial support. 1991 Mathematics Subject Classification: Primary 20 G 10; Secondary 19 D 55, 20 J 05, 55 R 40, 55 S 10.

which also induces a split surjection on all homotopy groups. Since the 2-torsion of K_{∗}(ℤ) is known by [W], it is easy to check that φ̃ and φ are actually homotopy equivalences. Consequently, we obtain for all primes p ≡ 3 or 5 mod 8 the pull-back diagram (see also Corollary 8 of [W])

(BGL(ℤ)^{+})_{ˆ2} × (S^{1})_{ˆ2} ⟶^{h′} BO_{ˆ2}
↓ f′_{p}  ↓ c
(F Ψ^{p})_{ˆ2}  ⟶^{b} BU_{ˆ2}

and the commutative diagram

SU_{ˆ2} ⟶^{η} (BGL(ℤ)^{+})_{ˆ2} ⟶^{h} BO_{ˆ2}
↓ ζ  ↓ f_{p}  ↓ c
U_{ˆ2} ⟶^{θ}  (F Ψ^{p})_{ˆ2}  ⟶^{b} BU_{ˆ2} ,

where both rows are fibrations: here f_{p} and h denote the composition of the inclusion (BGL(ℤ)^{+})_{ˆ2} ↪ (BGL(ℤ)^{+})_{ˆ2} × (S^{1})_{ˆ2} with f′_{p} and h′ respectively ( f_{p} is in fact the map induced by the reduction mod p : GL(ℤ) → GL(F_{p}) ), and ζ is the 2-completion of the inclusion SU ↪ U ≃ SU × S^{1} .

This diagram enables us to calculate the mod 2 cohomology of BGL(ℤ)^{+} : we first check easily (see Theorem 1) that there is a ring isomorphism

(∗)  H^{∗}(BGL(ℤ)^{+}; ℤ/2) ≅ H^{∗}(BO; ℤ/2) ⊗ H^{∗}(SU; ℤ/2) .

Recall that H^{∗}(BO; ℤ/2) ≅ ℤ/2[w_{1}, w_{2}, …] and H^{∗}(SU; ℤ/2) ≅ Λ(v_{3}, v_{5}, …) where deg(w_{j}) = j and deg(v_{2k−1}) = 2k − 1 . The main objective of this paper is to describe the generators of H^{∗}(BGL(ℤ)^{+}; ℤ/2) : the generators of the polynomial part are the Stiefel-Whitney classes, also denoted by w_{j} , coming from H^{∗}(BO; ℤ/2) via the homomorphism induced by h, and we identify the exterior generators u_{2k−1} of degree 2k − 1 in H^{∗}(BGL(ℤ)^{+}; ℤ/2) in terms of the image of the homomorphism

f_{p}^{∗} : H^{∗}(F Ψ^{p}; ℤ/2) ≅ H^{∗}(BGL(𝔽_{p})^{+}; ℤ/2) → H^{∗}(BGL(ℤ)^{+}; ℤ/2)

induced by the map f_{p} for p ≡ 5 mod 8 (see Definitions 5 and 9). We can prove that the definition of these classes does not depend on the choice of p by using the fact that they are primitive cohomology classes. The computation of the action of the Steenrod squares on the u_{2k−1}’s enables us to deduce in Corollary 12 that the above isomorphism (*) is actually an isomorphism of Hopf algebras and of modules over the Steenrod algebra (this last result can also be deduced by an indirect argument from the above diagram and Theorem 4.3 and Remark 4.5 of [M]). Moreover, we obtain an explicit formula relating the classes u_{2k−1} ∈ H^{∗}(BGL(ℤ)^{+}; ℤ/2) , for k ≥ 2 , to the image of the homomorphism f_{p}^{∗} for all primes ≡ 3 or 5 mod 8 (see Theorem 13). This provides a complete description of the mod 2 cohomology of the group GL(ℤ) . In the remainder of the paper we compute the mod 2 cohomology of the infinite special linear group SL(ℤ) and of the infinite Steinberg group St(ℤ) (see Corollary 14, Theorem 16 and Remark 17).

2. The mod 2 cohomology of the linear groups GL(ℤ) and SL(ℤ)

Theorem 1. There is a ring isomorphism

H^{∗}(BGL(ℤ)^{+}; ℤ/2) ≅ H^{∗}(BO; ℤ/2) ⊗ H^{∗}(SU; ℤ/2) .

Proof. Let Q denote the subgroup of diagonal matrices in GL(ℤ) and λ : BQ → BGL(ℤ)^{+} the map induced by the inclusion Q ↪ GL(ℤ) . It is known by Theorem 22.7 of [Bor] that the composition h λ : BQ → BO induces an injective homomorphism λ^{∗} h^{∗} : H^{∗}(BO; ℤ/2) ≅ ℤ/2[w_{1}, w_{2}, …] → H^{∗}(BQ; ℤ/2) ≅ ℤ/2[z_{1}, z_{2}, …] (with deg(z_{i}) = 1 ) and that λ^{∗} h^{∗}(w_{j}) = σ_{j} , the j-th elementary symmetric function in the z_{i}’s. This implies that the infinite loop map h induces an injective homomorphism h^{∗} : H^{∗}(BO; ℤ/2) → H^{∗}(BGL(ℤ)^{+}; ℤ/2) . Therefore, Theorem 15.2 of [Bor] shows that the Serre spectral sequence of the fibration

SU_{ˆ2} ⟶^{η} (BGL(ℤ)^{+})_{ˆ2} ⟶^{h} BO_{ˆ2}

collapses (see also Corollary 4.3 of [DF]) and we get additively the desired isomorphism. Since (BGL(ℤ)^{+})_{ˆ2} is an H-space, the maps λ and η produce an H-map

ψ : BQ × SU_{ˆ2} ⟶ (BGL(ℤ)^{+})_{ˆ2}

which induces an injective homomorphism

ψ^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) ⟶ H^{∗}(BQ; ℤ/2) ⊗ H^{∗}(SU; ℤ/2) .

Moreover, the image of ψ^{∗} is isomorphic to R ⊗ H^{∗}(SU; ℤ/2) , where R is the subalgebra of H^{∗}(BQ; ℤ/2) generated by the elementary symmetric functions σ_{j} . This provides the statement of the theorem.

Our next goal is to show that the isomorphism given by the above theorem is actually an isomorphism of modules over the Steenrod algebra. Therefore, we first need to identify the generators of H^{∗}(BGL(ℤ)^{+}; ℤ/2) and to understand the action of the Steenrod algebra on them. For j ≥ 1 let us write w_{j} ∈ H^{∗}(BGL(ℤ)^{+}; ℤ/2) for the image of the j-th universal Stiefel-Whitney class in H^{∗}(BO; ℤ/2) under the homomorphism h^{∗} : H^{∗}(BO; ℤ/2) → H^{∗}(BGL(ℤ)^{+}; ℤ/2) . The action of the Steenrod algebra on the Stiefel-Whitney classes is known by Wu’s formula (see for instance [MT], Part I, p. 141). It remains to identify the exterior generators of H^{∗}(BGL(ℤ)^{+}; ℤ/2) . This will be done by using the homomorphism f_{p}^{∗} : H^{∗}(F Ψ^{p}; ℤ/2) → H^{∗}(BGL(ℤ)^{+}; ℤ/2) induced by the map f_{p} .

Let us first recall some properties of H^{∗}(F Ψ^{p}; ℤ/2) ≅ H^{∗}(BGL(𝔽_{p})^{+}; ℤ/2) . According to Quillen’s calculation and notation (see [Q2]), if p is a prime ≡ 5 mod 8 , then

H^{∗}(F Ψ^{p}; ℤ/2) ≅ ℤ/2[c_{1}, c_{2}, …] ⊗ Λ(e_{1}, e_{2}, …) ,

where deg c_{j} = 2j and deg e_{k} = 2k − 1 ; if p is a prime ≡ 3 mod 8 , then H^{∗}(F Ψ^{p}; ℤ/2) is also generated by the classes c_{j} and e_{k} (j ≥ 1 , k ≥ 1) , but one has the relations

e_{k}^{2} = c_{2k−1} + ∑_{j=1}^{k−1} c_{j}c_{2k−1−j}

for k ≥ 1 , and H^{∗}(F Ψ^{p}; ℤ/2) is polynomial:

H^{∗}(F Ψ^{p}; ℤ/2) ≅ ℤ/2[e_{1}, e_{2}, …, c_{2}, c_{4}, …]

(see also Section IV.8 of [FP]). In both cases, c_{j} is the image under b^{∗} : H^{∗}(BU; ℤ/2) → H^{∗}(F Ψ^{p}; ℤ/2) of the reduction mod 2 of the j-th universal Chern class in H^{2j}(BU; ℤ) and a spectral sequence argument shows that θ^{∗} : H^{∗}(F Ψ^{p}; ℤ/2) → H^{∗}(U; ℤ/2) ≅ Λ(v_{1}, v_{2}, …) satisfies θ^{∗}(e_{k}) = v_{2k−1} for k ≥ 1 . For a prime p ≡ 3 or 5 mod 8 , consider the homomorphism f_{p}^{∗} : H^{∗}(F Ψ^{p}; ℤ/2) → H^{∗}(BGL(ℤ)^{+}; ℤ/2) induced by f_{p} . For all j ≥ 1 , we showed in Lemma 1.4 of [A1] that

f_{p}^{∗}(c_{j}) = w_{j}^{2}

and established in [A2] for k ≥ 2 the non-vanishing of the exterior class f_{p}^{∗}(e_{k}) if p ≡ 5 mod 8 , respectively of the exterior class

γ_{k} = f_{p}^{∗}(e_{k}) + w_{2k−1} + ∑_{j=1}^{k−1} w_{j}w_{2k−j−1}

of degree 2k − 1 if p ≡ 3 mod 8 .

Let us mention the effect of the Steenrod squares on these cohomology classes.

Lemma 2. (a) In H^{∗}(SU; ℤ/2) , Sq^{2i}v_{2k−1} = \binom{k−1}{i}v_{2k+2i−1} for k ≥ 2 , 1 ≤ i < k, and Sq^{2i−1}v_{2k−1} = 0 for k ≥ 2 , 1 ≤ i ≤ k.

(b) In H^{∗}(BGL(ℤ)^{+}; ℤ/2) , for any odd prime p, for k ≥ 1 and 1 ≤ i < k,

Sq^{2i}f_{p}^{∗}(e_{k}) = \binom{k−1}{i} f_{p}^{∗}(e_{k+i}) + ∑_{j=1}^{i} \binom{k−j−1}{i−j} (w_{j}^{2}f_{p}^{∗}(e_{k+i−j}) + w_{k+i−j}^{2}f_{p}^{∗}(e_{j})) .

(c) In H^{∗}(BGL(ℤ)^{+}; ℤ/2) , for k ≥ 1 and 1 ≤ i ≤ k,

Sq^{2i−1}f_{p}^{∗}(e_{k}) = \{ \begin{matrix}0 & , if p ≡ 1 mod 4 or if p ≡ 3 mod 4 and k − i is odd, \\ ∑_{j=0}^{i−1} \binom{k−j−1}{i−j−1} w_{j}^{2}w_{k+i−j−1}^{2} & , if p ≡ 3 mod 4 and k − i is even. \end{matrix}

(d) In H^{∗}(BGL(ℤ)^{+}; ℤ/2) , for any prime p ≡ 3 mod 8 and for k ≥ 1 ,

Sq^{2i}γ_{k} = \binom{k−1}{i} γ_{k+i} + ∑_{j=1}^{i} \binom{k−j−1}{i−j} (w_{j}^{2}γ_{k+i−j} + w_{k+i−j}^{2}γ_{j})

for 1 ≤ i < k and Sq^{2i−1}γ_{k} = 0 for 1 ≤ i ≤ k.

Proof. Lemma 4 of [A2] gives the following information on the action of the Steenrod squares on the classes e_{k} ∈ H^{∗}(F Ψ^{p}; ℤ/2) for k ≥ 1 : for any odd prime p and for 1 ≤ i < k,

Sq^{2i}e_{k} = \binom{k−1}{i} e_{k+i} + ∑_{j=1}^{i} \binom{k−j−1}{i−j} (c_{j} e_{k+i−j} + c_{k+i−j} e_{j}) ,

and for 1 ≤ i ≤ k,

Sq^{2i−1}e_{k} = \{ \begin{matrix}0 & , if p ≡ 1 mod 4 or if p ≡ 3 mod 4 and k − i is odd, \\ ∑_{j=0}^{i−1} \binom{k−j−1}{i−j−1} c_{j}c_{k+i−j−1} & , if p ≡ 3 mod 4 and k − i is even. \end{matrix}

The formula (a) is well known but can be deduce from the previous equalities because the composition ζ^{∗} θ^{∗} : H^{∗}(F Ψ^{p}; ℤ/2) → H^{∗}(SU; ℤ/2) satisfies ζ^{∗} θ^{∗}(e_{k}) = v_{2k−1} for k ≥ 2 and ζ^{∗} θ^{∗}(c_{j}) = 0 for j ≥ 1 . The statements (b) and (c) follow directly since Sq^{2i}f_{p}^{∗}(e_{k}) = f_{p}^{∗}(Sq^{2i}e_{k}) and f_{p}^{∗}(c_{j}) = w_{j}^{2} for j ≥ 1 . In order to get (d), let us consider again the homomorphism λ^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) → H^{∗}(BQ; ℤ/2) which is injective on ℤ/2[w_{1}, w_{2}, …] and trivial on exterior classes because H^{∗}(BQ; ℤ/2) is polynomial. If p ≡ 3 mod 8 , one has by the definition of γ_{k}

Sq^{2i}γ_{k} = Sq^{2i}f_{p}^{∗}(e_{k}) + Sq^{2i}w_{2k−1} + ∑_{j=1}^{k−1} Sq^{2i}(w_{j}w_{2k−j−1})

for 1 ≤ i < k. According to (b),

Sq^{2i}γ_{k} = \binom{k−1}{i} f_{p}^{∗}(e_{k+i}) + ∑_{j=1}^{i} \binom{k−j−1}{i−j} (w_{j}^{2}f_{p}^{∗}(e_{k+i−j}) + w_{k+i−j}^{2}f_{p}^{∗}(e_{j})) + (element of ℤ/2[w_{1}, w_{2}, …])

and consequently,

Sq^{2i}γ_{k} = \binom{k−1}{i} γ_{k+i} + ∑_{j=1}^{i} \binom{k−j−1}{i−j} (w_{j}^{2}γ_{k+i−j} + w_{k+i−j}^{2}γ_{j}) + (element of ℤ/2[w_{1}, w_{2}, …]) .

Since the classes γ_{k} are exterior, they belong to the kernel of λ^{∗} and λ^{∗}(Sq^{2i}γ_{k}) = 0 . However, the injectivity of λ^{∗} on Stiefel-Whitney classes implies that the element of ℤ/2[w_{1}, w_{2}, …] in the last formula vanishes. The assertion (c) shows that Sq^{2i−1}γ_{k} is an element of ℤ/2[w_{1}, w_{2}, …] and one deduces similarly that Sq^{2i−1}γ_{k} = 0 .

Our argument will be based on the understanding of the homomorphism

μ^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) → H^{∗}(BGL(ℤ)^{+} × BGL(ℤ)^{+}; ℤ/2) ≅ H^{∗}(BGL(ℤ)^{+}; ℤ/2) ⊗ H^{∗}(BGL(ℤ)^{+}; ℤ/2)

induced by the H-space structure μ of BGL(ℤ)^{+} .

Lemma 3. (a) For any j ≥ 1 ,

μ^{∗}(w_{j}) = ∑_{s=0}^{j} w_{s} ⊗ w_{j−s} .

(b) For any prime p ≡ 5 mod 8 and any integer k ≥ 2 ,

μ^{∗}(f_{p}^{∗}(e_{k})) = f_{p}^{∗}(e_{k}) ⊗ 1 + 1 ⊗ f_{p}^{∗}(e_{k}) + ∑_{ℓ=1}^{k−2}(w_{ℓ}^{2} ⊗ f_{p}^{∗}(e_{k−ℓ}) + f_{p}^{∗}(e_{k−ℓ}) ⊗ w_{ℓ}^{2}) .

(c) For any prime p ≡ 3 mod 8 and any integer k ≥ 2 ,

μ^{∗}(γ_{k}) = γ_{k} ⊗ 1 + 1 ⊗ γ_{k} + ∑_{ℓ=1}^{k−2}(w_{ℓ}^{2} ⊗ γ_{k−ℓ} + γ_{k−ℓ} ⊗ w_{ℓ}^{2}) .

Proof. Assertion (a) is known (see for instance [MT], Part I, p. 140). If ν denotes the H-space structure of F Ψ^{p} , Proposition 2 of [Q2] implies that

μ^{∗}(f_{p}^{∗}(e_{k})) = f_{p}^{∗}(ν^{∗}(e_{k})) = f_{p}^{∗}(∑_{ℓ=0}^{k}(c_{ℓ} ⊗ e_{k−ℓ} + e_{k−ℓ} ⊗ c_{ℓ})) = ∑_{ℓ=0}^{k}(w_{ℓ}^{2} ⊗ f_{p}^{∗}(e_{k−ℓ}) + f_{p}^{∗}(e_{k−ℓ}) ⊗ w_{ℓ}^{2})

for any odd prime p. If p ≡ 5 mod 8 , f_{p}^{∗}(e_{1}) vanishes since e_{1} is exterior and one gets immediately (b). If p ≡ 3 mod 8 , the definition of γ_{k} ,

γ_{k} = f_{p}^{∗}(e_{k}) + w_{2k−1} + ∑_{j=1}^{k−1} w_{j}w_{2k−j−1} ,

shows that

μ^{∗}(γ_{k}) = ∑_{ℓ=0}^{k}(w_{ℓ}^{2} ⊗ f_{p}^{∗}(e_{k−ℓ}) + f_{p}^{∗}(e_{k−ℓ}) ⊗ w_{ℓ}^{2}) + (element of ℤ/2[w_{1}, w_{2}, …]) .

Since p ≡ 3 mod 8 , it turns out that f_{p}^{∗}(e_{1}) = w_{1} and consequently that

μ^{∗}(γ_{k}) = γ_{k} ⊗ 1 + 1 ⊗ γ_{k} + ∑_{ℓ=1}^{k−2}(w_{ℓ}^{2} ⊗ γ_{k−ℓ} + γ_{k−ℓ} ⊗ w_{ℓ}^{2}) + (element of ℤ/2[w_{1}, w_{2}, …]) .

However, the element of ℤ/2[w_{1}, w_{2}, …] in that formula must be trivial since μ^{∗}(γ_{k}) is exterior. This implies the last assertion.

Now, let p be a prime ≡ 5 mod 8 and k an integer ≥ 2 . Consider an integer m ≥ k, C the cyclic group of order p − 1 and H^{∗}(BC^{m}; ℤ/2) ≅ ℤ/2[x_{1}, x_{2}, …, x_{m}] ⊗ Λ(y_{1}, y_{2}, …, y_{m}) with deg(x_{i}) = 2 and deg(y_{i}) = 1 for 1 ≤ i ≤ m, endowed with the differential d defined by d(x_{i}) = y_{i} and d(y_{i}) = 0 . Then, look at the homomorphism ρ : H^{∗}(F Ψ^{p}; ℤ/2) → H^{∗}(BC^{m}; ℤ/2) , introduced in [Q2], p. 563–565, which is injective in dimensions ≤ 2m (and in particular in dimensions ≤ 2k) since its kernel is the ideal generated by the elements c_{j} and e_{j} for j > m, and which fulfills ρ(c_{j}) = s_{j} and ρ(e_{j}) = d(s_{j}) for 1 ≤ j ≤ m, where s_{j} denotes the j-th elementary symmetric function in x_{1}, x_{2}, …, x_{m} . For k ≥ 1 , define the exterior class

ξ_{k} = ∑_{j=1}^{m} x_{j}^{k−1}y_{j} ∈ H^{2k−1}(BC^{m}; ℤ/2) .

Lemma 4. For p ≡ 5 mod 8 and for any k ≥ 2 , the class e_{k} ∈ H^{2k−1}(F Ψ^{p}; ℤ/2) satisfies

e_{k} = ρ^{−1}(ξ_{k}) + ∑_{j=1}^{k−1} c_{j}ρ^{−1}(ξ_{k−j}) .

Proof. Since s_{k} = ∑_{i_{1}<i_{2}<···<i_{k}} x_{i_{1}}x_{i_{2}} ⋯ x_{i_{k}} , one has d(s_{k}) = ∑_{i_{1}<i_{2}<···<i_{k}} ∑_{ℓ} x_{i_{1}} ⋯ x̂_{i_{ℓ}} ⋯ x_{i_{k}}y_{i_{ℓ}} . Then,

consider the difference

d(s_{k}) − s_{k−1}ξ_{1} = ∑_{i_{1}<i_{2}<···<i_{k}} ∑_{ℓ} x_{i_{1}} ⋯ x̂_{i_{ℓ}} ⋯ x_{i_{k}}y_{i_{ℓ}} − ∑_{i_{1}<i_{2}<···<i_{k−1}} x_{i_{1}}x_{i_{2}} ⋯ x_{i_{k−1}} ∑_{i} y_{i}

= ∑_{i_{1}<i_{2}<···<i_{k−1}} ∑_{ℓ} x_{i_{1}}x_{i_{2}} ⋯ x_{i_{ℓ}} ⋯ x_{i_{k−1}}y_{i_{ℓ}}

= ∑_{i_{1}<i_{2}<···<i_{k−1}} ∑_{ℓ} x_{i_{1}}x_{i_{2}} ⋯ x̂_{i_{ℓ}} ⋯ x_{i_{k−1}}x_{i_{ℓ}}y_{i_{ℓ}} .

From this formula, one may compute the difference

d(s_{k}) − s_{k−1}ξ_{1} − s_{k−2}ξ_{2} = ∑_{i_{1}<i_{2}<···<i_{k−2}} ∑_{ℓ} x_{i_{1}}x_{i_{2}} ⋯ x̂_{i_{ℓ}} ⋯ x_{i_{k−2}}x_{i_{ℓ}}^{2}y_{i_{ℓ}}

and obtain by induction

d(s_{k}) = ξ_{k} + ∑_{j=1}^{k−1} s_{j}ξ_{k−j}

for k ≥ 2 . Since ρ(e_{k}) = d(s_{k}) and ρ(c_{j}) = s_{j} , we get

ρ(e_{k}) = ξ_{k} + ∑_{j=1}^{k−1} ρ(c_{j})ξ_{k−j}

and the lemma follows from the injectivity of ρ in dimensions ≤ 2k.

Definition 5. Let p be a prime ≡ 5 mod 8 . For all integers k ≥ 2 , let us define the exterior class u_{2k−1}(p) = f_{p}^{∗}(ρ^{−1}(ξ_{k})) ∈ H^{2k−1}(BGL(ℤ)^{+}; ℤ/2) , where f_{p}^{∗} denotes the homomorphism H^{∗}(F Ψ^{p}; ℤ/2) → H^{∗}(BGL(ℤ)^{+}; ℤ/2) induced by f_{p} . Observe that this definition does not depend on the choice of m ≥ k. Notice also that f_{p}^{∗}(ρ^{−1}(ξ_{1})) = f_{p}^{∗}(e_{1}) = 0 .

Proposition 6. For any prime p ≡ 5 mod 8 and for k ≥ 2 , one has:

(a) u_{2k−1}(p) = f_{p}^{∗}(e_{k}) + ∑_{j=1}^{k−2} w_{j}^{2}u_{2k−2j−1}(p) ,

(b) the homomorphism η^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) → H^{∗}(SU; ℤ/2) fulfills η^{∗}(u_{2k−1}(p)) = v_{2k−1} .

Proof. Lemma 4 implies that

f_{p}^{∗}(e_{k}) = u_{2k−1}(p) + ∑_{j=1}^{k−2} w_{j}^{2}u_{2k−2j−1}(p)

since f_{p}^{∗}(ρ^{−1}(ξ_{1})) = 0 . Consequently, (b) follows directly from the commutativity of the diagram induced by the last diagram of the introduction

H^{∗}(SU; ℤ/2) ⟵^{η^{∗}} H^{∗}(BGL(ℤ)^{+}; ℤ/2) ⟵^{h^{∗}} H^{∗}(BO; ℤ/2)
↑ ζ^{∗}  ↑ f_{p}^{∗}
H^{∗}(U; ℤ/2) ⟵^{θ^{∗}}  H^{∗}(F Ψ^{p}; ℤ/2)

because η^{∗} f_{p}^{∗}(e_{k}) = ζ^{∗} θ^{∗}(e_{k}) = v_{2k−1} and η^{∗}(w_{j}) = 0 for all k ≥ 2 , j ≥ 1 .

Corollary 7. For any prime p ≡ 5 mod 8 and for any integer k ≥ 2 , the element u_{2k−1}(p) is a primitive cohomology class in H^{2k−1}(BGL(ℤ)^{+}; ℤ/2) .

Proof. We must show that the homomorphism μ^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) → H^{∗}(BGL(ℤ)^{+} × BGL(ℤ)^{+}; ℤ/2) satisfies μ^{∗}(u_{2k−1}(p)) = u_{2k−1}(p) ⊗ 1 + 1 ⊗ u_{2k−1}(p) . We proceed by induction on k. We just established in Proposition 6 that

u_{2k−1}(p) = f_{p}^{∗}(e_{k}) + ∑_{j=1}^{k−2} w_{j}^{2}u_{2k−2j−1}(p) .

For instance, u_{3}(p) = f_{p}^{∗}(e_{2}) and it follows from Lemma 3 (b) that μ^{∗}(u_{3}) = u_{3} ⊗ 1 + 1 ⊗ u_{3} . We then may deduce from Lemma 3 and the induction hypothesis that

\begin{aligned}μ^{∗}(u_{2k−1}(p)) & = & f_{p}^{∗}(e_{k}) ⊗ 1 + 1 ⊗ f_{p}^{∗}(e_{k}) + ∑_{ℓ=1}^{k−2}(w_{ℓ}^{2} ⊗ f_{p}^{∗}(e_{k−ℓ}) + f_{p}^{∗}(e_{k−ℓ}) ⊗ w_{ℓ}^{2}) \\ & & +∑_{j=1}^{k−2}(∑_{s=0}^{j} w_{s}^{2} ⊗ w_{j−s}^{2})(u_{2k−2j−1}(p) ⊗ 1 + 1 ⊗ u_{2k−2j−1}(p)) \end{aligned}

and therefore that

μ^{∗}(u_{2k−1}(p)) = u_{2k−1}(p) ⊗ 1 + 1 ⊗ u_{2k−1}(p) + ∑_{ℓ=1}^{k−2}(w_{ℓ}^{2} ⊗ u_{2k−2ℓ−1}(p) + u_{2k−2ℓ−1}(p) ⊗ w_{ℓ}^{2})

+∑_{ℓ=1}^{k−3} ∑_{t=1}^{k−ℓ−2}(w_{ℓ}^{2} ⊗ w_{t}^{2}u_{2k−2ℓ−2t−1}(p) + w_{t}^{2}u_{2k−2ℓ−2t−1}(p) ⊗ w_{ℓ}^{2})

+∑_{j=1}^{k−2}(w_{j}^{2} ⊗ u_{2k−2j−1}(p) + u_{2k−2j−1}(p) ⊗ w_{j}^{2})

+∑_{j=1}^{k−2} ∑_{s=1}^{j−1}(w_{s}^{2}u_{2k−2j−1}(p) ⊗ w_{j−s}^{2} + w_{s}^{2} ⊗ w_{j−s}^{2}u_{2k−2j−1}(p)) .

The last sum can be written as follows:

∑_{j=1}^{k−2} ∑_{s=1}^{j−1}(w_{s}^{2}u_{2k−2j−1}(p) ⊗ w_{j−s}^{2} + w_{s}^{2} ⊗ w_{j−s}^{2}u_{2k−2j−1}(p)) =

\begin{matrix}= & ∑_{j=1}^{k−2} ∑_{s=1}^{j−1}(w_{s}^{2} ⊗ w_{j−s}^{2}u_{2k−2j−1}(p) + w_{j−s}^{2}u_{2k−2j−1}(p) ⊗ w_{s}^{2}) \\ = & ∑_{s=1}^{k−3} ∑_{t=1}^{k−s−1}(w_{s}^{2} ⊗ w_{t}^{2}u_{2k−2s−2t−1}(p) + w_{t}^{2}u_{2k−2s−2t−1}(p) ⊗ w_{s}^{2}) . \end{matrix}

Consequently, μ^{∗}(u_{2k−1}(p)) = u_{2k−1}(p) ⊗ 1 + 1 ⊗ u_{2k−1}(p) and u_{2k−1}(p) is primitive.

Lemma 8. Consider the homomorphism η^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) → H^{∗}(SU; ℤ/2) . For k ≥ 2 , let u′_{2k−1} and u′′_{2k−1} be primitive exterior classes of degree 2k − 1 in H^{∗}(BGL(ℤ)^{+}; ℤ/2) such that η^{∗}(u′_{2k−1}) = η^{∗}(u′′_{2k−1}) = v_{2k−1} . Then u′_{2k−1} = u′′_{2k−1} .

Proof. Observe first that u′_{3} = u′′_{3} since there is only one exterior class of degree 3 in H^{∗}(BGL(ℤ)^{+}; ℤ/2) . Then, let us define ũ_{2k−1} = u′_{2k−1} − u′′_{2k−1} for all k ≥ 2 and prove by induction on k that ũ_{2k−1} = 0 . Since ũ_{2k−1} is exterior and belongs to the kernel of η^{∗} , Theorem 1 and the induction hypothesis show that one can write

ũ_{2k−1} = ∑_{s=3}^{2k−2} u′(s)w(s) ,

where u′(s) is an element of degree s in Λ(u′_{3}, u′_{5}, …, u′_{2k−3}) and w(s) an element of degree 2k − s − 1 in ℤ/2[w_{1}, w_{2}, …] . However, the primitivity of the classes u′_{2j−1} and Lemma 3 (a) provide an explicit computation of μ^{∗}(ũ_{2k−1}) which contradicts the primitivity of ũ_{2k−1} unless one has ũ_{2k−1} = 0 .

Definition 9. Proposition 6 (b), Corollary 7 and Lemma 8 enable us to conclude that the classes u_{2k−1}(p) do not depend on p. Therefore, for k ≥ 2 , we can define u_{2k−1} = u_{2k−1}(p) ∈ H^{2k−1}(BGL(ℤ)^{+}; ℤ/2) for any prime p ≡ 5 mod 8 . Since the image of u_{2k−1} under η^{∗} is v_{2k−1} , the classes u_{2k−1} are non-trivial algebraically independent exterior classes in H^{∗}(BGL(ℤ)^{+}; ℤ/2) . According to Theorem 1, we can write

H^{∗}(BGL(ℤ)^{+}; ℤ/2) ≅ ℤ/2[w_{2}, w_{3}, …] ⊗ Λ(u_{3}, u_{5}, …) .

Remark 10. The homomorphism ψ^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) → H^{∗}(BQ; ℤ/2) ⊗ H^{∗}(SU; ℤ/2) defined in the proof of Theorem 1 is a homomorphism of Hopf algebras since ψ is an H-map. It then follows from the primitivity of the classes v_{2k−1} in H^{∗}(SU; ℤ/2) , Corollary 7 and Definition 9 that the isomorphism provided by Theorem 1 is an isomorphism of Hopf algebras.

The action of the Steenrod algebra on the classes u_{2k−1} in H^{∗}(BGL(ℤ)^{+}; ℤ/2) is given by

Lemma 11. For all k ≥ 2 , Sq^{2i}u_{2k−1} = \binom{k−1}{i}u_{2k+2i−1} for 1 ≤ i < k and Sq^{2i−1}u_{2k−1} = 0 for 1 ≤ i ≤ k.

Proof. It is sufficient to prove the assertion for the classes u_{2k−1}(p) where p is any prime ≡ 5 mod 8 . This follows from the injectivity of the map ρ which was explained just before Lemma 4 (if m is large enough) and from the computations Sq^{2i−1}ξ_{k} = 0 and

Sq^{2i}ξ_{k} = ∑_{j=1}^{m} Sq^{2i}x_{j}^{k−1}y_{j} = ∑_{j=1}^{m} \binom{k−1}{i}x_{j}^{k+i−1}y_{j} = \binom{k−1}{i}ξ_{k+i} .

Theorem 1, Definition 9, Proposition 6 (b) and Lemmas 2 (a) and 11 imply directly the main result of the paper.

Corollary 12. The isomorphism of Hopf algebras

H^{∗}(BGL(ℤ)^{+}; ℤ/2) ≅ H^{∗}(BO; ℤ/2) ⊗ H^{∗}(SU; ℤ/2)

given by Theorem 1 and Remark 10 is an isomorphism of modules over the Steenrod algebra.

It is even possible to describe the classes u_{2k−1} in terms of the image of f_{p}^{∗} for all primes p ≡ 3 or 5 mod 8 .

Theorem 13. For k ≥ 2 , the classes u_{2k−1} ∈ H^{2k−1}(BGL(ℤ)^{+}; ℤ/2) satisfy

u_{2k−1} = \{ \begin{matrix}f_{p}^{∗}(e_{k}) + ∑_{j=1}^{k−2} w_{j}^{2}u_{2k−2j−1} , & if p ≡ 5 mod 8, \\ f_{p}^{∗}(e_{k}) + w_{2k−1} + ∑_{j=1}^{k−1} w_{j}w_{2k−j−1} + ∑_{j=1}^{k−2} w_{j}^{2}u_{2k−2j−1} , & if p ≡ 3 mod 8, \end{matrix}

where f_{p}^{∗} denotes the homomorphism H^{∗}(F Ψ^{p}; ℤ/2) ≅ H^{∗}(BGL(𝔽_{p})^{+}; ℤ/2) → H^{∗}(BGL(ℤ)^{+}; ℤ/2) induced by the reduction mod p : GL(ℤ) → GL(𝔽_{p}) .

Proof. If p ≡ 5 mod 8 , the statement is given by Proposition 6 (a). Observe in particular that u_{2k−1} can be written as follows: u_{2k−1} = F_{k}(f_{p}^{∗}(e_{2}), f_{p}^{∗}(e_{3}), …, f_{p}^{∗}(e_{k})) , where F_{k} is a polynomial with coefficients in ℤ/2[w_{1}, w_{2}, …] . If p ≡ 3 mod 8 , consider again

γ_{k} = f_{p}^{∗}(e_{k}) + w_{2k−1} + ∑_{j=1}^{k−1} w_{j}w_{2k−j−1}

and define û_{2k−1} = F_{k}(γ_{2}, γ_{3}, …, γ_{k}) . It is obvious that û_{2k−1} is an exterior class and easy to check as in the proof of Proposition 6 that η^{∗}(û_{2k−1}) = v_{2k−1} . Moreover, observe that the homomorphism μ^{∗} : H^{∗}(BGL(ℤ)^{+}; ℤ/2) → H^{∗}(BGL(ℤ)^{+} × BGL(ℤ)^{+}; ℤ/2) acts on γ_{k} (for p ≡ 3 mod 8 ) and on f_{p}^{∗}(e_{k}) (for p ≡ 5 mod 8 ) exactly in the same way according to Lemma 3 (b) and (c). Thus, the argument of the proof of Corollary 7 implies that u_{2k−1}(p) is also primitive if p ≡ 3 mod 8 . It finally follows from Lemma 8 that

u_{2k−1} = û_{2k−1} = F_{k}(γ_{2}, γ_{3}, …, γ_{k}) = γ_{k} + ∑_{j=1}^{k−2} w_{j}^{2}u_{2k−2j−1} .

It is known that BGL(ℤ)^{+} ≃ BSL(ℤ)^{+} × Bℤ/2 and one deduces immediately the calculation of the mod 2 cohomology of the space BSL(ℤ)^{+} (recall that H^{∗}(BSL(𝔽_{p})^{+}; ℤ/2) is obtained from H^{∗}(BGL(𝔽_{p})^{+}; ℤ/2) by dividing out e_{1} and c_{1} ):

Corollary 14. There is an isomorphism of modules over the Steenrod algebra

H^{∗}(BSL(ℤ)^{+}; ℤ/2) ≅ ℤ/2[w_{2}, w_{3}, …] ⊗ Λ(u_{3}, u_{5}, …) ,

where w_{k} and u_{2k−1} are also written for the image of w_{k} and u_{2k−1} under the homomorphism induced by the inclusion SL(ℤ) ↪ GL(ℤ) . The formulas for u_{2k−1} given by Theorem 13 do still hold but observe that the first Stiefel-Whitney class of SL(ℤ) is trivial.

Remark 15. The results of this section determine also the mod 2 cohomology of the groups GL(ℤ) and SL(ℤ) because H^{∗}(BG^{+}; ℤ/2) ≅ H^{∗}(G; ℤ/2) for G = GL(ℤ) or SL(ℤ) .

3. The mod 2 cohomology of the Steinberg group St(ℤ)

The goal of this last section is to compute H^{∗}(St(ℤ); ℤ/2) by looking at the universal central extension

ℤ/2 ≅ K_{2}(ℤ) ≻⟶ St(ℤ) ⟶^{π} SL(ℤ)

and at the associated Serre spectral sequence

E_{2}^{∗,∗} ≅ H^{∗}(SL(ℤ); ℤ/2) ⊗ H^{∗}(ℤ/2; ℤ/2) ⟹ H^{∗}(St(ℤ); ℤ/2) .

Let us use the notation Q_{0} = Sq^{1} and Q_{r} = Sq^{2^{r}} Q_{r−1} + Q_{r−1} Sq^{2^{r}} and observe that Q_{r}(w_{2}) = Sq^{2^{r}} Sq^{2^{r−1}} ⋯ Sq^{1}w_{2} because Sq^{2^{r}}w_{2} = 0 for r ≥ 2 and Sq^{1} Sq^{2}w_{2} = 0 .

Theorem 16. (a) There is an isomorphism of modules over the Steenrod algebra

H^{∗}(St(ℤ); ℤ/2) ≅ ℤ/2[w̄_{2}, w̄_{3}, …]/(w̄_{2}, Q_{r}(w̄_{2}), r ≥ 0) ⊗ Λ(ū_{3}, ū_{5}, …) ,

where w̄_{k} and ū_{2k−1} denote the image of w_{k} and u_{2k−1} under the homomorphism π^{∗} : H^{∗}(SL(ℤ); ℤ/2) → H^{∗}(St(ℤ); ℤ/2) .

(b) For k ≥ 2 ,

ū_{2k−1} = \{ \begin{matrix}f_{p}^{∗}(e_{k}) + ∑_{j=4}^{k−2} w̄_{j}^{2}ū_{2k−2j−1} , & if p ≡ 5 mod 8, \\ f_{p}^{∗}(e_{k}) + w̄_{2k−1} + ∑_{j=4}^{k−1} w̄_{j}w̄_{2k−j−1} + ∑_{j=4}^{k−2} w̄_{j}^{2}ū_{2k−2j−1} , & if p ≡ 3 mod 8, \end{matrix}

where f_{p}^{∗} is written here for the homomorphism H^{∗}(SL(𝔽_{p}); ℤ/2) → H^{∗}(St(ℤ); ℤ/2) induced by the reduction mod p : St(ℤ) → St(𝔽_{p}) ≅ SL(𝔽_{p}) .

Proof. Because H^{∗}(ℤ/2; ℤ/2) ≅ ℤ/2[z] with deg z = 1 , one can compute the differentials in the above spectral sequence: d_{2}(z) = w_{2} , d_{3}(z^{2}) = Sq^{1}d_{2}(z) = Sq^{1}w_{2} = w_{3} , d_{5}(z^{4}) = Sq^{2}d_{3}(z^{2}) = Sq^{2}w_{3} and inductively, d_{2^{r}+1}(z^{2^{r}}) = d_{2^{r}+1}(Q_{r−1}(z)) = Q_{r−1}(d_{2}(z)) = Q_{r−1}(w_{2}) = w_{2^{r}+1} + (decomposable element of ℤ/2[w_{2}, w_{3}, …]) by Wu’s formula ([MT], Part I, p. 141). Therefore, the sequence (w_{2}, Q_{0}(w_{2}), Q_{1}(w_{2}), …) is regular and we obtain E_{∞}^{s,t} = 0 if t > 0 and E_{∞}^{∗,0} ≅ H^{∗}(SL(ℤ); ℤ/2)/(w_{2}, Q_{r}(w_{2}), r ≥ 0) . This gives the mod 2 cohomology of St(ℤ) as described by statement (a) and assertion (b) follows directly from Corollary 14 (b) since w̄_{2} = w̄_{3} = 0 .

Remark 17. The above argument exhibits a surjective homomorphism from H^{∗}(BSL(ℤ)^{+}; ℤ/2) to H^{∗}(BSt(ℤ)^{+}; ℤ/2) . However, it is actually possible to find a nice map from BSt(ℤ)^{+} to the space BSpin inducing an injective homomorphism on mod 2 cohomology. More precisely, consider the map ε : BSL(ℤ)^{+} → BSL(ℝ)^{+} induced by the inclusion ℤ ↪ ℝ and the map κ : BSL(ℝ)^{+} → BSL(ℝ)^{top} ≃ BSO induced by the obvious map SL(ℝ) → SL(ℝ)^{top} , where the first group SL(ℝ) is endowed with the discrete topology and SL(ℝ)^{top} with the usual topology. Then, look at the following commutative diagram, where the rows are fibrations in which the maps α, α′ , α′′ are the second Postnikov sections of the corresponding spaces ( BSpin is the fiber of α′′ ), the maps ε̄ and κ̄ are the second Postnikov sections of ε and κ, and the vertical maps on the left are the restrictions of ε and κ to the fibers:

BSt(ℤ)^{+} ⟶^{π} BSL(ℤ)^{+} ⟶^{α} K(K_{2}ℤ, 2)
↓  ↓ ε  ↓ ε̄
BSt(ℝ)^{+} ⟶ BSL(ℝ)^{+} ⟶^{α′} K(K_{2}ℝ, 2)
↓  ↓ κ  ↓ κ̄
BSpin ⟶^{τ}  BSO  ⟶^{α′′} K(π_{2}BSO, 2)

The composition κ̄ ε̄ is a homotopy equivalence because κ̄_{∗} ε̄_{∗} : K_{2}ℤ → π_{2}BSO is an isomorphism (see Corollary 4.6 of [Br] or p. 25-26 of [Be]). Let us denote the composition κ ε by χ and its restriction to BSt(ℤ)^{+} by χ̃ : BSt(ℤ)^{+} → BSpin (note that χ is the universal cover of the map h defined in the introduction and that the fiber of the 2-completion of χ̃ is SU_{ˆ2} because of the last diagram of the introduction). We get the commutative diagram

Bℤ/2 ⟶ BSt(ℤ)^{+} ⟶^{π} BSL(ℤ)^{+}
↓ ≃  ↓ χ̃  ↓ χ
Bℤ/2 ⟶ BSpin ⟶^{τ} BSO .

The ring structure of the mod 2 cohomology of BSpin is known by Proposition 6.5 of [Q1]:

H^{∗}(BSpin; ℤ/2) ≅ ℤ/2[w̃_{2}, w̃_{3}, …]/(w̃_{2}, Q_{r}(w̃_{2}), r ≥ 0) ,

where the w̃_{k}’s are written here for the image of the universal Stiefel-Whitney classes under the homomorphism τ^{∗} : H^{∗}(BSO; ℤ/2) → H^{∗}(BSpin; ℤ/2) . Since χ^{∗} : H^{∗}(BSO; ℤ/2) → H^{∗}(BSL(ℤ)^{+}; ℤ/2) is injective, the map χ̃ induces an injective homomorphism of modules over the Steenrod algebra

χ̃^{∗} : H^{∗}(BSpin; ℤ/2) → H^{∗}(BSt(ℤ)^{+}; ℤ/2) .

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Dominique Arlettaz Institut de mathématiques Université de Lausanne 1015 Lausanne, Switzerland e-mail: dominique.arlettaz@ima.unil.ch Mamoru Mimura Department of Mathematics Faculty of Science Okayama University Okayama, Japan 700 e-mail: mimura@math.okayama-u.ac.jp

Koji Nakahata Institut de mathématiques Université de Lausanne 1015 Lausanne, Switzerland e-mail: koji.nakahata@ima.unil.ch Nobuaki Yagita Faculty of education Ibaraki University Mito Ibaraki, Japan e-mail: yagita@mito.ipc.ibaraki.ac.jp