Integral cohomology and Chern classes of the special linear group over the ring of integers

By DOMINIQUE ARLETTAZ Institut de mathématiques, Université de Lausanne, 1015 Lausanne, Switzerland

                           e-mail: dominique.arlettaz@ima.unil.ch

CHRISTIAN AUSONI Departement Mathematik, HG, ETH-Zentrum, 8092 Zürich, Switzerland

                                e-mail: ausoni@math.ethz.ch

MAMORU MIMURA Department of Mathematics, Faculty of Science, Okayama University, Okayama, Japan 700

                            e-mail: mimura@math.okayama-u.ac.jp

and NOBUAKI YAGITA Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

                           e-mail: yagita@mito.ipc.ibaraki.ac.jp

Abstract

This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(ℤ) over the ring of integers ℤ. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups.

1. Introduction

The Hopf algebra structure of the mod 2 cohomology of the infinite special and general linear groups SL(ℤ) and GL(ℤ) over the ring of integers ℤ has been completely determined in [AMNY] as a module over the Steenrod algebra. For instance, H^{∗}(SL(ℤ); ℤ/2) ≅ H^{∗}(BSL(ℤ)^{+}; ℤ/2) is the tensor product of a polynomial algebra with an exterior algebra:

\begin{aligned}H^{∗}(BSL(ℤ)^{+}; ℤ/2) & ≅ & H^{∗}(BSO; ℤ/2) ⊗ H^{∗}(SU; ℤ/2) \\ & ≅ & ℤ/2[w_{2}, w_{3}, …, w_{j}, …] ⊗ Λ_{ℤ/2}(u_{3}, u_{5}, …, u_{2k−1}, …) , \end{aligned}

where deg(w_{j}) = j and deg(u_{2k−1}) = 2k − 1 .

The first goal of this paper is to investigate the mod 2 cohomological Bockstein spectral sequence

E_{1}^{∗}(BSL(ℤ)^{+}) ≅ H^{∗}(BSL(ℤ)^{+}; ℤ/2) ⟹ (H^{∗}(BSL(ℤ)^{+}; ℤ)/torsion) ⊗ ℤ/2

2000 Mathematics Subject Classification: Primary 20 G 10; Secondary 19 D 55, 55 R 40, 55 T 99, 57 R 20.

of the space BSL(ℤ)^{+} (see Sections 1 – 5 of [Brd]). By using the mod 2 Bockstein spectral sequences of the spaces BSO and BSL(𝔽_{p})^{+} (for a prime p ≡ 5 mod 8 ) and the maps h : BSL(ℤ)^{+} → BSO and f_{p} : BSL(ℤ)^{+} → BSL(𝔽_{p})^{+} induced by the inclusion ℤ ↪ ℝ and by the reduction mod p respectively, we compute the terms E_{r}^{∗}(BSL(ℤ)^{+}) and the differentials d_{r} for all r ≥ 1 (see Theorem 4.3 and Corollary 4.4). Of course, this detects the 2-torsion of the integral cohomology H^{∗}(BSL(ℤ)^{+}; ℤ) ≅ H^{∗}(SL(ℤ); ℤ) of the special linear group SL(ℤ) . Theorem 4.7 actually provides an explicit additive presentation of the 2-torsion of H^{∗}(BSL(ℤ)^{+}; ℤ) by generators and relations (see also Remark 4.8 for some partial information on the multiplicative structure). It turns out that H^{∗}(BSL(ℤ)^{+}; ℤ) contains no cyclic direct summand of order 4 and that the set of all non-trivial elements of Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) is in one-to-one correspondence with an additive basis of H^{∗}(BSL(ℤ)^{+}; ℤ)/torsion (see Corollary 4.6). Moreover, we are able to understand the effect of the induced homomorphisms h^{∗} : H^{∗}(BSO; ℤ) → H^{∗}(BSL(ℤ)^{+}; ℤ) and f_{p}^{∗} : H^{∗}(BSL(𝔽_{p})^{+}; ℤ) → H^{∗}(BSL(ℤ)^{+}, ℤ) on the 2-torsion elements: Theorem 4.9 asserts in particular that h^{∗} is injective on the elements of order 2 and that f_{p}^{∗} is injective on all cyclic direct summands of order 2^{r} with r ≥ 3 .

Notice that it is easy to extend these results to the integral cohomology H^{∗}(GL(ℤ); ℤ) ≅ H^{∗}(BGL(ℤ)^{+}; ℤ) of the general linear group GL(ℤ) because of the homotopy equivalence BGL(ℤ)^{+} ≃ BSL(ℤ)^{+} × Bℤ/2 (see for example [Ar1], Lemma 1.2).

As a consequence, we obtain the exact order of all Chern classes c_{n}(SL(ℤ)) ∈ H^{2n}(SL(ℤ); ℤ) of the inclusion SL(ℤ) ↪ GL(ℂ) (see Proposition 5.2 and Theorem 5.3) and deduce the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups (see Corollary 5.6).

The paper is organized as follows. Sections 2 and 3 present the mod 2 Bockstein spectral sequence for the spaces BSO and BSL(𝔽_{p})^{+} respectively. The mod 2 Bockstein spectral sequence and the 2-torsion of the integral cohomology of BSL(ℤ)^{+} are computed in Section 4. Finally, Section 5 is devoted to the investigation of the order of the Chern classes of integral and rational representations of discrete groups.

2. The mod 2 Bockstein spectral sequence for BSO

The determination of H^{∗}(BSL(ℤ)^{+}; ℤ/2) is based on cohomological calculations involving the pull-back diagram

(BSL(ℤ)^{+})_{ˆ2} ⟶^{h} BSO_{ˆ2}
↓ f_{p}  ↓ c
(BSL(𝔽_{p})^{+})_{ˆ2} ⟶^{b} BSU_{ˆ2} ,

where (−)_{ˆ2} denotes the completion at the prime 2 , p any prime ≡ 3 or 5 mod 8 , h the map induced by the inclusion ℤ ↪ ℝ, f_{p} the map induced by the reduction mod p : ℤ → → 𝔽_{p} , c the complexification and b the Brauer lift, and where the homotopy fibers of both horizontal maps are SU_{ˆ2} (for the details of that construction, see [AMNY], where the argument is presented for GL(ℤ) instead of SL(ℤ) , or Chapter 3 of [Au]; notice also that an unstable version of this computation is given in [He]). In order to go through the mod 2 Bockstein spectral sequence for BSL(ℤ)^{+} , we shall first consider the mod 2 Bockstein spectral sequence for the spaces BSO and BSL(𝔽_{p})^{+} , and the homomorphisms induced in cohomology by the maps h and f_{p} .

Let us start by looking at the mod 2 Bockstein spectral sequence

E_{1}^{∗}(BSO) ≅ H^{∗}(BSO; ℤ/2) ≅ ℤ/2[w_{2}, w_{3}, …, w_{j}, …] ⟹ (H^{∗}(BSO; ℤ)/torsion) ⊗ ℤ/2 .

Its first differential is d_{1} = Sq^{1} and we know by Wu’s formula (see for instance [MT], Part I, p.138, Theorem 5.12) that for i ≥ 1 , Sq^{1}(w_{2i}) = w_{2i+1} , Sq^{1}(w_{2i+1}) = 0 , and that Sq^{1}(w_{j}^{2}) = 0 for j ≥ 1 . Thus, we may deduce that

E_{2}^{∗}(BSO) ≅ ℤ/2[w_{2}^{2}, w_{4}^{2}, …, w_{2i}^{2}, …]

is concentrated in degrees ≡ 0 mod 4 . Since d_{r} is of degree 1 , it is then obvious that d_{r} = 0 for all r ≥ 2 . Consequently, we have proved the following result.

Proposition 2.1.

(a) The mod 2 Bockstein spectral sequence for BSO has the property that E_{1}^{∗}(BSO) ≅ ℤ/2[w_{j}, j ≥ 2] and E_{r}^{∗}(BSO) ≅ E_{∞}^{∗}(BSO) ≅ ℤ/2[w_{2i}^{2}, i ≥ 1] for all r ≥ 2 .

(b) All non-trivial 2-torsion elements of H^{∗}(BSO; ℤ) have order exactly equal to 2 .

(c) H^{∗}(BSO; ℤ)/torsion ≅ ℤ[p_{4i}, i ≥ 1] , where p_{4i} is of degree 4i and represents an element of H^{4i}(BSO; ℤ) whose reduction mod 2 is w_{2i}^{2} ∈ H^{4i}(BSO; ℤ/2) .

Remark 2.2. The additive and multiplicative structures of the 2-torsion of H^{∗}(BSO; ℤ) has been obtained a long time ago in [Brn], Theorem 1.5, and [F], Theorem 1 (see also [Bo], Theorem 24.7 and Proposition 25.6, [CV], Theorem 1, and [ThE], Theorem A). For completeness, let us recall here its additive structure, which can also be determined by the argument we shall use in the next sections (see Lemma 3.6 and Theorems 3.7 and 4.7): if Ψ denotes the graded ℤ-algebra ℤ[q_{2i+1}, i ≥ 1] ⊗ ℤ[p_{4i}, i ≥ 1] with deg(q_{2i+1}) = 2i + 1 and deg(p_{4i}) = 4i, then the 2-torsion subgroup of H^{∗}(BSO; ℤ) is additively isomorphic to the graded Ψ-module generated by

{G_{A} | A running over all non-empty finite subsets of N_{1} = ℕ − {0}}

with relations generated by

{2G_{A} , ∑_{i∈A} q_{2i+1}G_{A−{i}}} .

Here, the element G_{{i_{1},...,i_{t}}} ∈ H^{∗}(BSO; ℤ) is of degree 2(∑_{j=1}^{t} i_{j}) + 1 and reduces mod 2 to the class ∑_{j=1}^{t} w_{2i_{j}+1}w_{2i_{1}} ⋯ w_{2i_{(j−1)}}w_{2i_{(j+1)}} ⋯ w_{2i_{t}} ∈ H^{∗}(BSO; ℤ/2) .

3. The mod 2 Bockstein spectral sequence for BSL(𝔽_{p})^{+}

In this section, let us consider the space BSL(𝔽_{p})^{+} for any prime number p ≡ 5 mod 8 . Its mod 2 cohomology is H^{∗}(BSL(𝔽_{p})^{+}; ℤ/2) ≅ ℤ/2[c_{2}, c_{3}, …, c_{k}, …] ⊗ Λ_{ℤ/2}(e_{2}, e_{3}, …, e_{k}, …) with deg(c_{k}) = 2k and deg(e_{k}) = 2k − 1 (see Theorem 1 of [Q]). The first differential of its mod 2 Bockstein spectral sequence

E_{1}^{∗}(BSL(𝔽_{p})^{+}) ≅ ℤ/2[c_{k}, k ≥ 2] ⊗ Λ_{ℤ/2}(e_{k}, k ≥ 2) ⟹ (H^{∗}(BSL(𝔽_{p})^{+}; ℤ)/torsion) ⊗ ℤ/2

is trivial since d_{1}(c_{k}) = Sq^{1}(c_{k}) = 0 and d_{1}(e_{k}) = Sq^{1}(e_{k}) = 0 for all k ≥ 2 according to Lemmas 3 and 4 of [Ar4]. Thus,

E_{2}^{∗}(BSL(𝔽_{p})^{+}) ≅ E_{1}^{∗}(BSL(𝔽_{p})^{+}) .

In order to understand the higher differentials d_{r} , let us recall the definition of d_{r} (see Section 1 of [Brd]). If x ∈ E_{r}^{n}(BSL(𝔽_{p})^{+}) , then there is an element x̃ ∈ H^{n}(BSL(𝔽_{p})^{+}; ℤ/2^{r}) such that the homomorphism θ_{r} : H^{n}(BSL(𝔽_{p})^{+}; ℤ/2^{r}) → H^{n}(BSL(𝔽_{p})^{+}; ℤ/2) induced by the natural surjection ℤ/2^{r} → → ℤ/2 sends x̃ onto x. Let β_{r} : H^{n}(BSL(𝔽_{p})^{+}; ℤ/2^{r}) → H^{n+1}(BSL(𝔽_{p})^{+}; ℤ) denote the Bockstein homomorphism associated with the short exact sequence

0 ⟶ ℤ ⟶^{·2^{r}} ℤ ⟶ ℤ/2^{r} ⟶ 0

and red_{2} : H^{n+1}(BSL(𝔽_{p})^{+}; ℤ) → H^{n+1}(BSL(𝔽_{p})^{+}; ℤ/2) the reduction mod 2. Then, the differential d_{r} : E_{r}^{n}(BSL(𝔽_{p})^{+}) → E_{r}^{n+1}(BSL(𝔽_{p})^{+}) is defined by

d_{r}(x) = red_{2}(β_{r}(x̃)) .

Let us apply this to the case of the space BSL(𝔽_{p})^{+} .

Definition 3.1. For any integer r ≥ 2 , let N_{r} = {k ∈ ℕ | v_{2}(k) = r − 2}, where v_{2}(−) is the 2-adic valuation (in other words, N_{r} = {k = 2^{r−1}i + 2^{r−2} | i ≥ 0}).

Remark 3.2. For any prime p ≡ 5 mod 8 and any integer r ≥ 2 , N_{r} = {k ∈ ℕ | v_{2}(p^{k} − 1) = r}. In order to check this, it is sufficient to show that v_{2}(p^{k} − 1) = v_{2}(k) + 2 for any positive integer k. Let us write

p = 4m + 1 with m odd. Then p^{k} − 1 = ∑_{t=1}^{k} \binom{k}{t} 4^{t}m^{t} . For t ≥ 2 , one has

v_{2}(\binom{k}{t} 4^{t}m^{t}) = 2t + v_{2}(\frac{k(k − 1) ⋯ (k − t + 1)}{t!}) ≥ 2t + v_{2}(k) − v_{2}(t!) ≥ v_{2}(k) + t + 1 ≥ v_{2}(k) + 3 ,

since v_{2}(t!) ≤ t − 1 . This implies that v_{2}(p^{k} − 1) = v_{2}(4km) = v_{2}(k) + 2 .

Lemma 3.3. Let p be any prime ≡ 5 mod 8 and r be any integer ≥ 2 . If k ∈ N_{r} , then the class e_{k} belongs to E_{s}^{2k−1}(BSL(𝔽_{p})^{+}) for all s ≤ r and d_{r}(e_{k}) = c_{k} ∈ E_{r}^{2k}(BSL(𝔽_{p})^{+}) .

Proof. If k belongs to N_{r} , then r = v_{2}(p^{k} − 1) by Remark 3.2. Thus, according to Section 3 of [Q], e_{k} is the image of an element ẽ_{k} ∈ H^{2k−1}(BSL(𝔽_{p})^{+}; ℤ/2^{r}) under the homomorphism θ_{r} and consequently, e_{k} ∈ E_{r}^{2k−1}(BSL(𝔽_{p})^{+}) . Moreover, it follows from Lemma 5 of [Q] that d_{r}(e_{k}) = red_{2}(β_{r}(ẽ_{k})) = c_{k} .

We get the complete calculation of the mod 2 Bockstein spectral sequence for the space BSL(𝔽_{p})^{+} .

Theorem 3.4. For any prime p ≡ 5 mod 8 , the mod 2 Bockstein spectral sequence for BSL(𝔽_{p})^{+} satisfies:

(a) E_{2}^{∗}(BSL(𝔽_{p})^{+}) ≅ E_{1}^{∗}(BSL(𝔽_{p})^{+}) ≅ ℤ/2[c_{k}, k ≥ 2] ⊗ Λ_{ℤ/2}(e_{k}, k ≥ 2) and d_{2}(e_{k}) = c_{k} whenever k ∈ N_{2} = {odd positive integers},

(b) for any r ≥ 3 , E_{r}^{∗}(BSL(𝔽_{p})^{+}) ≅ ℤ/2[c_{k} , k ∈ N_{s} for s ≥ r] ⊗ Λ_{ℤ/2}(e_{k} , k ∈ N_{s} for s ≥ r) and d_{r}(e_{k}) = c_{k} whenever k ∈ N_{r} ,

(c) E_{∞}^{∗}(BSL(𝔽_{p})^{+}) = 0 .

By looking at the differential graded ℤ/2-algebras

F_{s} = ℤ/2[c_{k}, 2 ≤ k ∈ N_{s}] ⊗ Λ_{ℤ/2}(e_{k}, 2 ≤ k ∈ N_{s}) , δ_{s}(e_{k}) = c_{k} , for s ≥ 2 ,

one can write the E_{r}-terms of the mod 2 Bockstein spectral sequence of BSL(𝔽_{p})^{+} as follows.

Corollary 3.5. For any prime p ≡ 5 mod 8 and for any integer r ≥ 2 , E_{r}^{∗}(BSL(𝔽_{p})^{+}) ≅ ⨂_{s≥r} F_{s} with the differential d_{r} = δ_{r} on F_{r} and d_{r} = 0 on F_{s} when s > r.

The knowledge of the mod 2 Bockstein spectral sequence determines the additive structure of the 2-torsion of the integral cohomology H^{∗}(BSL(𝔽_{p})^{+}; ℤ) since the elements of the image of d_{r} detect the elements of order 2^{r} in H^{∗}(BSL(𝔽_{p})^{+}; ℤ) . Let us start with the following observation.

Lemma 3.6. Consider a set N and the differential graded ℤ/2-algebra

D_{N} = ℤ/2[x_{n}, n ∈ N] ⊗ Λ_{ℤ/2}(y_{n}, n ∈ N) ,

where the differential is a derivation δ given by δ(y_{n}) = x_{n} . Then the image of δ is the ℤ/2[x_{n}, n ∈ N]- module generated by

{H_{A} = ∑_{a∈A} x_{a} ∏_{b∈A−{a}} y_{b} | A running over all non-empty finite subsets of N}

with the relations generated by {∑_{a∈A} x_{a}H_{A−{a}} | A running over all non-empty finite subsets of N} .

Proof. Let us write P for the polynomial tensor factor ℤ/2[x_{n}, n ∈ N] of D_{N} . Since δ^{2} = 0 , one has δ(P) = 0 and the fact that δ is a derivation shows that D_{N} → → Im(δ) is a morphism of P -modules. The elements of the form ∏_{a∈A} y_{a} , where A runs over all non-empty finite subsets of N , generate D_{N} as a P -module. Therefore, the image of δ is generated, as a P -module, by the elements H_{A} = δ(∏_{a∈A} y_{a}) . We get obviously the relations ∑_{a∈A} x_{a}H_{A−{a}} = δ^{2}(∏_{a∈A} y_{a}) = 0 and there are no other relations because H_{∗}(D_{N}, δ) ≅ ℤ/2 .

Let us deduce the following explicit description of the additive structure of the 2-torsion of H^{∗}(BSL(𝔽_{p})^{+}; ℤ) . Observe in particular that there is no direct summand of order 2 in H^{∗}(BSL(𝔽_{p})^{+}; ℤ) .

Theorem 3.7. Let p be any prime ≡ 5 mod 8 and consider the graded ℤ-algebra

Φ = ℤ[a_{k}, k ≥ 2] ⊗ Λ_{ℤ}(b_{k}, k even ≥ 2) ,

where deg(a_{k}) = 2k and deg(b_{k}) = 2k − 1 . As a graded abelian group, the 2-torsion subgroup of H^{∗}(BSL(𝔽_{p})^{+}; ℤ) is additively isomorphic to the graded Φ-module generated by

{H_{A,r} | r ≥ 2 , A running over all non-empty finite subsets of N_{r}}

with relations generated by

2^{r}H_{A,r},
a_{k}H_{A,r}  for all k ∈ N_{s} with 2 ≤ s < r,
b_{k}H_{A,r}  for all k ∈ N_{s} with 2 ≤ s ≤ r,
∑_{k∈A} a_{k}H_{A−{k},r} for all r ≥ 2, A ⊂ N_{r}.

The element H_{{k_{1},...,k_{t}},r} ∈ H^{∗}(BSL(𝔽_{p})^{+}; ℤ) is of degree 2(∑_{j=1}^{t} k_{j}) − t + 1 and reduces mod 2 to the class ∑_{j=1}^{t} c_{k_{j}}e_{k_{1}} ⋯ e_{k_{(j−1)}}e_{k_{(j+1)}} ⋯ e_{k_{t}} ∈ H^{∗}(BSL(𝔽_{p})^{+}; ℤ/2) .

Proof. Let us denote by P_{r} the polynomial tensor factor ℤ/2[c_{k}, 2 ≤ k ∈ N_{r}] of F_{r} for r ≥ 2 . According to Corollary 3.5, the image of d_{r} is

Im(d_{r}) = Im(δ_{r}) ⊗ (⨂_{s>r} F_{s})

and Lemma 3.6 implies that Im(d_{r}) is the (P_{r} ⊗ (⨂_{s>r} F_{s}))-module generated by the H_{A,r}’s, where A runs over all non-empty finite subsets of N_{r} , with the relations given by Lemma 3.6. By definition of Φ , the generators a_{k} and b_{k} of the ℤ-algebra Φ are in one-to-one correspondence with the classes c_{k} and e_{k} respectively, which generate the ℤ/2-algebra

P_{2} ⊗ (⨂_{s>2} F_{s}) ≅ ℤ/2[c_{k}, k ≥ 2] ⊗ Λ_{ℤ/2}(e_{k}, k even ≥ 2) .

The assertion then follows by gluing together the information on Im(d_{r}) for r ≥ 2 .

Remark 3.8. The additive structure of H^{∗}(BSL(𝔽_{p})^{+}; ℤ) has been already calculated in [Hu], but in a completely different way.

4. The mod 2 Bockstein spectral sequence for BSL(ℤ)^{+}

Finally, let us investigate the mod 2 Bockstein spectral sequence

E_{1}^{∗}(BSL(ℤ)^{+}) ≅ ℤ/2[w_{j}, j ≥ 2] ⊗ Λ_{ℤ/2}(u_{2k−1}, k ≥ 2) ⟹ (H^{∗}(BSL(ℤ)^{+}; ℤ)/torsion) ⊗ ℤ/2

for the mod 2 cohomology of the space BSL(ℤ)^{+} . Since the induced homomorphism h^{∗} : H^{∗}(BSO; ℤ/2) → H^{∗}(BSL(ℤ)^{+}; ℤ/2) sends the Stiefel-Whitney classes w_{j} ∈ H^{j}(BSO; ℤ/2) onto the corresponding classes w_{j} ∈ H^{j}(BSL(ℤ)^{+}; ℤ/2) , we have again Sq^{1}(w_{2i}) = w_{2i+1} , Sq^{1}(w_{2i+1}) = 0 for i ≥ 1 and we know from Lemma 12 of [AMNY] that Sq^{1}(u_{2k−1}) = 0 for k ≥ 2 . Therefore, we obtain the E_{2}-term as follows:

E_{2}^{∗}(BSL(ℤ)^{+}) ≅ E_{2}^{∗}(BSO) ⊗ Λ_{ℤ/2}(u_{2k−1}, k ≥ 2) ≅ ℤ/2[w_{2i}^{2}, i ≥ 1] ⊗ Λ_{ℤ/2}(u_{2k−1}, k ≥ 2) .

Because of the naturality of the mod 2 Bockstein spectral sequence with respect to h^{∗} , we may deduce from Section 1 that all higher differentials d_{r} are trivial on ℤ/2[w_{2i}^{2}, i ≥ 1] , r ≥ 2 .

Lemma 4.1. For all positive integers r and i, one has d_{r}(u_{4i+1}) = 0 in E_{r}^{4i+2}(BSL(ℤ)^{+}) .

Proof. This is true if r = 1 . For r ≥ 2 , d_{r}(u_{4i+1}) is of the form d_{r}(u_{4i+1}) = ∑_{s} w(s) ⊗ u(s) , where w(s) is a product of classes w_{2i}^{2} ( i ≥ 1 ) and u(s) a product of classes u_{2k−1} ( k ≥ 2 ). According to Proposition 7 of [AMNY], the classes u_{2k−1} ( k ≥ 2 ) are primitive cohomology classes, in other words, μ^{∗}(u_{2k−1}) = u_{2k−1} ⊗ 1 + 1 ⊗ u_{2k−1} , where μ^{∗} is the coproduct H^{∗}(BSL(ℤ)^{+}; ℤ/2) → H^{∗}(BSL(ℤ)^{+} × BSL(ℤ)^{+}; ℤ/2) provided by the H-space structure of BSL(ℤ)^{+} . In particular, it follows from the fact that E_{r}^{∗}(BSL(ℤ)^{+}) is a differential Hopf algebra (see Proposition 4.7 of [Brd]) that μ^{∗}(d_{r}(u_{4i+1})) = d_{r}(μ^{∗}(u_{4i+1})) and consequently that d_{r}(u_{4i+1}) is primitive. However, for ∑_{s} w(s) ⊗ u(s) to be primitive, it is necessary to have u(s) primitive, which is only possible if u(s) = 1 or u(s) = u_{2k−1} for some k. In both cases, the element w(s) ⊗ u(s) cannot lie in degree 4i + 2 since deg(w(s)) ≡ 0 mod 4 . Consequently, the sum must be empty and we get d_{r}(u_{4i+1}) = 0 .

Now, let us consider any prime p ≡ 5 mod 8 and the homomorphism

f_{p}^{∗} : H^{∗}(BSL(𝔽_{p})^{+}; ℤ/2) ≅ ℤ/2[c_{k}, k ≥ 2] ⊗ Λ_{ℤ/2}(e_{k}, k ≥ 2) ⟶ H^{∗}(BSL(ℤ)^{+}; ℤ/2)

induced by the reduction mod p. We shall replace the generators u_{4i−1} (for i ≥ 1 ) of the exterior subalgebra Λ_{ℤ/2}(u_{2k−1}, k ≥ 2) of H^{∗}(BSL(ℤ)^{+}; ℤ/2) by

ε_{4i−1} = f_{p}^{∗}(e_{2i}) = u_{4i−1} + ∑_{j=2}^{2i−2} w_{j}^{2}u_{4i−2j−1} ∈ H^{4i−1}(BSL(ℤ)^{+}; ℤ/2)

(see Theorem 13 of [AMNY]). Thus, the first two terms of the mod 2 Bockstein spectral sequence of BSL(ℤ)^{+} can be expressed as

E_{1}^{∗}(BSL(ℤ)^{+}) ≅ H^{∗}(BSL(ℤ)^{+}; ℤ/2) ≅ ℤ/2[w_{j}, j ≥ 2] ⊗ Λ_{ℤ/2}(ε_{4i−1}, i ≥ 1) ⊗ Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) ,
E_{2}^{∗}(BSL(ℤ)^{+}) ≅ ℤ/2[w_{2i}^{2}, i ≥ 1] ⊗ Λ_{ℤ/2}(ε_{4i−1}, i ≥ 1) ⊗ Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) .

Let us consider again the sets of integers N_{r} = {k = 2^{r−1}i + 2^{r−2} | i ≥ 0} introduced in Definition 3.1.

Lemma 4.2. For any r ≥ 3 , if k ∈ N_{r} , then ε_{2k−1} belongs to E_{s}^{∗}(BSL(ℤ)^{+}) for all s ≤ r and d_{r}(ε_{2k−1}) = w_{k}^{2} .

Proof. By naturality of the mod 2 Bockstein spectral sequence with respect to f_{p}^{∗} , this follows from the equality

d_{r}(e_{k}) = c_{k} for k ∈ N_{r}

given by Lemma 3.3, from the definition ε_{2k−1} = f_{p}^{∗}(e_{k}) (where k is even since k ∈ N_{r} with r ≥ 3 ) and the formula f_{p}^{∗}(c_{k}) = w_{k}^{2} (see Lemma 1.4 of [Ar3]).

This argument implies also the vanishing of d_{2} , because w_{k}^{2} = 0 in E_{2}^{∗}(BSL(ℤ)^{+}) when k belongs to N_{2} = {odd positive integers}. Let us summarize the information we obtain on E_{r}^{∗}(BSL(ℤ)^{+}) .

Theorem 4.3. The mod 2 Bockstein spectral sequence for BSL(ℤ)^{+} satisfies:

(a) E_{1}^{∗}(BSL(ℤ)^{+}) ≅ H^{∗}(BSL(ℤ)^{+}; ℤ/2) ≅ ℤ/2[w_{j}, j ≥ 2] ⊗ Λ_{ℤ/2}(ε_{4i−1}, i ≥ 1) ⊗ Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) and d_{1}(w_{2i}) = w_{2i+1} , d_{1}(w_{2i+1}) = 0 , and d_{1} is trivial on all classes ε_{4i−1} and u_{4i+1} (i ≥ 1) .

(b) E_{2}^{∗}(BSL(ℤ)^{+}) ≅ E_{3}^{∗}(BSL(ℤ)^{+}) ≅ ℤ/2[w_{k}^{2}, k even ≥ 2] ⊗ Λ_{ℤ/2}(ε_{2k−1}, k even ≥ 2) ⊗ Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) .

(c) For any r ≥ 3 ,

E_{r}^{∗}(BSL(ℤ)^{+}) ≅ ℤ/2[w_{k}^{2} , k ∈ N_{s} for s ≥ r] ⊗ Λ_{ℤ/2}(ε_{2k−1} , k ∈ N_{s} for s ≥ r) ⊗ Λ_{ℤ/2}(u_{4i+1}, i ≥ 1)

and d_{r}(ε_{2k−1}) = w_{k}^{2} whenever k ∈ N_{r} .

(d) E_{∞}^{∗}(BSL(ℤ)^{+}) ≅ Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) .

The calculation of the differentials in that mod 2 Bockstein spectral sequence enables us to split its E_{1}-term as a tensor product of differential graded ℤ/2-algebras:

E_{1}^{∗}(BSL(ℤ)^{+}) ≅ D_{1} ⊗ (⨂_{s≥3} D_{s}) ⊗ D_{∞} ,

where

D_{1} = ℤ/2[w_{2i+1}, i ≥ 1] ⊗ Λ_{ℤ/2}(w_{2i}, i ≥ 1) , δ_{1}(w_{2i}) = w_{2i+1} ,
D_{s} = ℤ/2[w_{k}^{2}, k ∈ N_{s}] ⊗ Λ_{ℤ/2}(ε_{2k−1}, k ∈ N_{s}) , δ_{s}(ε_{2k−1}) = w_{k}^{2} , for s ≥ 3 ,
D_{∞} = Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) , δ_{∞} = 0 .

The spectral sequence can then be described in the following simple way.

Corollary 4.4. The mod 2 Bockstein spectral sequence for BSL(ℤ)^{+} satisfies:

(a) E_{1}^{∗}(BSL(ℤ)^{+}) ≅ D_{1} ⊗ (⨂_{s≥3} D_{s}) ⊗ D_{∞} and the first differential is d_{1} = δ_{1} on D_{1} and d_{1} = 0 on D_{s} when 3 ≤ s ≤ ∞.

(b) E_{2}^{∗}(BSL(ℤ)^{+}) ≅ E_{3}^{∗}(BSL(ℤ)^{+}) and for r ≥ 3 , E_{r}^{∗}(BSL(ℤ)^{+}) ≅ (⨂_{s≥r} D_{s})⊗D_{∞} with the differential d_{r} = δ_{r} on D_{r} and d_{r} = 0 on D_{s} when r < s ≤ ∞.

(d) E_{∞}^{∗}(BSL(ℤ)^{+}) ≅ D_{∞} .

Remark 4.5. Let us mention that the mod 2 Bockstein spectral sequence for the group SL_{3}(ℤ[\frac{1}{2}]) has been recently computed by H.-W. Henn in Section 4.3 of [He].

The following interesting observations are immediate consequences of Theorem 4.3 (b) and (d) and Corollary 4.4 (b) and (d).

Corollary 4.6.

(a) There is no cyclic direct summand of order 4 in H^{∗}(BSL(ℤ)^{+}; ℤ) .

(b) The set of all non-trivial elements of Λ_{ℤ/2}(u_{4i+1}, i ≥ 1) is in one-to-one correspondence with an additive basis of H^{∗}(BSL(ℤ)^{+}; ℤ)/torsion .

By applying again Lemma 3.6, we get an explicit description of the additive structure of the 2-torsion of the integral cohomology H^{∗}(BSL(ℤ)^{+}; ℤ) . In order to formulate the main result of this section, let us use again the notation introduced in Remark 2.2 and Definition 3.1: N_{1} = ℕ−{0} and N_{r} = {k = 2^{r−1}i+2^{r−2} | i ≥ 0} for r ≥ 3 .

Theorem 4.7. Consider the graded ℤ-algebra

Ω = ℤ[ω_{k,1}, k ∈ N_{1}] ⊗ ℤ[ω_{k,r}, k ∈ N_{r} with r ≥ 3] ⊗ Λ_{ℤ}(z_{2k−1}, k ≥ 2) ,

where deg(ω_{k,1}) = 2k + 1 , deg(ω_{k,r}) = 2k when r ≥ 3 and deg(z_{2k−1}) = 2k − 1 . As a graded abelian group, the 2-torsion subgroup of H^{∗}(BSL(ℤ)^{+}; ℤ) is additively isomorphic to the graded Ω-module generated by

{J_{A,r} | r = 1 or r ≥ 3 , A running over all non-empty finite subsets of N_{r}}

with relations generated by 2^{r}J_{A,r},

ω_{k,s}J_{A,r}  for all s < r and all k ∈ N_{s}, when r ≥ 3,
z_{2k−1}J_{A,r}  for all k ∈ N_{s} with 3 ≤ s ≤ r, when r ≥ 3,
∑_{k∈A} ω_{k,r}J_{A−{k},r} for r = 1 and r ≥ 3, A ⊂ N_{r}.

The element J_{{k_{1},...,k_{t}},1} ∈ H^{∗}(BSL(ℤ)^{+}; ℤ) is of degree 2(∑_{j=1}^{t} k_{j}) + 1 and reduces mod 2 to the class ∑_{j=1}^{t} w_{2k_{j}+1}w_{2k_{1}} ⋯ w_{2k_{(j−1)}}w_{2k_{(j+1)}} ⋯ w_{2k_{t}} ∈ H^{∗}(BSL(ℤ)^{+}; ℤ/2) . For r ≥ 3 , the element J_{{k_{1},...,k_{t}},r} ∈ H^{∗}(BSL(ℤ)^{+}; ℤ) is of degree 2(∑_{j=1}^{t} k_{j}) − t + 1 and reduces mod 2 to the class ∑_{j=1}^{t} w_{k_{j}}^{2}ε_{2k_{1}−1} ⋯ ε_{2k_{(j−1)}−1}ε_{2k_{(j+1)}−1} ⋯ ε_{2k_{t}−1} ∈ H^{∗}(BSL(ℤ)^{+}; ℤ/2) .

Proof. For r = 1 or r ≥ 3 , let us call P_{r} the polynomial tensor factor of D_{r} . Because of the splitting of E_{r}^{∗}(BSL(ℤ)^{+}) given by Corollary 4.4, the image of d_{r} is

Im(d_{r}) = Im(δ_{r}) ⊗ (⨂_{s>r} D_{s}) ⊗ D_{∞}

and Lemma 3.6 implies that Im(d_{r}) is the (P_{r} ⊗ (⨂_{s>r} D_{s}) ⊗ D_{∞})-module generated by the J_{A,r}’s, where A runs over all non-empty finite subsets of N_{r} , with the relations provided by Lemma 3.6. Now, let us denote by Ω the graded ℤ-algebra Ω = ℤ[ω_{k,1}, k ∈ N_{1}] ⊗ ℤ[ω_{k,r}, k ∈ N_{r} with r ≥ 3] ⊗ Λ_{ℤ}(z_{2k−1}, k ≥ 2) whose generators are in one-to-one correspondence with those of

P_{1} ⊗ (⨂_{s≥3} D_{s}) ⊗ D_{∞} ≅

ℤ/2[w_{2k+1}, k ≥ 1] ⊗ ℤ/2[w_{k}^{2}, k even ≥ 2] ⊗ Λ_{ℤ/2}(ε_{2k−1}, k even ≥ 2) ⊗ Λ_{ℤ/2}(u_{2k−1}, k odd ≥ 3)

as follows: ω_{k,1} corresponds to w_{2k+1} , ω_{k,r} to w_{k}^{2} when r ≥ 3 and k ∈ N_{r} , z_{2k−1} to ε_{2k−1} when k is even and to u_{2k−1} when k is odd. The assertion of the theorem then follows by gluing together the information on the elements of order 2^{r} in H^{∗}(BSL(ℤ)^{+}; ℤ) given by the determination of Im(d_{r}) for r ≥ 1 .

Remark 4.8. The isomorphism established in Theorem 4.7 is an additive isomorphism: for instance, for 3 ≤ s < r, k ∈ N_{s} and A a non-empty finite subset of N_{r} , the product of the elements of H^{∗}(BSL(ℤ)^{+}; ℤ) corresponding to J_{{k},s} and J_{A,r} under that isomorphism is non-trivial, even if the reduction mod 2 of J_{{k},s} is w_{k}^{2} , which is the generator of D_{s} corresponding to the generator ω_{k,s} of Ω (that product is actually an element of order 2^{s} in H^{∗}(BSL(ℤ)^{+}; ℤ) which reduces mod 2 to the reduction mod 2 of (∑_{k∈A} ω_{k,r} ∏_{i∈A−{k}} z_{2i−1}) J_{{k},s} , where (∑_{k∈A} ω_{k,r} ∏_{i∈A−{k}} z_{2i−1}) ∈ Ω ). More generally, the above mod 2 Bockstein spectral sequence calculation provides the following multiplicative relations mod 2 between the additive generators of the 2-torsion subgroup of H^{∗}(BSL(ℤ)^{+}; ℤ) given by Theorem 4.7. If 1 ≤ s < r, then it is obvious that

J_{A,r}J_{B,s} ≡ (∑_{k∈A} ω_{k,r} ∏_{i∈A−{k}} z_{2i−1}) J_{B,s} mod 2 .

If r ≥ 3 and if red_{2} denotes again the reduction mod 2, one has

red_{2}(J_{A,r}J_{B,r}) = red_{2}(J_{A,r}) δ_{r}( ∏_{j∈B} ε_{2j−1})
= δ_{r}(red_{2}(J_{A,r}) ∏_{j∈B} ε_{2j−1})
= δ_{r}(∑_{k∈A} w_{k}^{2} ∏_{i∈A−{k}}ε_{2i−1} ∏_{j∈B} ε_{2j−1}) .

The fact that the classes ε_{2i−1} are exterior implies that

red_{2}(J_{A,r}J_{B,r}) = δ_{r}( ∑_{k∈A such that A−{k}∩B=∅} w_{k}^{2} ∏_{i∈A−{k}∪B} ε_{2i−1}) = ∑_{k∈A such that A−{k}∩B=∅} w_{k}^{2} δ_{r}( ∏_{i∈A−{k}∪B} ε_{2i−1})

and finally that

J_{A,r}J_{B,r} ≡ ∑_{k∈A such that A−{k}∩B=∅} ω_{k,r} J_{A−{k}∪B,r} mod 2 .

If r = 1 , the formula

J_{A,1}J_{B,1} ≡ ∑_{k∈A such that A−{k}∩B=∅} ω_{k,1} J_{A−{k}∪B,1} mod 2

does still hold but the classes ε_{2i−1} should be replaced by w_{2i} in the argument.

Finally, the above computation helps us to understand, at the prime 2 , the homomorphisms h^{∗} : H^{∗}(BSO; ℤ) → H^{∗}(BSL(ℤ)^{+}; ℤ) and f_{p}^{∗} : H^{∗}(BSL(𝔽_{p}); ℤ) → H^{∗}(BSL(ℤ)^{+}; ℤ) induced by the inclusion ℤ ↪ ℝ and by the reduction mod p, when p ≡ 5 mod 8 .

Theorem 4.9.

(a) The homomorphism h^{∗} : H^{∗}(BSO; ℤ) → H^{∗}(BSL(ℤ)^{+}; ℤ) is injective on the torsion classes of H^{∗}(BSO; ℤ) .

(b) For every generator of infinite order p_{2k} in H^{2k}(BSO; ℤ) with k even, h^{∗}(p_{2k}) is an element of order 2^{r} if k belongs to N_{r} , r ≥ 3 (up to odd torsion).

(c) For p ≡ 5 mod 8 , the image of any generator of any cyclic direct summand of order 4 in H^{∗}(BSL(𝔽_{p})^{+}; ℤ) under the homomorphism f_{p}^{∗} : H^{∗}(BSL(𝔽_{p})^{+}; ℤ) → H^{∗}(BSL(ℤ)^{+}; ℤ) has order 2 in H^{∗}(BSL(ℤ)^{+}; ℤ) .

(d) For any r ≥ 3 , the homomorphism f_{p}^{∗} (with p ≡ 5 mod 8 ) is injective on all cyclic direct summands of order 2^{r} in H^{∗}(BSL(𝔽_{p})^{+}; ℤ) .

Proof. Assertion (a) is obvious since the Stiefel-Whitney classes w_{j} ∈ H^{j}(BSO; ℤ) correspond to the elements w_{j} ∈ H^{j}(BSL(ℤ)^{+}; ℤ) via h^{∗} . The fact that the reduction mod 2 of p_{2k} is w_{k}^{2} ∈ H^{2k}(BSO; ℤ/2) , by Proposition 2.1 (c), and that d_{r}(ε_{2k−1}) = w_{k}^{2} if k ∈ N_{r} , according to Theorem 4.3 (c), implies (b). According to Theorem 3.7, the generators of the cyclic direct summands of order 2^{r} in H^{∗}(BSL(𝔽_{p})^{+}; ℤ) belong to the Φ-module generated by the H_{A,r} ’s, where A = {k_{1}, …, k_{t}} is a finite subset of N_{r} and one has:

f_{p}^{∗}(red_{2}(H_{A,r})) = f_{p}^{∗}(∑_{j=1}^{t} c_{k_{j}}e_{k_{1}} ⋯ e_{k_{(j−1)}}e_{k_{(j+1)}} ⋯ e_{k_{t}}) = ∑_{j=1}^{t} w_{k_{j}}^{2}ε_{2k_{1}−1} ⋯ ε_{2k_{(j−1)}−1}ε_{2k_{(j+1)}−1} ⋯ ε_{2k_{t}−1}

since f_{p}^{∗}(c_{k_{j}}) = w_{k_{j}}^{2} by Lemma 1.4 of [Ar3]. If r = 2 , N_{2} = {odd positive integers} and Assertion (c) follows from the equality

d_{1}(w_{k_{j}−1}w_{k_{j}}ε_{2k_{1}−1} ⋯ ε_{2k_{(j−1)}−1}ε_{2k_{(j+1)}−1} ⋯ ε_{2k_{t}−1}) = w_{k_{j}}^{2}ε_{2k_{1}−1} ⋯ ε_{2k_{(j−1)}−1}ε_{2k_{(j+1)}−1} ⋯ ε_{2k_{t}−1}

when k_{j} is odd. If r ≥ 3 , then Assertion (d) is a consequence of

d_{r}( ∏_{j=1}^{t} ε_{2k_{j}−1}) = ∑_{j=1}^{t} w_{k_{j}}^{2}ε_{2k_{1}−1} ⋯ ε_{2k_{(j−1)}−1}ε_{2k_{(j+1)}−1} ⋯ ε_{2k_{t}−1}

in E_{r}^{∗}(BSL(ℤ)^{+}) .

5. Chern classes of integral representations of groups

For n ≥ 1 , let us call c_{n}(SL(ℤ)) ∈ H^{2n}(SL(ℤ); ℤ) the n-th Chern class of the inclusion σ : SL(ℤ) ↪ GL(ℂ) , i.e., c_{n}(SL(ℤ)) = σ^{∗}(\overline{c}_{n}) , where \overline{c}_{n} denotes the n-th universal Chern class of degree 2n in H^{∗}(BGL(ℂ); ℤ) ≅ ℤ[\overline{c}_{1}, \overline{c}_{2}, …, \overline{c}_{n}, …] and σ^{∗} : H^{∗}(BGL(ℂ); ℤ) → H^{∗}(BSL(ℤ); ℤ) ≅ H^{∗}(SL(ℤ); ℤ) the homomorphism induced by σ. The order of c_{n}(SL(ℤ)) has been only determined up to a factor 2.

Definition 5.1. For any positive even integer n, let E_{n} be the denominator of \frac{B_{n}}{n} , where B_{n} is the n-th Bernoulli number; for instance, B_{2} = \frac{1}{6} , B_{4} = −\frac{1}{30} , B_{6} = \frac{1}{42} , and E_{2} = 12 , E_{4} = 120 , E_{6} = 252 .

Proposition 5.2. The Chern classes c_{n}(SL(ℤ)) are all torsion classes. If n is odd, then c_{n}(SL(ℤ)) is of order 2 in H^{2n}(SL(ℤ); ℤ) . If n is even, then the order of c_{n}(SL(ℤ)) in H^{2n}(SL(ℤ); ℤ) is equal to 2E_{n} when n ≡ 2 mod 4 , and to E_{n} or 2E_{n} when n ≡ 0 mod 4 .

Proof. See [EM1], Section 5, [EM2], Main Theorem, and [Ar1], Einleitung and Korollar 2.5.

The determination of the exact order of c_{n}(SL(ℤ)) in the case where n ≡ 0 mod 4 now follows from the mod 2 Bockstein spectral sequence calculations presented in Theorem 4.3 and Corollary 4.4.

Theorem 5.3. For any positive even integer n, the Chern class c_{n}(SL(ℤ)) is a torsion class of order 2E_{n} in H^{2n}(SL(ℤ); ℤ) .

Proof. The odd-primary part of the order of c_{n}(SL(ℤ)) is given by Proposition 5.2. For all integers r ≥ 3 , we know from Lemma 4.2 that d_{r}(ε_{2n−1}) = w_{n}^{2} when n ∈ N_{r} . Therefore, the fact that red_{2}(c_{n}(SL(ℤ))) = w_{n}^{2} for all n ≥ 1 (see [MT], Part I, p.137, Theorem 5.11) implies that for n even, c_{n}(SL(ℤ)) is of order 2^{r} in H^{n}(SL(ℤ); ℤ) when n ∈ N_{r} (r ≥ 3) . On the other hand, according to von Staudt’s theorem (see [BS], p.384, Theorem 4), 2^{t} divides E_{n} if and only if 2^{t−1} divides n. Thus, if n ∈ N_{r} = {n ∈ ℕ | v_{2}(n) = r−2}, then the 2-primary part of E_{n} is 2^{r−1} . Consequently, the 2-primary parts of the order of c_{n}(SL(ℤ)) and of the integer 2E_{n} coincide for any even integer n ≥ 2 .

Remark 5.4. The same result holds for the Chern classes c_{n}(GL(ℤ)) of the general linear group GL(ℤ) since there is a homotopy equivalence BGL(ℤ)^{+} ≃ BSL(ℤ)^{+} × Bℤ/2 (see for instance [Ar1], Lemma 1.2).

The knowledge of the order of the Chern classes of SL(ℤ) produces the following result on the Chern classes of the linear groups over the field of rationals ℚ.

Corollary 5.5. The Chern classes c_{n}(SL(ℚ)) and c_{n}(GL(ℚ)) are all torsion classes. If n is odd, they are of order 2 . If n is even, the order of c_{n}(SL(ℚ)) and of c_{n}(GL(ℚ)) is equal to 2E_{n} .

Proof. Since the order of c_{n}(SL(ℚ)) and of c_{n}(GL(ℚ)) is a positive multiple of the order of c_{n}(SL(ℤ)) , a lower bound for it is given by Proposition 5.2 and Theorem 5.3. The assertion then follows from Theorem 11 of [Ar2].

For any complex representation ρ : G → GL(ℂ) of any discrete group G, the Chern classes of ρ are c_{n}(ρ) = ρ^{∗}(\overline{c}_{n}) ∈ H^{2n}(G; ℤ) , where ρ^{∗} is the induced homomorphism H^{2n}(BGL(ℂ); ℤ) → H^{2n}(BG; ℤ) ≅ H^{2n}(G; ℤ) . Of course, the above calculations produce the following consequence for any integral representation ρ : G → GL(ℤ) ↪ GL(ℂ) or any rational representation ρ : G → GL(ℚ) ↪ GL(ℂ) of any discrete group G.

Corollary 5.6. The best upper bound for the order of the n-th Chern class c_{n}(ρ) of any integral or rational representation ρ of any discrete group G is equal to 2 when n is odd and to 2E_{n} when n is even.

Proof. Since ρ is an integral representation, respectively a rational representation, c_{n}(ρ) is the image of c_{n}(GL(R)) under the induced homomorphism H^{2n}(GL(R); ℤ) → H^{2n}(G; ℤ) , where R = ℤ, respectively R = ℚ. Consequently, the order of c_{n}(ρ) divides the order of c_{n}(GL(R)) which has been obtained in Proposition 5.2, Theorem 5.3, Remark 5.4 and Corollary 5.5. It turns out that the order of c_{n}(SL(ℤ)) is the best possible upper bound since one can choose G = SL(ℤ) and ρ the inclusion into GL(ℂ) .

Remark 5.7. The assertion of Corollary 5.6 is of particular interest because the best upper bound for the order of the n-th Chern class of any integral representation of any finite group is smaller, i.e., only equal to E_{n} (see [EM1], Theorem 4.12, and [ThC], p. 89).

Acknowledgements. The third and fourth authors would like to thank the Swiss National Science Foundation for financial support when they visited the University of Lausanne and the JAMI of the Johns Hopkins University for its hospitality during the last stage of writing the manuscript.

References

[Ar1] D. Arlettaz. Chern-Klassen von ganzzahligen und rationalen Darstellungen diskreter Gruppen. Math. Z. 187 (1984), 49–60.

[Ar2] D. Arlettaz. On the homology of the special linear group over a number fields. Comment. Math. Helv. 61 (1986), 556-564.

[Ar3] D. Arlettaz. Torsion classes in the cohomology of congruence subgroups. Math. Proc. Cambridge Philos. Soc. 105 (1989), 241–248.

[Ar4] D. Arlettaz. A note on the mod 2 cohomology of SL(ℤ) . In: Algebraic Topology Poznań 1989, Proceedings, Lecture Notes in Math. 1474 (1991), 365–370.

[AMNY] D. Arlettaz, M. Mimura, K. Nakahata and N. Yagita. The mod 2 cohomology of the linear groups over the ring of integers. Proc. Amer. Math. Soc. 127 (1999), 2199–2212.

[Au] C. Ausoni. Propriétés homotopiques de la K-théorie algébrique des entiers. Ph.D. thesis, Université de Lausanne (1998).

[Bo] A. Borel. Topics in the homology theory of fibre bundles. Lecture Notes in Math. 36 (Springer 1967).

[BS] Z. Borevich and I. Shafarevich. Number Theory (Academic Press 1973).

[Brd] W. Browder. Torsion in H-spaces. Ann. of Math. 74 (1961), 24–51.

[Brn] E. Brown. The cohomology of BSO_{n} and BO_{n} with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), 283–288.

[CV] M. Čadek and J. Vanžura. The cohomology rings of BO(n) and BSO(n) with ℤ_{2^{m}} coefficients. J. Math. Kyoto Univ. 35 (1995), 35–41.

[EM1] B. Eckmann and G. Mislin. Chern classes of group representations over a number field. Compositio Math. 44 (1981), 41–65.

[EM2] B. Eckmann and G. Mislin. Profinite Chern classes of group representations. In: Topological topics, London Math. Soc. Lecture Note Ser. 86 (1983), 103–119.

[F] M. Feshbach. The integral cohomology rings of the classifying spaces of O(n) and SO(n) . Indiana Univ. Math. J. 32 (1983), 511–516.

[He] H.-W. Henn. The cohomology of SL(3, ℤ[1/2]) . K-Theory 16 (1999), 299–359.

[Hu] J. Huebschmann. The cohomology of F Ψ^{p} . The additive structure. J. Pure Appl. Algebra 45 (1987), 73–91.

[MT] M. Mimura and H. Toda. Topology of Lie groups I and II. Translations of Math. Monographs 91 (AMS 1991).

[Q] D. Quillen. On the cohomology and K-theory of the general linear groups over a finite field. Ann. of Math. 96 (1972), 552–586.

[ThC] C. Thomas. Characteristic classes and the cohomology of finite groups. Cambridge Stud. Adv. Math. 9 (Cambridge University Press 1986).

[ThE] E. Thomas. On the cohomology of the real Grassmann complexes and the characteristic classes of n-plane bundles. Trans. Amer. Math. Soc. 96 (1960), 67–89.