Stephan Raidt
email: stephan.raidt@gipsa-lab.inpg.fr

Gérard Bailly
email: gerard.bailly@gipsa-lab.inpg.fr

Frédéric Elisei
email: frederic.elisei@gipsa-lab.inpg.fr

Analyzing Gaze during Face-to-Face Interaction

Keywords: face-to-face interaction, talking head, gaze control model

published or not. The documents may come

Gaze is an essential component of face-to-face interaction. When interacting with a human interlocutor, the gaze patterns of an Embodied Conversational Agent (ECA) carry important cues not only for signaling the ECA's own communicative intentions but also its awareness of the environment as well as of the cognitive and emotional state of its interlocutors. The aim of our research is to understand the functions of gaze in social interaction and to develop a coherent gaze control model for our talking heads taking into account the results of detailed multimodal scene analysis. Bilvi and Pelachaud [START_REF] Bilvi | Communicative and statistical eye gaze predictions[END_REF] propose a gaze model for dyadic conversations. Textual input to the system augmented with tags indicating communicative functions drives a statistical model to generate eye movements alternating between direct and averted gaze. Lee et al [START_REF] Lee | Eyes alive[END_REF] propose also a similar statistical model based on analysis of video recordings of monologues uttered by one subject. Both models take into account the cognitive activity of the ECA (e.g. speaking vs. listening, etc) but do not integrate any detailed scene analysis: they do not determine exactly where their target speakers are looking. Itti et al. [START_REF] Itti | Realistic avatar eye and head animation using a neurobiological model of visual attention[END_REF] developed a gaze control model coupled with a visual attention system that detects salient and pertinent points of interest in a natural scene and triggers exogenous saccades. There is however no detection nor separate treatment of faces in this system. We developed an experimental platform where two subjects can interact via a crossed camera-screen setup, with the aim to give interlocutors the impression to be facing each other across a table. Video and audio signals as well as gaze directions are recorded during the interaction for later analysis. The experiment involves dyadic interactions between a female reference subject (for whom we build a virtual speaking clone) and several naïve subjects with same social status and sex. The interaction consists in a sentence-repeating game. One partner utters semantically unpredictable sentences that the other is asked to repeat. Roles are further exchanged. With this rather restricted scenario we try to capture the main elements of face-to-face interaction and to enhance the gaze cues of mutual attention. It also imposes a clear chaining of cognitive states (reading, speaking, listening, waiting, thinking…) and simplifies complex negotiation of turn taking and state dependent gaze analysis. According to our knowledge this is the first experimental setup that monitors both subjects during such a mediated face-to-face interaction. Based on the audiovisual and gaze data saccades and fixations of each speaker with reference to the respective position of the face of their interlocutor are automatically computed. Analysis of these data clearly confirms the triangular pattern of fixations scanning the eyes and the mouth previously obtained by Vatikiotis-Bateson, Eigsti et al [START_REF] Vatikiotis-Bateson | Eye movement of perceivers during audiovisual speech perception[END_REF] for perception of prerecorded audiovisual speech. The fixations towards four regions of interest (left and right eyes, mouth, and face) have been further distinguished and impact of role and cognitive state of each interlocutor are examined. We show that both factors have a significant impact on distributions of gaze fixations and blinking rate of our reference subject. We show for example that speakers never fixate the mouth of their interlocutors when speaking and that blink rate is accelerated when 'speaking', whereas 'reading' and 'listening' slow it down and often inhibit blinks. Following general results obtained by Gullberg and Holmqvist [START_REF] Gullberg | Visual attention towards gestures in face-to-face interaction vs on screen[END_REF] we also show that pre-recorded stimuli produce significantly different gaze behavior of the interlocutors compared to live interaction. Based on the measured data we built a first gaze control model for our talking head by training a Hidden Markov Model. Given a succession of cognitive states with associated durations it computes parameters describing the fixations of the ECA towards the various regions on the face of its interlocutor. An initial state in each HMM has been added to cope with the particular distribution of the first fixation. The observation probabilities determine the duration of the fixation emitted by the HMM at each transition. They are computed from fixations gathered from the interactions. Fixations to the mouth are for instance longer than fixations to the eyes


J O Ël Blot

RANK THEOREM IN INFINITE DIMENSION AND LAGRANGE MULTIPLIERS

Keywords: Rank theorem, Lagrange multipliers, Banach spaces. Classification MSC 2010: 49K27, 49K30

We use an extension to the infinite dimension of the Rank Theorem of the differential calculus to establish a Lagrange theorem for optimization problems in Banach spaces. We provide an application to variational problems on a space of bounded sequences under equality constraints.

Introduction

Firstly we recall the extension of the rank theorem to the infinite dimension as it is established in [START_REF] Blot | The rank theorem in infinite dimension[END_REF] (Section 3). Secondly we establish a theorem of existence of Lagrange multipliers in Banach spaces for maximization problems under equality constraints (Section 4) like the following one (M ) Maximize J(x) subject to f (x) = 0 where the functional J and the mapping f are defined on an open subset of a Banach space, and the mapping f takes its values in a Banach space. Among the classical multiplier rules, when the differential of f at the solution is surjective we obtain the existence of Lagrange multipliers without a multiplier before the criterion, i.e. x being a solution of (M ), we have DJ(x) = λ • Df (x) where λ belongs to the topological dual space of the co-domain of f , and when we have only the closedness of the image of the differential of f at the solution, there exists a multiplier rule with a multiplier before the criterion, i.e. λ 0 DJ(x) = λ•Df (x) with (λ 0 , λ) = (0, 0) and λ 0 ∈ R. Our aim is to establish a necessary optimality condition in the form DJ(x) = λ • Df (x) wihtout assuming the surjectivity of Df (x). To reach this aim we establish a result by using assumptions which are issued from a generalization to the infinite-dimensional Banach spaces of the classical Rank Theorem. Thirdly, in Section 5, we establish results on the space of the bounded sequences with values in a Banach space to prepare the following section. Lastly, in Section 6, we apply our theorem on Lagrange multipliers to a variational problem in infinite horizon and in discrete time on a space of bounded sequences under equality constraints.

Notation

When E and F are sets, when f : E → F is a mapping, when E 0 ⊂ E and F 0 ⊂ F are such that f (E 0 ) ⊂ F 0 , we define the abridgement of f (relatively to E 0 Date: January, 28, 2018.

1 and F 0 ) as abf ; E 0 → F 0 by setting abf (x) := f (x); this notation comes from [START_REF] Rohlin | Premier cours de topologie (French)[END_REF] (p. 12). When X is a topological space, O(X) denotes the topology of X, and when x ∈ X, O x (X) denotes the set of the open neighborhoods of x in X. The topological interior is denoted by Int. When E is a Banach space and E 1 , E 2 are closed vector subspaces of E, the writting

E = E 1 ⊕ a E 2 (respectively E = E 1 ⊕E 2 ) means the algebraic (respectively topological) direct sum, i.e. the mapping (x 1 , x 2 ) → x 1 + x 2 , from E 1 × E 2 into E
is an isomorhism of vector spaces (respectively an isomorphism of topological vector spaces). When E 1 is a closed vector subspace of E, to say that E 1 is topologically complemented in E means that there exists a closed vector subspace of E, say E 2 , such that

E = E 1 ⊕ E 2 .
The letter D denotes the Fréchet differentiation, and when i ∈ {1, 2}, D i denotes the partial Fréchet differentiation of a mapping defined on a product space E 1 × E 2 with respect to the i th variable. When X and Y are real Banach spaces, L(X, Y ) denotes the space of the linear continuous mappings from X into Y , and if L ∈ L(X, Y ), its norm is

L L := sup{ L(x) : x ∈ X, x ≤ 1}. The topological dual space of X, L(X, R), is denoted by X * . When L ∈ L(X, Y ), the adjoint of L is L * ∈ L(Y * , X * ) defined by L * ϕ := ϕ • L for all ϕ ∈ Y * . When M ⊂ X, the orthogonal of M is M ⊥ := {ϕ ∈ X * : ∀x ∈ M, ϕ(x) = 0}. When A ∈ O(X), C 1 (A, Y ) denotes the space of the continuously Fréchet differen- tiable mappings from A into Y . When f : A → Y is a mapping and y ∈ Y , we set [f = y] := {x ∈ A : f (x) = y}.
When M ⊂ X and a ∈ M , the tangent space of M at a is T a M := {α ′ (0) : α ∈ C 1 ((-ǫ, ǫ), M ), α(0) = a}.

We consider Ω as N or N * := N \ {0}. When E is a set, E Ω denotes the space of the sequences defined on Ω with values in E. An element of E Ω will be denoted by x = (x t ) t∈Ω . When E is a normed space, x ∈ ℓ ∞ (Ω, E) denotes the set of the bounded sequences from Ω into E; we set

x ∞ := sup t∈Ω x t . When A ⊂ E, ℓ ∞ (Ω, A) := {x ∈ ℓ ∞ (Ω, E) : ∀t ∈ Ω, x t ∈ A}. We also write B ∞ (x, r) := {u ∈ ℓ ∞ (Ω, E) : u -x ∞ < r}. ℓ 1 (Ω, E) denotes the space of the sequences x = (x t ) t∈Ω ∈ E Ω such that t∈Ω p t < +∞. c 0 (Ω, E) denotes the space of the sequences x = (x t ) t∈Ω ∈ E Ω such that lim t→+∞ x t = 0.

The rank theorem in infinite dimension

In this section we recall the Rank Theorem in infinite dimension and we establish several consequences of this theorem which are useful for the sequel. Under the assumptions of the Rank Theorem we describe the tangent space of a level set. As we indicate in [START_REF] Blot | The rank theorem in infinite dimension[END_REF], there exist other generalizations of the classical Rank Theorem to the infinite-dimensional Banach spaces.

X and Y are real Banach spaces. The following result is given in [START_REF] Blot | Le théorème du rang en dimension infinie (French)[END_REF] and it is established in Theorem 1 and in Theorem 5 of [START_REF] Blot | The rank theorem in infinite dimension[END_REF].

Theorem 3.1. Let A ∈ O(X), f ∈ C 1 (A, Y ), x ∈ A,
and we set ŷ := f (x). We assume that E 2 := KerDf (x) is topologically complemented in X and F 1 := ImDf (x) is closed and topologically complemented in Y ; and so X = E 1 ⊕E 2 where E 1 is a closed vector subspace of X, and Y = F 1 ⊕ F 2 where F 2 is a closed vector subspace of Y . We also assume that the following condition is fulfilled:

( * ) ∃A 0 ∈ O x(A) s.t. ∀x ∈ A 0 , ImDf (x) ∩ F 2 = {0}.
Then the following assertions hold.

(i) ∃V 1 ∈ O D1f1(x) -1 (ŷ1) (E 1 ), ∃V 2 ∈ O x2 (E 2 ), ∃B ∈ O x(X ), ∃ψ : V 1 × V 2 → B a C 1 diffeormorphism, ∃W ∈ O (ŷ1,0) (X), ∃Ω 1 ∈ O ŷ1 (F 1 ), ∃Ω 2 ∈ O ŷ2 (F 2 ), ∃φ : W → Ω 1 × Ω 2 a C 1 diffeomorphism such that φ -1 • f • ψ = Df (x) on V 1 × V 2 . (ii) ∃G 1 ∈ O x1 (E 1 ), ∃G 2 ∈ O x2 (E 2 ), ∃ξ ∈ C 1 (G 2 , G 1 ) such that [f = ŷ] ∩ (G 1 × G 2 ) = {(ξ(x 2 ), x 2 ) : x 2 ∈ G 2 }.
Note that the equality of the conclusion (i) can be rewritten as follows (cf. Theorem 1 of [START_REF] Blot | The rank theorem in infinite dimension[END_REF])

∀(x 1 , x 2 ) ∈ V 1 × V 2 , φ -1 • f • ψ(x 1 , x 2 ) = (D 1 f 1 (x)x 1 , 0) (3.1)
which a kind of local linearization of f . When X and Y are finite-dimensional, the assumption ( * ) of Theorem 3.1 is equivalent to the constancy of the rank of Df (x) on a neighborhood of x, cf. Proposition 4 in [START_REF] Blot | The rank theorem in infinite dimension[END_REF]. We can provide a heuristic meaning of the condition ( * ): when x is closed to x, due to the continuity of Df , Df (x) is closed to Df (x) and ImDf (x) cannot be "smaller" than ImDf (x); this is a consequence of the openness of the set of the invertible linear continuous operators in the normed space of the linear continuous operators. And so to have ImDf (x) isomorphic to ImDf (x), it suffices to forbid ImDf (x) to be too "big". The condition ImDf (x) ∩ F 2 = {0} is a way to forbid ImDf (x) to be too "big". Now we describe consequences of Theorem 3.1.

Proposition 3.2. In the setting and under the assumptions of Theorem 3.1, the following assertions hold.

(i) Df 2 (x) = 0. (ii) D 2 f 1 (x) = 0. (iii) ξ(x 2 ) = x1 . (iv) Dξ(x 2 ) = -D 1 f 1 (x) -1 • D 2 f 1 (x) = 0. Proof. π 2 : Y → F 2 and p 2 : X → E 2 denote the projections. (i) f 2 = π 2 • f implies Df 2 (x) = π 2 • Df (x) = 0 since ImDf (x) ∩ F 2 = {0}. (ii) D 2 f 1 (x) = Df 1 (x) • p 2 = 0 since E 2 := KerDf (x).
(iii) From Theorem 3.1(ii), since f (x 1 , x2 ) = ŷ, we necessarily obtain x1 = ξ(x 2 ).

(iv) Using (ii) of Theorem 3.1, we have, for all

x 2 ∈ G 2 , f 1 (ξ(x 2 ), x 2 ) = ŷ2 .
Differentiating this equality with respect to x 2 at x2 , we obtain 0 = D 1 f (x) • Dξ(x 2 ) + D 2 f (x) which implies the announced formulas.

Proposition 3.3. In the setting and under the assumptions of Theorem 3.1 we have

T x[f = ŷ] = KerDf (x). Proof. Let v ∈ T x[f = ŷ].
Hence there exists c ∈ C 1 ((-ǫ, ǫ), X), where ǫ > 0, such that c((-ǫ, ǫ)) ⊂ [f = ŷ], c(0) = x and c ′ (0) = v. Since f (c(θ)) = ŷ for all θ ∈ (-ǫ, ǫ), differentiating this equality with respect to θ at 0, we obtain 0 = Df (c(0))c ′ (0) = Df (x)v, and so v ∈ KerDf (x). We have proven that

T x[f = ŷ] ⊂ KerDf (x).
To prove the inverse inclusion, we consider v = (v 1 , v 2 ) ∈ KerDf (x). Therefore we have v 1 = 0 after the definition of E 1 , E 2 , and so v = (0, v 2 ). Using Theorem 3.1(ii), since G 2 is open, there exists ǫ > 0 such that, for all θ ∈ (-ǫ, ǫ), we have x2 + θv 2 ∈ G 2 . We define c : (-ǫ, ǫ) → X by setting c(θ

) := (ξ(x 2 + θv 2 ), x2 + θv 2 ) ∈ [f = ŷ] ∩ (G 1 × G 2 )
. Note that we have c(0) = (ξ(x 2 ), x2 ) = x after Proposition 3.2(iii). Note that we have c ′ (0) = (Dξ(x 2 )v 2 , v 2 ) = (0, v 2 ) = v after Proposition 3.2(iv) and so we have

v ∈ T x[f = ŷ]. We have proven that KerDf (x) ⊂ T x[f = ŷ],
ansd so we have proven the announced equality.

Lagrange multipliers in Banach spaces

In this section we establish a theorem of existence of Lagrange multipliers for problem (M ) (written in Introduction) by using the Rank Theorem in infinite dimension. The interest of this result is to avoid a surjectivity on the differential of the equality constraint and nevertheless to avoid the presence of a multiplier before the criterion. Let X and Y be real Banach spaces, A ∈ O(A), J : A → R be a functional, and f : A → Y be a mapping. We consider the problem (M ).

Theorem 4.1. Let x be a local solution of the problem (M ). We assume that the following conditions are fulfilled.

(a) J is Fréchet differentiable at x and f is of class

C 1 on a neighborhood of x. (b) E 2 := KerDf (x) is topologically complemented in X; i.e. X = E 1 ⊕ E 2
where E 1 is a closed vector subspace of X. F 1 := ImDf (x) is closed and topologically complemented in

Y ; i.e. Y = F 1 ⊕ F 2 where F 2 is a closed vector subspace of Y . (c) There exists A 0 ∈ O x(A) such that, for all x ∈ A 0 , ImDf (x) ∩ F 2 = {0}. Then there exists λ ∈ Y * such that DJ(x) = λ • Df (x).
Proof. Using Theorem 4.1, we know that the set [f = 0] is a C 1 manifold around x, and since J is of class C 1 at x, the first-order necessary optimality condition is DJ(x)h = 0 when h ∈ T x[f = 0]. Using Proposition 3.3, we know that T x[f = 0] = KerDf (x), hence we obtain DJ(x)h = 0 when h ∈ KerDf (x), i.e., DJ(x) ∈ (KerDf (x)) ⊥ . Since ImDf (x) is closed, using Theorem 2.19 in [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF] (p. 46), we have (KerDf (x)) ⊥ = ImDf (x) * , and then we have

DJ(x) ∈ Im(Df (x)) * , hence there exists λ ∈ Y * such that DJ(x) = Df (x) * λ = λ • Df (x).
Notice that in Theorem 4.1 we have not a multiplier before the differential of the criterion. Such a result in finite-dimensional spaces is proven in [START_REF] Janin | Directional derivative of the marginal function in nonlinear programming[END_REF] (Proposition 2.3).

On spaces of bounded sequences

In this section we establish several results on the space of the bounded sequences to prepare the using of the theorem on the Lagrange multipliers of the previous section in such sequence spaces. First we recall a result on the characterization of the closedness of linear operators. Proposition 5.1. Let X and Y be two real Banach spaces and L ∈ L(X , Y). The two following assertions are equivalent.

(i) ImL is closed in Y.

(ii) ∃c(L) > 0, ∀y ∈ ImL, ∃x y ∈ X s.t. L(x y ) = y and x y ≤ c(L) y .

We can find two different proofs of this result in [START_REF] Bachir | Infinite dimensional multipliers and Pontryagin principles for discretetime problems[END_REF] (Lemma 3.4) and in [6] (Lemma 2.1).

Remark 5.2. About the constant c(L), under (i), when in addition we assume that KerL is topologically complemented in X , i.e. X = KerL ⊕ X 1 , where X 1 is a closed vector subspace of X , we can consider the abridgement abL : X 1 → ImL, abL(x) := L(x), which is an isomorphism of Banach spaces, and we have c(L) = (abL) -1 L . Let X and Y be real Banach spaces. Let (T t ) t∈N * ∈ L(X, Y ) N * . We consider the three following conditions on this family of linear operators.

(C1) sup t∈N * T t L < +∞. (C2) For all t ∈ N * , KerT t is topologically complemented in X, i.e. there exists a closed vector subspace of X, say S t , such that X = KerT t ⊕ S t , and ImT t is closed and topologocally complemented in Y , i.e. there exists a closed vector subspace of Y , say W t , such that Y = ImT t ⊕ W t . (C3) ĉ := sup t∈N * c(T t ) < +∞. Notice that (C2) is automatically fulfilled when dimX < +∞ and dimY < +∞. We associate to this family the four following sequence spaces.

K := {x ∈ ℓ ∞ (N * , X) : ∀t ∈ N * , x t ∈ KerT t }. S := {x ∈ ℓ ∞ (N * , X) : ∀t ∈ N * , x t ∈ S t }. I := {y ∈ ℓ ∞ (N * , Y ) : ∀t ∈ N * , y t ∈ ImT t }. W := {y ∈ ℓ ∞ (N * , Y ) : ∀t ∈ N * , y t ∈ W t }.
The condition (C1), which is equivalent to the boundeness of (T t (x t )) t∈N * when x is bounded after the Banach-Steinhaus theorem, permits us to define the operator

T : ℓ ∞ (N * , X) → ℓ ∞ (N * , Y ), T (x) := (T y (x t )) t∈N * .
(5.1)

Lemma 5.3. Under (C1, C2, C3) the following assertions hold.

(i) ℓ ∞ (N * , X) = KerT ⊕ S. (ii) ℓ ∞ (N * , Y ) = ImT ⊕ W.
Proof. For each t ∈ N * , we consider the mappings π t : ℓ ∞ (N * , X) → X defined by π t (x) := x t , and the mapping ̟ t : ℓ ∞ (N * , Y ) → Y , defined by ̟ t (y) := y t . Clearly we have π t ∈ L(ℓ ∞ (N * , X), X) and ̟ t ∈ L(ℓ ∞ (N * , Y ), Y ). Note that we have

K = t∈N * π -1 t (KerT t ), S = t∈N * π -1 t (S t ), I = t∈N * ̟ -1 t (ImT t ), W = t∈N * ̟ -1 t (W t ).
Since x is a solution of (V), proceeding by contradiction, it is easy to see that xt is a solution of (R t ). Under our assumptions we can use Theorem 4.1 on (R t ) and assert that there exists q t ∈ Y * such the following equality holds.

β t-1 D 2 u t-1 xt-1 , xt ) + β t u t (x t , xt+1 ) = q t • Dg t (x t ). ( 6.1)
Under (A11), we know that, for every y ∈ Y , there exists a unique (z, w) ∈ ImDg t (x t ) × W t such that y = z + w. We define p t ∈ Y * by setting p t (y) := q t (z). Hence we deduce from (6.1) the following equality: Notice that (A3), (A4), (A8), (A9) and (A11) are automatically fulfilled when dimX < +∞ and dimY < +∞. When dimY < +∞, using a represention of the topological dual space of ℓ ∞ (N * , Y * ) (as a direct of ℓ 1 (N * , Y * ) and of another subspace) which is given in [START_REF] Aliprantis | Infinite dimensional analysis[END_REF] Chapter 15, Section 15.8) we can process as in [START_REF] Blot | Infinite horizon optimal control in the discrete-time framework[END_REF] (Chapter 3) and to obtain the sequence (p t ) tb∈N * as the component in ℓ 1 (N * , Y * ) of an element of ℓ ∞ (N * , Y * ) * .

β t-1 D 2 u t-1 (x t-1 , xt ) + β t D 1 u t (x t , xt+1 ) = p t • Dg t (x t ). ( 6

  .2)Now we ought to prove that (p t ) t∈N * ∈ ℓ 1 (N * , Y * ). Let y ∈ ℓ ∞ (N * , Y ). Using Proposition 5.8, there exists x ∈ ℓ ∞ (N * , X) and w ∈ W such that y = DG(x)x + w, i.e. y t = Dg t (x t )x t + w t . Hence we havep t (y t ) = p t • Dg t (x t )x t + p t (w t ) = p t • Dg t (x t )x t + 0 = β t-1 D 2 u t-1 (x t-1 , xt )x t + β t D 1 u t (x t , xt+1 )x tThe sequences (D 2 u t-1 (x t-1 , xt )) t∈N * and (D 1 u t (x t , xt+1 )) t∈N * are bounded after (A4), and since (x t ) t∈N * is bounded, the sequences (D 2 u t-1 (x t-1 , xt )x t ) t∈N * and (D 1 u t (x t , xt+1 )x t ) t∈N * are bounded. Since β ∈ (0, 1), the series t≥1 β t-1 D 2 u t (x t-1 , xt )x t and t≥0 β t D 1 u t (x t , xt+1 )x t are convergent in R, and consequently the series t≥1 p t (y t ) is convergent in R. Since c 0 (N * , Y ) ⊂ ℓ ∞ (N * , Y ), we obtain that the series t≥1 p t (y t ) is convergentin R when y ∈ c 0 (N * , Y ). Using [12](assertion (α) in page 247), we can assert that (p t ) t∈N * ∈ ℓ 1 (N

* , Y * ).

Acknowledgements. I thank the reviewers very much for helping me improve the contents of the paper, especially the one who kept me from making a mathematical mistake and whose suggestions were very useful and constructive.

Due to the continuity of the π t and ̟ t , these four spaces are intersections of closed subsets which implies that K and S are closed vector subspaces of ℓ ∞ (N * , X) and I and W are closed vector subspaces of ℓ ∞ (N * , Y ).

It is easy to verify that ℓ ∞ (N * X) = K ⊕ a S, ℓ ∞ (N * , Y ) = I ⊕ a W.

Since ℓ ∞ (N * , X) and ℓ ∞ (N * , Y ) are Banach spaces and since K, S, I, W are closed vector subspaces, from the Inverse Mapping Theorem of Banach-Schauder, [START_REF] Lang | Real and functional analysis[END_REF] (Corollary 1.5, p. 388), we obtain

We easily verify that K = KerT , and from (5.2) we obtain (i).

About the image, we see that ImT ⊂ I. if y ∈ I then, for all t ∈ N * , we have y t ∈ ImT t . From (C2), using Proposition 5.1, we can say that there exists x t,yt ∈ X such that T t (x t,yt ) = y t and x t,yt ≤ c(T t ) y t . Hence, setting x y := (x t,yt ) t∈N * , we have T (x y ) = y and x y ∞ ≤ ĉ y , therefore x y ∈ ℓ(N * , X) and T (x y ) = y, i.e. y ∈ ImT . We have proven that I ⊂ ImT , and consequently I = ImT From (5.2) we obtain (ii).

Lemma 5.4. Let A ∈ O(X). Then the following assertions hold.

2 ) ⊂ Intℓ ∞ (N * , A). Proof. Assertion (i) is proven in [START_REF] Blot | On the smoothness of optimal paths[END_REF] (Lemma A.1.1). To prove (ii), if u ∈ t∈N B(x t , r

2 ), then we have u tx t < r 2 , and for all v ∈ A c , we have

which implies d(u t , A c ) ≥ r 2 > 0, and so u ∈ Intℓ ∞ (N * , A). Definition 5.5. Let A ∈ O(X) and g t : A → Y be a mapping for all t ∈ N * . The sequence (g t ) t∈N * is said uniformly equicontinuous on the bounded subsets of

Proposition 5.6. Let A ∈ O(X) and g t : A → Y be a mapping for all t ∈ N * . We assume that the following condition is fulfilled:

This condition permits to define the operator

If, in addition we assume that (g t ) t∈N is uniformly equicontinuous in the bounded

Proof. We arbitrarily fix x ∈ Intℓ ∞ (N * , A). Hence there exists r > 0 such that

where η B,ǫ is provided by Definition 5.5. Hence we have z t ∈ B since z tx t < r for all t ∈ N * , and we have z tx t < η B,ǫ which implies g t (z t )g t (x t ) < ǫ. Hence we have G(z) -G(x) ∞ ≤ ǫ. We have proven that G is continuous at x. Proposition 5.7. Let A ∈ O(X) and (g t ) t∈N * ∈ C 1 (A, Y ) N * . We assume that the following conditions are fulfilled.

(c) (Dg t ) t∈N * is uniformly equicontinuous on the bounded subsets of A.

We consider the operator G :

Then the following assertions hold for all x ∈ Intℓ ∞ (N * , A).

(i) G is Fréchet differentiable at x, and, for all v ∈ ℓ ∞ (N * , X), we have

Proof. We arbitrarily fix x ∈ Intℓ ∞ (N * , A). Using (b), the linear operator T : ℓ ∞ (N * , X) → ℓ ∞ (N * Y ), defined by T h := (Dg t (x t )h t ) t∈ * N , is well-defined and continuous. We can use Lemma 5.4(ii) and assert that t∈N * B(x t , r

2 ) ⊂ Intℓ ∞ (N * , A). Using the set B := {u ∈ A : ∃t u ∈ N * , ux t < r}, we have yet seen that B is bounded in A and we have B(x t , r

2 ) ⊂ B. We arbitrarily fix ǫ > 0, and, using assumption (c), we consider η B,ǫ > 0 provided by Definition 5.5 for the sequence (Dg t ) t∈N * . We arbitrarily fix h ∈ ℓ ∞ (N * , X) such that h ∞ < min{ r 2 , η B,ǫ }. Hence, ∀t ∈ N * ,

2 ), we have z tx t < η B,ǫ which implies Dg t (z t ) -Dg t (x t ) L < ǫ. Now using the Mean Value Inequality as established in [START_REF] Alexeev | Commande optimale (French) MIR[END_REF]

Hence we have proven that G is Fréchet differentiable at x and that DG(x)h = (Dg t (x t )h t ) t∈N * . Applying Proposition 5.6 to DG, we obtain the continuity of DG. Proposition 5.8. Let A ∈ O(X) and (g t ) t∈N * ∈ C 1 (A, Y ) N * . We assume that the conditions (a), (b), (c) of Proposition 5.7 are fulfilled. Let x = (x t ) t∈N * ∈ Intℓ ∞ (N * , A). We assume that the following conditions are fulfilled.

(d) sup t∈N * Dg(x t ) L < +∞.

(e) For all t ∈ N * there exist a closed vector subspace of X, say S t , and a closed vector subspace of Y , say W t , such that X = KerDg(x t ) ⊕ S t and

Proof. This result is a consequence of Proposition 5.7 and of Lemma 5.3 with T t = Dg(x t ).

Notice that (e) is automatically fulfilled when dimX < +∞ and dimY < +∞.

A variational problem

In this section we consider a maximization problem in infinite horizon and in discrete time under holonomic constraints. The unknown variable is a bounded sequence with values in a real Banach space X. Using the results of the previous sections we obtain a first-order necessary optimality condition in the form of a nonhomogeneous Euler-Lagrange equation. Let A be a nonempty subset of X. For all t ∈ N, let u t : A × A → R and for all t ∈ N * let g t : A → Y be functions, where Y is a real Banach space. We fix a vector σ ∈ A, a real number β ∈ (0, 1) and we consider the following variational problem under holonomic constraints.

The problems of Calculus of Variations or of Optimal Control in discrete time and in infinite horizon are very usual in Economics and in Management; see for instance [START_REF] Blot | Infinite horizon optimal control in the discrete-time framework[END_REF], [START_REF] Bachir | Infinite dimensional multipliers and Pontryagin principles for discretetime problems[END_REF], [START_REF] Blot | On the smoothness of optimal paths[END_REF] and references therein.

Theorem 6.1. Let x be a solution of (V). We assume that the following conditions are fulfilled

) (Dg t ) t∈N * is uniformly equicontinuous on the bounded subsets of A.

(A11) For all t ∈ N * , there exists a closed vector subspace of X, say S t , such that X = KerDg t (x t ) ⊕ S t , and there exists a closed vector subspace of Y , say W t , such that Y = ImDg t (x t ) ⊕ W t . (A12) sup t∈N * c(Dg t (x t )) < +∞. (A13) ∃r > 0, ∀x ∈ B ∞ (x, r), ∀t ∈ N * , ImDg t (x t ) ∩ W t = {0}.

Then there exists a sequence (p t ) t∈N * ∈ ℓ 1 (N * , Y * ) which satisfies, for all t ∈ N * , the following equality

Proof. For each t ∈ N * we consider the following static problem. (R t ) Maximize J t (x t ) := β t-1 u t-1 (x t-1 , x t ) + β t u t (x t , xt+1 ) subject to x t ∈ A, g t (x t ) = 0.


Hans-Werner Henn

ON THE MOD-2 COHOMOLOGY OF SL 3 (Z[ 1 2 , i])

Let Γ = SL 3 (Z[ 1 2 , i]), let X be any mod-2 acyclic Γ-CW complex on which Γ acts with finite stabilizers and let Xs be the 2-singular locus of X. We calculate the mod-2 cohomology of the Borel construction of Xs with respect to the action of Γ. This cohomology coincides with the mod-2 cohomology of Γ in cohomological degrees bigger than 8 and the result is compatible with a conjecture of Quillen which predicts the structure of the cohomology ring H * (Γ; F 2 ).

Introduction

A major motivation for studying the mod-2 cohomology of SL 3 (Z[ 1 2 , i]) comes from a conjecture of Quillen (Conjecture 14.7 of [Q1]) which concerns the structure of the modp cohomology of GL n (Λ) where Λ is a ring of S-integers in a number field such that p is invertible in Λ and Λ contains a primitive p-th root of unity ζ p . The conjecture stipulates that under these assumptions H * (GL n (Λ); Z/p) is free over the polynomial algebra Z/p[c 1 , . . . , c n ] where the c i are the mod-p Chern classes associated to an embedding of Λ into the complex numbers. In the sequel we will denote this conjecture by C(n, Λ, p).

We will show in Theorem 5.1 that for Λ = Z[ 1 2 , i] conjecture C(n, Λ, 2) is equivalent to the existence of an isomorphism

H * (GL n (Z[ 1 2 , i]); F 2 ) ∼ = F 2 [c 1 , .
. . , c n ] ⊗ E(e 1 , e 1 , . . . , e 2n-1 , e 2n-1 )

where the classes c i are the Chern classes of the tautological n-dimensional complex representation of GL n (Z[ 1 2 , i]), E denotes an exterior algebra and the classes e 2i-1 , e 2i-1 are of cohomological degree 2i -1 for i = 1, . . . , n.

Conjecture C(n, Z[ 1 2 , i], 2) is trivially true for n = 1 and has been verified for n = 2 in [W]. On the other hand, Dwyer's method in [D] using étale approximations X n for the homotopy type of the 2-completion of BGL n (Z[ 1 2 ]) and comparing the set of homotopy classes of [BP, X n ] with that of [BP, BGL n (Z[ 1 2 ])] for suitable cyclic groups of order 2 n can be adapted to disprove C(16, Z[ 1 2 , i], 2). We will not dwell on this in this paper. However, we note that étale approximations can also be used to show that if C(n, Z[ 1 2 , i], 2) fails then C(2n, Z[ 1 2 ], 2) fails as well [HL]. We also note that C(n, Z[ 1 2 ], 2) is known to be true for n = 2 by [M] and n = 3 by [H2] but is known to be false for n = 32 by [D] and even for n ≥ 14 [HL].

In this paper we give a partial calculation of H * (SL 3 (Z[ 1 2 , i]); F 2 ) and make a first step in an attempt to study conjecture C(3, Z[ 1 2 , i], 2). We propose the same strategy as the one which was used in the case of SL 3 (Z[ 1 2 ]). In a first step one uses a centralizer spectral sequence introduced in [H1] in order to calculate the mod-2 Borel cohomology H * G (X s ; F 2 ) where X is any mod-2 acyclic G-CW complex on which a suitable discrete group G acts with finite stabilizers and X s is the 2-singular locus of X, i.e. the subcomplex consisting of all points for which the isotropy group of the action of G is of even order. For

G = SL 3 (Z[ 1 2 ])
this step was carried out in [H1] and for G = SL 3 (Z[ 1 2 , i]) it is carried out in this paper. The precise form of X does not really matter in this step.

The second step involves a very laborious analysis of the relative mod-2 Borel cohomology H * G (X, X s ; F 2 ) and of the connecting homomorphism for the Borel cohomology of the pair (X, X s ). In the case of G = SL 3 (Z[ 1 2 ]) this was carried out by hand in [H2]. A by hand calculation looks forbidding in the case of G = SL 3 (Z[ 1 2 , i]) and this paper makes no attempt on such a calculation. However, we do make some comments on what is likely to be involved in such an attempt.

Here are the main results of this paper. In these results the elements b 2 respectively b 3 are of degree 4 resp. 6. They are given as Chern classes of the tautological 3-dimensional complex representation of SL 3 (Z[ 1 2 , i]). The indices of the other elements give their cohomological degrees. These elements come from Quillen's exterior cohomology classes in the cohomology of GL 3 (F p ) for suitable primes p, for example for p = 5 (cf. section 3.2 for more details). Furthermore Σ n denotes n-fold suspension so that Σ 4 F 2 is a one dimensional F 2 -vector space concentrated in degree 4.

Theorem 1.1. Let Γ = SL 3 (Z[ 1 2 , i]
) and let X be any mod-2 acyclic Γ-CW complex such that the isotropy group of each cell is finite. Then the centralizer spectral sequence of [H1]

lim s A * (Γ) H t C Γ (E); F 2 ) =⇒ H s+t Γ (X s ; F 2 ) collapses at E 2 and gives a short exact sequence 0 → Σ 4 F 2 ⊕ Σ 4 F 2 ⊕ Σ 7 F 2 → H * Γ (X s ; F 2 ) → F 2 [b 2 , b 3 ] ⊗ E(d 3 , d 3 , d 5 , d 5 )
→ 0 in which the second map is a map of graded algebras.

Next let

ψ : H * (Γ; F 2 ) = H * Γ (X; F 2 ) → H * Γ (X s ; F 2 ) → F 2 [b 2 , b 3 ] ⊗ E(d 3 , d 3 , d 5 , d 5
) be the composition of the map induced by the inclusion X s ⊂ X and the epimorphism of Theorem 1.1.

Theorem 1.2. Let Γ = SL 3 (Z[ 1 2 , i]
) and X be as in the previous theorem.

a) If SD 3 (Z[ 1 2 , i]
) denotes the subgroup of diagonal matrices of Γ then the target of ψ can be identified with a subalgebra of H * (SD 3 (Z[ 1 2 , i]); F 2 ) and ψ is induced by the restriction homomorphism H * (BΓ;

F 2 ) → H * (SD 3 (Z[ 1 2 , i]); F 2 ).
b) There exists a map of graded F 2 -algebras

ϕ : F 2 [c 2 , c 3 ] ⊗ E(e 3 , e 3 , e 5 , e 5 ) → H * (Γ; F 2 )
with e i and e i of degree 2i -1 such that the composition of ϕ with ψ is the isomorphism which sends c i to b i , i = 2, 3, e i to d i and e i to d i , i = 3, 5.

c) The homomorphism ψ is surjective in all degrees, an isomorphism in degrees * > 8 and its kernel is finite dimensional in degrees * ≤ 8.

Remark 1.3. In section 5 we will discuss the relation of Theorem 1.2 with a conjecture of Quillen on the structure of the cohomology of H * (GL(n, Λ); F 2 ) for rings of S-integers Λ in a number field satisfying suitable assumptions (cf. 14.7 of [Q1]). This conjecture would hold in the case of n = 3 and Λ = Z[ 1 2 , i] if the maps ψ and ψ of part (b) of Theorem 1.2 turned out to be isomorphisms (cf. Proposition 5.5).

The following result is an immediate consequence of Theorem 1.2.

Mod-2 cohomology of SL 3 (Z[ 1 2 , i]) 3 Corollary 1.4. Let Γ = SL 3 (Z[ 1 2 , i]
) and X be as in Theorem 1.1. Then the following conditions are equivalent.

a) The restriction homomorphism H * (BΓ;

F 2 ) → H * (SD 3 (Z[ 1 2 , i]); F 2 ) is injective and H * (BΓ; F 2 ) is isomorphic as a graded F 2 -algebra to F 2 [b 2 , b 3 ] ⊗ E(d 3 , d 3 , d 5 , d 5 ).
b) There is an isomorphism

H * Γ (X, X s ; F 2 ) ∼ = Σ 5 F 2 ⊕ Σ 5 F 2 ⊕ Σ 8 F 2 and the connecting homomorphism H * Γ (X s ; F 2 ) → H * +1 Γ (X, X s ; F 2 ) is surjective.
The paper is organized as follows. In section 2 we recall the centralizer spectral sequence and in section 3 we prove Theorem 1.1 and Theorem 1.2. In Section 4 we make some comments on step 2 of the program of a complete calculation of H * (Γ; F 2 ). Finally in section 5 we discuss the relation with Quillen's conjecture.

The author gratefully acknowledges numerous enlightening conversations with Jean Lannes over many years on the topics discussed in this paper.

The centralizer spectral sequence

We recall the centralizer spectral sequence introduced in [H1].

Let G be a discrete group and let p be a fixed prime. Let A(G) be the category whose objects are the elementary abelian p-subgroups E of G, i.e. subgroups which are isomorphic to (Z/p) k for some integer k; if E 1 and E 2 are elementary abelian p-subgroups of G, then the set of morphisms from E 1 to E 2 in A(G) consists precisely of those group homomorphisms α : E 1 → E 2 for which there exists an element g ∈ G with α(e) = geg -1 for all e ∈ E 1 . Let A * (G) be the full subcategory of A(G) whose objects are the non-trivial elementary abelian p-subgroups.

For an elementary abelian p-subgroup we denote its centralizer in G by C G (E). Then the assignment E → H * (C G (E); F p ) determines a functor from A * (G) to the category E of graded F p -vector spaces. The inverse limit functor is a left exact functor from the functor category E A * (G) to E. Its right derived functors are denoted by lim s . The p-rank r p (G) of a group G is defined as the supremum of all k such that G contains a subgroup isomorphic to (Z/p) k .

For a G-space X and a fixed prime p we denote by X s the p-singular locus, i.e. the subspace of X consisting of points whose isotropy group contains an element of order p. Let EG be the total space of the universal principal G-bundle. The mod-p cohomology of the Borel construction EG × G X of a G space X will be denoted H * G (X; F p ). The following result is a special case of part (a) of Corollary 0.4 of [H1].

Theorem 2.1. Let G be a discrete group and assume there exists a finite dimensional modp acyclic G-CW complex X such that the isotropy group of each cell is finite. Then there exists a cohomological second quadrant spectral sequence

E s,t 2 = lim s A * (G) H t (C G (E); F p ) =⇒ H s+t G (X s ; F p ) with E s,t 2 = 0 if s ≥ r p (G)
and t ≥ 0. Remark 2.2. The edge homomorphism in this spectral sequence is a map of algebras

H * G (X s ; F p ) → lim A * (G) H * (C G (E); F p ) which is given as follows.
Let X E be the fixed points for the action of E on X. The G-action on X restricts to an action of the centralizer C G (E) on X E and the G-equivariant maps

G × C G (E) X E → X s , (g, x) → gx . for E ∈ A * (G) induce compatible maps in Borel cohomology H * G (X s ; F 2 ) → H * G (G × C G (E) X E ; F 2 ) ∼ = H * C G (E) (X E ; F 2 ) ∼ = H * (C G (E); F 2 )
which assemble to give the map to the inverse limit. Here we have used that by classical Smith theory X E is mod p-acyclic if X is mod-p acyclic and hence we get canonical isomorphisms

H * C G (E) (X E ; F 2 ) ∼ = H * C G (E) ( * ; F 2 ) ∼ = H * (C G (E); F 2 ).
Furthermore the composition

(2.1) H * (G; F p ) = H * G (X; F p ) → H * G (X s ; F 2 ) → H * (C G (E); F 2 ) is induced by the inclusions C G (E) → G as E varies through A * (G).
In [H1] we have used this spectral sequence in the case p = 2 and G = SL 3 (Z). Here we will use it in the case p = 2 and G = SL(3, Z[ 1 2 , i]). In both cases we have r 2 (G) = 2 and hence the spectral sequence collapses at E 2 and degenerates into a short exact sequence

(2.2) 0 → lim 1 A * (G) H t (C G (E); F 2 ) → H t+1 G (X s ; F 2 ) → lim A * (G) H t+1 (C G (E); F 2 ) → 0 .
3. The centralizer spectral sequence for SL 3 (Z[ 1 2 , i])

3.1. The Quillen category. Let K be any number field, let O K be its ring of integers and consider the ring of S-integers O K [ 1 2 ]. Then, up to equivalence, the Quillen category of

G := SL 3 (O K [ 1 2 ]
) for the prime 2 is independant of K. In fact, because 2 is invertible every elementary abelian 2-subgroup is conjugate to a diagonal subgroup, and hence A * (G) has a skeleton, say A, with exactly two objects, say E 1 and E 2 of rank 1 and 2, respectively. We take E 1 to be the subgroup generated by the diagonal matrix whose first two diagonal entries are -1 and whose third diagonal entry is 1, and E 2 to be the subgroup of all diagonal matrices with diagonal entries 1 or -1 and determinant 1.

The automorphism group of E 1 is trivial, of course, while Aut A (E 2 ) is isomorphic to the group of all abstract automorphisms of E 2 which we can identify with S 3 , the symmetric group on three elements. There are three morphisms from E 1 to E 2 and Aut A (E 2 ) acts transitively on them.

The centralizers and their cohomology. For the centralizers in

H := GL 3 (O K [ 1 2 ]) we find C H (E 1 ) ∼ = GL 2 (O K [ 1 2 ]) × GL 1 (O K [ 1 2 ]) resp. C H (E 2 ) ∼ = D 3 (O K [ 1 2 ]) if D n (O K [ 1 2 ]) denotes the subgroup of diagonal matrices in GL n (O K [ 1 2 ]). This implies C G (E 1 ) ∼ = GL 2 (O K [ 1 2 ]), C G (E 2 ) ∼ = D 2 (O K [ 1 2 ]) ∼ = O K [ 1 2 ] × × O K [ 1 2 ] × .
From now on we specialize to the case

K = Q 2 [i] where we have O K [ 1 2 ] = Z[ 1 2 , i].
In this case the cohomology of the centralizers is explicitly known. In the sequel we abbreviate SL 3 (Z[ 1 2 , i]) by Γ.

The cohomology of C Γ (E 2

). There is an isomorphism of groups

Z/4 × Z ∼ = Z[ 1 2 , i] × , (n, m) → i n (1 + i) m
and therefore we get an isomorphism

(3.1) H * (C Γ (E 2 ); F 2 ) ∼ = H * (Z[ 1 2 , i] × × Z[ 1 2 , i] × ; F 2 ) ∼ = F 2 [y 1 , y 2 ] ⊗ E(x 1 , x 1 , x 2 , x 2 )
with y 1 and y 2 in degree 2 and the other generators in degree 1. We agree to choose the generators so that y 1 , x 1 and x 1 come from the first factor with x 1 and x 1 being the dual basis to the basis of

H 1 (Z[ 1 2 , i] × ; F 2 ) ∼ = Z[ 1 2 , i]) × / Z[ 1 2 , i]) × 2 ∼ = Z/2 × Z/2
given by the image of i and (1 + i) in the mod-2 reduction of the abelian group GL 1 (Z[ 1 2 , i]) and y 1 coming from H 2 (Z/4; F 2 ); likewise with y 2 , x 2 and x 2 coming from the second factor.

3.2.2. The cohomology of C Γ (E 1 ). This cohomology has been calculated in [W]. In fact, from Theorem 1 of [W] we know

(3.2) H * (C Γ (E 1 ); F 2 ) ∼ = H * (GL 2 (Z[ 1 2 , i]); F 2 ) ∼ = F 2 [c 1 , c 2 ] ⊗ E(e 1
, e 1 , e 3 , e 3 ) .

In the sequel we give a short summary of this calculation. The classes e 1 , e 1 , e 3 and e 3 are pulled back from Quillen's exterior classes q 1 and q 3 [Q2] in

(3.3) H * (GL 2 (F 5 ); F 2 ) ∼ = F 2 [c 1 , c 2 ] ⊗ E(q 1 , q 3 ) via two ring homomorphisms (3.4) π : Z[ 1 2 , i] → F 5 , π : Z[ 1 2 , i] → F 5 .
We choose π such that i is sent to 3 and π such that i is sent to 2.

Then consider the two commutative diagrams (with horizontal arrows induced by inclusion and vertical arrows induced by π resp. π )

(3.5) D 2 (Z[ 1 2 ]) → GL 2 (Z[ 1 2 ]) ↓ ↓ D 2 (F 5 ) → GL 2 (F 5 ) .
By abuse of notation we can write

(3.6) H * (D 2 (F 5 ); F 2 ) ∼ = H * (F × 5 × F × 5 ; F 2 ) ∼ = F 2 [y 1 , y 2 ] ⊗ E(x 1 , x 2 ) with y 1 ∈ H 2 (F × 5 ; F 2 ) and x 1 ∈ H 2 (F × 5 ;
F 2 ) coming from the first factor and likewise with y 2 and x 2 coming from the second factor. Then these ring homomorphisms induce two homomorphisms

π * , π * : H * (D 2 (F 5 ); F 2 )) → H * (D 2 (Z[ 1 2 ]); F 2 )
which in term of the isomorphisms (3.6) and (3.1) are explicitly given by

(3.7) π * (y i ) = y i = π * (y i ), π * (x i ) = x i , π * (x i ) = x i + x i for i = 1, 2 .
The cohomology of GL 2 (F 5 ) is detected by restriction to the cohomology of diagonal matrices and restriction is given explicitly as follows:

(3.8)

c 1 → y 1 + y 2 , c 2 → y 1 y 2 , q 1 → x 1 + x 2 q 3 → y 1 x 2 + y 2 x 1 .
Then e 1 , e 1 , e 3 , e 3 are defined via (3.9) e 1 = π * (q 1 ), e 3 = π * (q 3 ), e 1 = π * (q 1 ), e 3 = π * (q 3 ) .

If c 1 and c 2 are the Chern classes of the tautological 2-dimensional complex representation of GL 2 (Z[ 1 2 ], i), then the restriction homomorphism from H * (GL 2 (Z[ 1 2 , i]); F 2 ) to the cohomology of the subgroup of diagonal matrices is injective and by using (3.5) and (3.8) we see that it is explicitly given by (3.10)

c 1 → y 1 + y 2 c 2 → y 1 y 2 e 1 → x 1 + x 2 e 3 → y 1 x 2 + y 2 x 1 e 1 → x 1 + x 1 + x 2 + x 2 e 3 → y 1 (x 2 + x 2 ) + y 2 (x 1 + x 1 ) .
3.2.3. Functoriality. We note that together with the isomorphisms (3.1) and (3.2) the restriction (3.10) also describes the map

α * : H * (C Γ (E 1 ); F 2 ) → H * (C Γ (E 2 ); F 2 ) induced from the standard inclusion of E 1 into E 2 .
To finish the description of H * (C Γ (-); F 2 ) as a functor on A it remains to describe the action of the symetric group

Aut A (E 2 ) ∼ = S 3 of rank 3 on H * (C Γ (E 2 ); F 2 ) ∼ = F 2 [y 1 , y 2 ] ⊗ Λ(x 1 , x 1 , x 2 , x 2
) and because of the multiplicative structure we need it only on the generators.

If τ ∈ Aut A (E 2 ) corresponds to permuting the factors in C Γ (E 2 ) ∼ = GL 1 (Z[ 1 2 , i]) × GL 1 (Z[ 1 2 , i]) then (3.11) τ * (y 1 ) = y 2 τ * (x 1 ) = x 2 τ * (x 1 ) = x 2 τ * (y 2 ) = y 1 τ * (x 2 ) = x 1 τ * (x 2 ) = x 1
and if σ ∈ Aut A (E 2 ) corresponds to the cyclic permutation of the diagonal entries (in suitable order) then

(3.12) σ * (y 1 ) = y 2 σ * (x 1 ) = x 2 σ * (x 1 ) = x 2 σ * (y 2 ) = y 1 + y 2 σ * (x 2 ) = x 1 + x 2 σ * (x 2 ) = x 1 + x 2 .
3.3. Calculating the limit and its derived functors. In Proposition 4.3 of [H1] we showed that for any functor F from A to Z (2) -modules there is an exact sequence

(3.13) 0 → lim A F → F (E 1 ) ϕ -→ Hom Z[S3] (St Z , F (E 2 )) → lim 1 A F → 0 where St Z is the Z[S 3
] module given by the kernel of the augmentation Z[S 3 /S 2 ] → Z, and if a and b are chosen to give an integral basis of St Z on which τ and σ act via

(3.14) τ * (a) = b τ * (b) = a σ * (a) = -b σ * (b) = a -b then ϕ(x)(a) = α * (x) -(σ * ) 2 α * (x) and ϕ(x)(b) = α * (x) -σ * α * (x) if x ∈ F (E 1 ).
Because in our case the functor takes values in F 2 -vector spaces we can replace Hom Z[S3] by Hom F2[S3] and St Z by its mod-2 reduction. The following elementary lemma is needed in the analysis of the third term in the exact sequence (3.13).

St ⊗ St ∼ = F 2 [S 3 /A 3 ] ⊕ St
where A 3 denotes the alternating group on three letters. In fact, the decomposition is given by

St ⊗ St ∼ = Im(id + σ * + σ 2 * ) ⊕ Ker(id + σ * + σ 2 * )
and the first summand is isomorphic to F 2 [S 3 /A 3 ] while the second summand is isomorphic to St.

b) The tensor product An easy calculation shows that in the case of St ⊗ St both submodules are non-trivial and this together with the fact these submodules must be projective proves the claim. b) Again each of the factors in the tensor product is a projective F 2 [S 3 ]-module, hence the tensor product is a projective F 2 [S 3 ]-module. Because σ acts as the identity on F 2 [S 3 /A 3 ] we see that the idempotent e acts trivially on the tensor product and this forces the tensor product to be isomorphic to St ⊕ St.

F 2 [S 3 /A 3 ] ⊗ St is isomorphic to St ⊕ St.
Lemma 3.2. The Poincaré series χ 2 of Hom F2[S3 ](St, F 2 [y 1 , y 2 ]⊗E(x 1 , x 1 , x 2 , x 2 )) is given by χ 2 = 2t 2 (1 + 3t 2 + 3t 4 + t 6 ) + 2t(1 + 2t 2 + 2t 4 + 2t 6 + t 8 ) (1 -t 4 )(1 -t 6 ) .
Proof. The isomorphism of (3.1) is an isomorphism of F 2 [S 3 ]-modules where the action of S 3 is given by (3.11) and (3.12). In particular we see that

H 1 (GL 1 (Z[ 1 2 , i]) × GL 1 (Z[ 1 2 , i]); F 2 ) is isomorphic to St ⊕ St generated by x 1 , x 1 , x 2 , x 2 .
The exterior powers of H 1 are given as

E k (x 1 , x 2 , x 1 , x 2 ) ∼ = E k (St ⊕ St) ∼ = k j=0 E j St ⊗ E k-j St and, because E k (St) is isomorphic to Σ k F 2 if k = 0, 2, isomorphic to ΣSt if k = 1,
and trivially otherwise, we obtain

E k (x 1 , x 2 , x 1 , x 2 ) ∼ =          Σ k F 2 k = 0, 4 Σ k (St ⊕ St) k = 1, 3 Σ 2 F 2 ⊕ Σ 2 (St ⊗ St) ⊕ Σ 2 F 2 k = 2 0 k = 0, 1, 2, 3, 4
where F 2 denotes the trivial F 2 [S 3 ]-module whose additive structure is that of F 2 .

Therefore the Poincaré series χ 2 of Hom F2

[S3] (St, H * (C G (E 2 ); F 2 )) decomposes according to the decomposition of Λ(x 1 , x 2 , x 1 , x 2 ) as sum (3.15) χ 2 := (1 + 2t 2 + t 4 )χ 2,0 + t 2 χ 2,1 + 2(t + t 3 )χ 2,2
where χ 2,0 is the Poincaré series of Hom F2

[S3] (St, F 2 [y 1 , y 2 ]), χ 2,1 is the Poincaré series of Hom F2[S3] (St, St ⊗ St ⊗ F 2 [y 1 , y 2 ]) and χ 2,2 is that of Hom F2[S3] (St, St ⊗ F 2 [y 1 , y 2 ]).
Furthermore it is well known (and elementary to verify) that there is an isomorphism of

F 2 [S 3 ]-modules St ⊕ St ⊕ F 2 [S 3 /A 3 ] ∼ = F 2 [S 3 ] and therefore an isomorphism F 2 [y 1 , y 2 ] ∼ = Hom F2[S3] (St ⊕ St ⊕ F 2 [S 3 /A 3 ], F 2 [y 1 , y 2 ]) ∼ = Hom F2[S3] (St, F 2 [y 1 , y 2 ]) ⊕2 ⊕ F 2 [y 1 , y 2 ] A3 .
Together with the elementary fact that the generators 1 andy 3 1 + y 1 y 2 2 + y 3 2 of degree 0 resp. 6 this implies

A 3 -invariants F 2 [y 1 , y 2 ] A3 form a free module over F 2 [y 1 , y 2 ] S 3 ∼ = F 2 [c 2 , c 3 ] on two
2χ 2,0 + 1 + t 6 (1 -t 4 )(1 -t 6 ) = 1 (1 -t 2 ) 2 and hence (3.16) χ 2,0 = t 2 (1 -t 2 )(1 -t 6 )
.

It is elementary to check that St and

F 2 [S 3 /A 3 ] are both self-dual F 2 [S 3 ]-modules and hence Lemma 3.1 gives St ⊗ St * ∼ = F 2 [S 3 /A 3 ] ⊕ St and St ⊗ St * ⊗ St * ∼ = (F 2 [S 3 /A 3 ] ⊕ St) ⊗ St * ∼ = (F 2 [S 3 /A 3 ] ⊗ St) ⊕ (St ⊗ St) ∼ = St ⊕ St ⊕ St ⊕ F 2 [S 3 /A 3 ] . Therefore, if χ F2[y1,y2] A 3 denotes the Poincaré series of the A 3 -invariants then (3.17) χ 2,1 = 3χ 2,0 + χ F2[y1,y2] A 3 = 3t 2 (1 -t 2 )(1 -t 6 ) + 1 + t 6 (1 -t 4 )(1 -t 6 ) = 1 + 3t 2 + 3t 4 + t 6 (1 -t 4 )(1 -t 6 ) (3.18) χ 2,2 = χ 2,0 + χ F2[y1,y2] A 3 = t 2 (1 -t 2 )(1 -t 6 ) + 1 + t 6 (1 -t 4 )(1 -t 6 ) = 1 + t 2 + t 4 + t 6
(1 -t 4 )(1 -t 6 ) .

Finally (3.15), (3.16), (3.17) and (3.18) give

χ 2 = (1 + 2t 2 + t 4 )t 2 (1 + t 2 ) + t 2 (1 + 3t 2 + 3t 4 + t 6 ) + 2(t + t 3 )(1 + t 2 + t 4 + t 6 ) (1 -t 4 )(1 -t 6 ) = 2t 2 (1 + 3t 2 + 3t 4 + t 6 ) + 2t(1 + 2t 2 + 2t 4 + 2t 6 + t 8 ) (1 -t 4 )(1 -t 6 ) ,
and this finishes the proof.

Theorem 1.1 is now an immediate consequence of Theorem 2.1 and the following result.

Proposition 3.3. Let p = 2 and Γ = SL 3 (Z[ 1 2 , i]).
a) There is an isomorphism of graded F 2 -algebras

lim A * (Γ) H * (C Γ (E); F 2 ) ∼ = F 2 [b 2 , b 3 ] ⊗ E(d 3 , d 3 , d 5 , d 5 ) .
Furthermore, if we identify this limit with a subalgebra of

H * (C Γ (E 1 ); F 2 ) ∼ = F 2 [c 1 , c 2 ] ⊗ E(e 1 , e 1 , e 3 , e 3 ) then b 2 = c 2 1 + c 2 b 3 = c 1 c 2 d 3 = e 3 d 5 = c 1 e 3 + c 2 e 1 d 3 = e 3 d 5 = c 1 e 3 + c 2 e 1 . Mod-2 cohomology of SL 3 (Z[ 1 2 , i]) 9
b) There is an isomorphism of graded F 2 -vector spaces Now we use the exact sequence (3.13) and the description of ϕ to determine the inverse limit. Because α * is injective, we see that if we identify H * (C Γ (E 1 ); F 2 ) with its image in H * (C Γ (E 2 ); F 2 ) then the inverse limit can be identified with the intersection of the image of α * with the invariants in F 2 [y 1 , y 2 ] ⊗ E(x 1 , x 1 , x 2 , x 2 ) with respect to the action of the cyclic group of order 3 of Aut A (E 2 ) ∼ = S 3 generated by σ. This action has been described in (3.12) and with these formulas it is straightfoward to check that the elements

lim 1 A * (Γ) H * (C Γ (E); F 2 ) ∼ = Σ 3 F 2 ⊕ Σ 3 F 2 ⊕ Σ 6 F 2 . c) For any s > 1 lim s A * (Γ) H * (C Γ (E); F 2 ) = 0 .
(3.19) b 2 = y 2 1 + y 1 y 2 + y 2 2 b 3 = y 1 y 2 (y 1 + y 2 ) d 3 = y 1 x 2 + y 2 x 1 d 5 = (y 1 + y 2 )(y 1 x 2 + y 2 x 1 ) + y 1 y 2 (x 1 + x 2 ) = y 2 1 x 2 + y 2 2 x 1 d 3 = y 1 (x 2 + x 2 ) + y 2 (x 1 + x 1 ) d 5 = (y 1 + y 2 )(y 1 (x 2 + x 2 ) + y 2 (x 1 + x 1 )) + y 1 y 2 (x 1 + x 1 + x 2 + x 2 ) = y 2 1 (x 2 + x 2 ) + y 2 2 (x 1 + x 1
) . all belong to the inverse limit. Now consider the following Poincaré series

χ 0 := n≥0 dim F2 (F 2 [b 2 , b 3 ] ⊗ E(e 3 , e 3 , e 5 , e 5 ) n )t n = (1+t 3 ) 2 (1+t 5 ) 2 (1-t 4 )(1-t 6 ) χ 1 := n≥0 dim F2 H n (C Γ (E 1 ); F 2 )t n = (1+t) 2 (1+t 3 ) 2 (1-t 2 )(1-t 4 ) χ 2 := 2t 2 (1+3t 2 +3t 4 +t 6 )+2t(1+2t 2 +2t 4 +2t 6 +t 8 ) (1-t 4 )(1-t 6 )
.

Then we have the following identity

χ 0 + χ 2 -χ 1 = p (1 -t 4 )(1 -t 6 ) with p = (1 + t 3 ) 2 (1 + t 5 ) 2 + 2t 2 (1 + 3t 2 + 3t 4 + t 6 ) + 2t(1 + 2t 2 + 2t 4 + 2t 6 + t 8 ) -(1 + t) 2 (1 + t 3 ) 2 (1 + t 2 + t 4 ) = 2t 3 + t 6 -2t 7 -2t 9 -t 10 -t 12 + 2t 13 + t 16 = (2t 3 + t 6 )(1 -t 4 )(1 -t 6 ) and therefore (3.20) χ 0 + χ 2 = χ 1 + (2t 3 + t 6 ) .
to the natural action of S 3 on the cohomology of diagonal matrices H

* (D 3 (F 5 ); F 2 ) ∼ = F 2 [y 1 , y 2 , y 3 ] ⊗ E(x 1 , x 2 , x 3 ). Explicitly we get c 1 → 0, c 2 → y 2 1 + y 1 y 2 + y 2 2 , c 3 → y 1 y 2 (y 1 + y 2 ) q 1 → 0, q 3 → y 1 x 2 + y 2 x 1 , q 5 → y 2 1 x 2 + y 2 2 x 1 and if i denotes the inclusion GL 2 (Z[ 1 2 , i]) ⊂ SL 3 (Z[ 1 2 , i]) ⊂ GL 3 (Z[ 1 2 , i]) then (3.7) and (3.19) imply ψ(ϕ(e 3 )) = i * (π * (q 3 )) = π * j * (q 3 ) = π * (y 1 x 2 + y 2 x 1 ) = d 3 ψ(ϕ(e 5 )) = i * (π * (q 5 )) = π * j * (q 5 ) = π * (y 2 1 x 2 + y 2 2 x 1 ) = d 5 ψ(ϕ(e 3 )) = i * (π * (q 3 )) = π * j * (q 3 ) = π * (y 1 x 2 + y 2 x 1 ) = d 3 ψ(ϕ(e 5 )) = i * (π * (q 5 )) = π * j * (q 5 ) = π * (y 2 1 x 2 + y 2 2
x 1 ) = d 5 where we have identified the target of ψ with a subalgebra of H * (GL 2 (Z[ 1 2 , i]; F 2 ) and the latter via restriction with a subalgebra of F 2 [y 1 , y 2 ] ⊗ E(x 1 x 1 , x 3 , x 3 ).

c) The space X can be taken to be the product of symmetric space X ∞ := SL 3 (C)/SU (2) and the Bruhat-Tits building X 2 for SL 3 (Q 2 [i]). Now SL 3 (Q 2 [i])\X 2 is a 2-simplex (cf. [B]) and the projection map X → X 2 induces a map

SL 3 (Q 2 [i])\X → SL 3 (Q 2 [i])\X 2
whose fibres have the homotopy type of a 6-dimensional SL 3 (Z[ 1 2 , i])-invariant deformation retract (cf. section 4). Therefore we get H n G (X, X s ; F 2 ) = 0 if n > 8 and the inclusion X s ⊂ X induces an isomorphism H n G (X; F 2 ) ∼ = H n G (X s ; F 2 ) if n > 8. Then part c) simply follows from a) except for the finiteness statement for the kernel for which we refer to (4.1) and (4.2) below.

Comments on step 2

The situation for p = 2 and G = SL 3 (Z[ 1 2 , i]) is analogous to the situation for p = 2 and G = SL 3 (Z[ 1 2 ]) for which step 2 was carried out in [H2] via a detailed study of the relative cohomology H * G (X, X s ; F 2 ) for X equal to the product of the symmetric space X ∞ := SL 3 (R)/SO(3) with the Bruhat-Tits building X 2 for SL 3 (Q 2 ); the spaces involved had a few hundred cells and the calculation was painful. In the case of SL 3 (Z[ 1 2 , i]) with X the product of SL 3 (C)/SU (3) with the Bruhat-Tits building for SL 3 (Q 2 [i]) the calculational complexity of the second step is much more involved and an explicit calculation by hand does not look feasible. However, in recent years there have been a lot of machine aided calculations of the cohomology of various arithmetic groups (for example [GG], [BRW]) and a machine aided calculation seems to be within reach.

The natural strategy for undertaking this second step is to follow the same path as in [H2]. The equivariant cohomology H * Γ (X, X s ; F 2 ) can be studied via the spectral sequence of the projection map p : X = X ∞ × X 2 → X 2 . This gives a spectral sequence with (4.1)

E s,t 1 ∼ = σ∈Λs H t Γσ (X ∞ , X ∞,s ; F 2 ) =⇒ H s+t Γ (X, X s ; F 2 ) .
Here Λ s indexes the s-dimensional cells in the orbit space of X 2 with respect to the action of Γ. The orbit space is a 2-simplex, i.e. Λ 0 and Λ 1 contain 3 elements and Λ 2 is a singleton. Furthermore Γ σ is the isotropy group of a chosen representative in X 2 of the cell σ in the quotient space. For fixed s all s-dimensional cells have isomorphic isotropy groups because

  St be the F 2 [S 3 ]-module given as the kernel of the augmentation F 2 [S 3 /S 2 ] → F 2 . The tensor product St ⊗ St decomposes as F 2 [S 3 ]-module canonically as

  Proof. a) It is well known that St is a projective F 2 [S 3 ]-module, hence St ⊗ St is also projective. It is also well known that every projective indecomposable F 2 [S 3 ]-module is isomorphic to either St or F 2 [S 3 /A 3 ]. Both modules can be distinguished by the fact that e := id + σ * + σ * acts trivially on St and as the identity on F 2 [S 3 /A 3 ]. Furthemore e is a central idempotent in F 2 [S 3 ] and hence each F 2 [S 3 ]-module M decomposes as direct sum of F 2 [S 3 ]-modules M ∼ = Im(e : M → M ) ⊕ Ker(e : M → M ) .

  Proof. a) It is straightforward to check that the subalgebra of F 2 [c 1 , c 2 ] ⊗ E(e 1 , e 1 , e 3 , e 3 ) generated by the elements c 2 1 + c 2 , c 1 c 2 , e 3 , e 3 , c 1 e 3 + c 2 e 1 , c 1 e 3 + c 2 e 1 is isomorphic to the tensor product of a polynomial algebra on two generators b 2 and b 3 of degree 4 and 6 and an exterior algebra on 4 generators d 3 , d 3 , d 5 and d 5 of degree 3, 3, 5 and 5. In fact, it is clear that c 2 1 +c 2 and c 1 c 2 are algebraically independant and the elements e 3 , e 3 , c 1 e 3 +c 2 e 1 , c 1 e 3 + c 2 e 1 are exterior classes; their product is given as c 2 2 e 3 e 3 e 1 e 1 = 0, and this implies easily that the exterior monomials in these elements are linearly independant over the polynomial algebra generated by c 2 1 + c 2 , c 1 c 2 . From now on we identify b 2 , b 3 , d 3 , d 3 , d 5 and d 5 with c 2 1 + c 2 , c 1 c 2 , e 3 , e 3 , c 1 e 3 + c 2 e 1 and c 1 e 3 + c 2 e 1 .

Hans-Werner Henn

This together with the fact that lim A * (Γ) H * (C Γ (E); F 2 ) contains a subalgebra which is isomorphic to F 2 [b 2 , b 3 ] ⊗ Λ(d 3 , d 3 , d 5 , d 5 ) already implies that the sequence

in which the left hand arrow is given by inclusion is exact except possibly in dimensions 3 and 6.

In order to complete the proof of a) it is now enough to verify that in degrees 3 and 6 the inverse limit is not bigger than F 2 [b 2 , b 3 ] ⊗ E(d 3 , d 3 , d 5 , d 5 ). We leave this straightforward verification to the reader.

Then b) follows immediately from (a) together with (3.20) and the exact sequence (3.13), and (c) follows from Theorem 2.1 and the fact that r 2 (G) = 2.

We can now give the proof of Theorem 1.2.

Proof. a) The exact sequence of Theorem 1.1 is obtained from the exact sequence (2.2) via Proposition 3.3. Therefore the epimorphism of Theorem 1.1 is the edge homomorphism of the centralizer spectral sequence. The result then follows from (2.1) by observing that we have identified the target of the edge homomorphism with the subalgebra F 2 [b 2 , b 3 ] ⊗ E(d 3 , d 3 , d 5 , d 5 ) of H * (C Γ (E 1 ); F 2 ) and by recalling that

b) The two ring homomorphisms π, π :

. By [Q2] we have

We get a well defined homomorphism of F 2 -graded algebras

by sending c i to the i-th Chern class of the tautological 3-dimensional representation of Γ and by declaring ϕ(e i ) = π * (q i ) and ϕ(e i ) = π * (q i ) for i = 3, 5. The classes q 1 resp. q 3 resp. q 5 are the symmetrisations of x 1 resp. y 1 x 2 resp. y 1 y 2 x 3 with respect to the natural action of S 3 on H * (GL 3 (F 5 );

Next we determine the composition ψφ. The universal Chern classes c i are the elementary symmetric polynomials in variables, say y i , and the inclusion GL 2 (C) ⊂ SL 3 (C) ⊂ GL 3 (C) imposes the relation y 1 + y 2 + y 3 = 0. This implies that the behaviour of ψ on Chern classes is given by

In order to determine the composition ψϕ on the classes e 3 , e 3 , e 5 and e 5 we calculate at the level of F 5 and use naturality with respect to the homomorphisms induced by π and π . In fact, the inclusion

which is easily determined from (5.1) below by imposing the relations y 1 + y 2 + y 3 = 0 and x 1 + x 2 + x 3 = 0 on the symmetrisation of the classes y 1 x 2 resp. y 1 y 2 x 3 with respect the Γ-action on the Bruhat-Tits building is the restriction of a natural action of GL 3 (Z[ 1 2 , i]) on X 2 and this action is transitive on the set of s-dimensional cells (cf. [B]). Therefore all isotropy subgroups for the action on X 2 are, up to isomorphism, subgroups of SL 3 (Z[i]) which itself appears as isotropy group of a 0-dimensional cell in X 2 . The isotropy groups of 1-dimensional and 2-dimensional cells are isomorphic to well-known congruence subgroups of SL 3 (Z[i]). By the Soulé-Lannes method the fibre X ∞ of the projection map p admits a 6-dimensional SL 3 (Z[i])-equivariant deformation retract (the space of "wellrounded hermitean forms" modulo arithmetic equivalence) with compact quotient (cf. [Ash]) and therefore we have

The E 1 -term of this spectral sequence should be accessible to machine calculation. The spectral sequence will necessarily degenerate at E 3 and the calculation of the d 1 -differential and, if necessary the d 2 -differential, is likely to need human intervention, as it was necessary in the case of SL(3, Z[ 1 2 ]) (cf. section 3.4 of [H2]). Likewise the calculation of the connecting homomorphism for the mod-2 Borel cohomology of the pair (X, X s ) is likely to require human intervention.

Relation to Quillen's conjecture

The next result gives gives 2 reformulations of Quillen's conjecture which we had briefly discussed in the introduction. The classes e 2k-1 , e 2k-1 figuring in part c) will be introduced in (5.1) below.

Theorem 5.1. Suppose n ≥ 2. Then the following statements are equivalent.

where the c i are the mod-2 Chern classes of the tautological n-dimensional complex representation of GL n (Z[ 1 2 , i]).

b) The restriction homomorphism

c) There are isomorphisms

where the classes c k are the Chern classes of the tautological n-dimensional complex representation of GL n (Z[ 1 2 , i]) and the classes e 2k-1 , e 2k-1 are of cohomological degree 2k -1 for k = 1, . . . , n.

Proof. It is trivial that (c) implies (a).

In order to show that (a) implies (b) we observe that D n (Z[ 1 2 , i]) is the centralizer of the unique, up to conjugacy, maximal elementary abelian 2-subgroup

given by the subgroup of diagonal matrices of order 2. Now consider the top Dickson invariant ω in H * (BGL n (C); F 2 ), i.e. the class whose restriction to H * B(

) of rank less than n. If (a) holds then the image of ω is not a zero divisor in H * (GL n (Z[ 1 2 , i]); F 2 ) and hence Corollary I.5.8 of [HLS] implies that the restriction to the centralizer of E n is injective.

The implication (b) ⇒ (c) follows from Proposition 5.3 below.

Before we go on we introduce the classes e 2k-1 and e 2k-1 . As in the case of GL 2 they are obtained from Quillen's classes q 2k-1 ∈ H 2k-1 (GL n (F 5 ); F 2 ) [Q2] which restrict in the cohomology of diagonal matrices in F 5 to the symmetrization of the class y 1 . . . y k-1 x k where y k is of cohomological degree 2 corresponding to the k-th factor in the product n k=1 F × 5 and x k is of cohomological degree 1 of the same factor. Then we define (5.1) e 2k-1 := π * (q 2k-1 ), e 2k-1 := π * (q 2k-1 )

where π, π are the two ring homomorphisms Z[ 1 2 , i] → F 5 with π sending i to 3 and π sending i to 2 which we considered earlier in section 3. If we identify the mod-2 cohomology

. . , n of degree 2 and x k , x k , k = 1, . . . , n of degree 1 where as before we choose x k and x k to be the basis which is dual to the basis of the k-th factor in

given by the classes of i and 1 + i then we get the following lemma which generalizes (3.10) and whose straighforward proof we leave to the reader.

Lemma 5.2. The class e 2k-1 restricts in the cohomology of the subgroup of diagonal matrices H * (D n (Z[ 1 2 , i]; F 2 )) to the symmetrization of y 1 . . . y k-1 x k and the class e 2k-1 restricts to the symmetrization of y 1 . . . y k-1 (x k + x k ).

The following result determines the image of the restriction homomorphism and shows that (b) implies (c) in Theorem 5.1. It resembles results of Mitchell [M] for GL n (Z[ 1 2 ]) for p = 2 and of Anton [START_REF] Anton | On a conjecture of Quillen at the prime 3[END_REF] for GL n (Z[ 1 3 , ζ 3 ]) for p = 3.

Proposition 5.3. Let n ≥ 1 be an integer. The image of the restriction map

. . , e 2n-1 , e 2n-1 ) .

Here we have identified the Chern classes c i and the classes e 2i-1 and e 2i-1 with their image via i * . The images of the elements c i are, of course, the elementary symmetric polynomials in the y i and the images of the classes e 2i-1 and e 2i-1 have been determined in Lemma 5.2. We remark that even though i * need not be injective, it is injective on the subalgebra of H * (GL n (Z[ 1 2 , i]); F 2 ) generated by the classes c i , e 2i-1 and e 2i-1 , 1 ≤ i ≤ n.

This proposition is an analogue of Proposition 3.6 of [START_REF] Anton | An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup[END_REF]. Its proof uses crucially condition (5.3) below, which also plays a central role in [START_REF] Anton | An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup[END_REF].

Proof. In this proof we denote the subalgebra We will show B n ⊂ C n for n ≥ 2 by induction on n. This will be done in three steps.

From the inclusions

given by matrix block sum and the identifications of

and by induction hypothesis the latter subalgebra is equal to

2. The monomial basis in

is in bijection with the set S(n) of sequences

where the a i are integers ≥ 0 and ε i,j ∈ {0, 1} for i = 1, 2 and 1 ≤ j ≤ n. More precisely to I we associate the monomial

We equip S(n) with the lexicographical order and denote it by < n . This order has the property that for each 1 ≤ k < n it agrees with the lexicographical order on S(k) × S(n -k) if S(k) and S(n -k) are equipped with the orders < k and < n-k and S(n) is identified with S(k) × S(n -k) via concetanation of sequences.

In the sequel we replace the symmetrizations of the elements y 1 . . . y i-1 (x i + x i ), i = 1, . . . , n, by the symmetrization of y 1 . . . y i-1 x i and by abuse of notation we continue to denote them by e 2i-1 . This does not change the subalgebra C n . This subalgebra

has a monomial basis which is in bijection with the set T (n) of sequences

where the k i are integers ≥ 0 and φ i,j ∈ {0, 1} for i = 1, 2 and 1 ≤ j ≤ n. More precisely to K we associate the monomial We define a map α : T (n) → S(n) by associating to K ∈ T (n) the largest monomial in S(n) which occurs in the decomposition of c K as linear combination of elements x I with I ∈ S(n). The proof of the following result is elementary and is left to the reader.

Lemma 5.4. The map α is explicitly given by

with

From this lemma it is obvious that α is injective and a sequence

is in the image of α if and only if we have

In particular, if an element x is in C n then the maximal sequence which appears in the decomposition of x as a linear combination of the monomials x I with I ∈ S(n) satisfies (5.3) for all 1 ≤ j < n. Likewise, if x is in C i ⊗ C n-i then this maximal sequence is equal to the maximal sequence which appears in the decomposition of x as a linear combination of the monomials x I with I ∈ S(k) × S(n -k) and hence it satisfies (5.3) for all 1 ≤ j < i and i + 1 ≤ j < n.

3. Now let x be a homogeneous element of B n and let I 0 be the maximal sequence in S(n) appearing in the decomposition of x as a linear combination of the monomials x I with I ∈ S(n). By (5.2) we have x ∈ C n-1 ⊗ C 1 and x ∈ C n-2 ⊗ C 2 , and I 0 remains the maximal sequence in S(n -1) × S(1) resp. S(n -2) × S(2) appearing in the decomposition of x as a linear combination of the monomials x I with I ∈ S(n -1) × S(1) resp. I ∈ S(n -2) × S(2). Hence I 0 satisfies conditions (5.3) for 1 ≤ j < n -1 resp. 1 ≤ j < n -2 and j = n -1. In particular condition condition (5.3) holds for all 1 ≤ j < n and therefore there exists K 0 ∈ T (n) such that α(K 0 ) = I 0 . Then x -c K0 is still in B n and the maximal sequence appearing in the decomposition of x -c K0 is smaller than that of x. By iterating this procedure we see that x belongs to C n .

Finally we relate C(3, Z[ 1 2 , i], 2) to the behaviour of the restriction homomorphism H * (Γ; F 2 ) → H * (C Γ (E 2 ); F 2 ) .

For this we observe that the subgroups Γ = SL 3 (Z[ 1 2 , i]) and the center Z ∼ = Z[ 1 2 , i] × of GL 3 (Z[ 1 2 , i]) have trivial intersection and their product is the kernel of the homomorphism

given as the composition of the determinant with the natural quotient map. Therefore the spectral sequence of the extension Proof. The quotient Z/3 ∼ = (Z[ 1 2 , i]) × /(Z[ 1 2 , i]) × ) 3 acts clearly trivially on H * (Z; F 2 ) and on the image of the homomorphism ϕ of Theorem 1.2. Hence, the corollary follows immediately from (5.4) and Theorem 1.2.