On the Mod-2 Cohomology of SL 3 (z[ 1 2 , i])

TL;DR

This paper computes the mod-2 cohomology of the group SL 3 over Z[1/2, i], revealing its structure in high degrees and supporting Quillen's conjecture about the cohomology ring.

Contribution

It provides the first explicit calculation of the mod-2 cohomology for SL 3 over Z[1/2, i], confirming predictions about its structure in high degrees.

Findings

Cohomology matches that of the group in degrees >8

Supports Quillen's conjecture on cohomology ring structure

Provides explicit cohomology calculations for a specific arithmetic group

Abstract

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

Equations

$$H^{\ast }(G{L}_n(\mathbb{Z}[\frac{1}{2},i]);{\mathbb{F}}_2)\cong {\mathbb{F}}_2[c_1,\dots ,c_n]\otimes E(e_1,e_1^{\prime },\dots ,e_{2n-1},e_{2n-1}^{\prime })$$

$${\lim }_{{\mathcal{A}}_{\ast }(\Gamma )}^s{H}^t{C}_{\Gamma }(E);{\mathbb{F}}_2)⟹H_{\Gamma }^{s+t}(X_s;{\mathbb{F}}_2)$$

$$0\to {\Sigma }^4{\mathbb{F}}_2\oplus {\Sigma }^4{\mathbb{F}}_2\oplus {\Sigma }^7{\mathbb{F}}_2\to H_{\Gamma }^{\ast }(X_s;{\mathbb{F}}_2)\to {\mathbb{F}}_2[b_2,b_3]\otimes E(d_3,d_3^{\prime },d_5,d_5^{\prime })\to 0$$

$$\psi :H^{\ast }(\Gamma ;{\mathbb{F}}_2)=H_{\Gamma }^{\ast }(X;{\mathbb{F}}_2)\to H_{\Gamma }^{\ast }(X_s;{\mathbb{F}}_2)\to {\mathbb{F}}_2[b_2,b_3]\otimes E(d_3,d_3^{\prime },d_5,d_5^{\prime })$$

$$\phi :{\mathbb{F}}_2[c_2,c_3]\otimes E(e_3,e_3^{\prime },e_5,e_5^{\prime })\to H^{\ast }(\Gamma ;{\mathbb{F}}_2)$$

$$H_{\Gamma }^{\ast }(X,X_s;{\mathbb{F}}_2)\cong {\Sigma }^5{\mathbb{F}}_2\oplus {\Sigma }^5{\mathbb{F}}_2\oplus {\Sigma }^8{\mathbb{F}}_2$$

$$E_2^{s,t}={\lim }_{{\mathcal{A}}_{\ast }(G)}^s{H}^t(C_G(E);{\mathbb{F}}_p)⟹H_G^{s+t}(X_s;{\mathbb{F}}_p)$$

$$H_G^{\ast }(X_s;{\mathbb{F}}_p)\to {\lim }_{{\mathcal{A}}_{\ast }(G)}{H}^{\ast }(C_G(E);{\mathbb{F}}_p)$$

$$G{\times }_{C_G(E)}{X}^E\to X_s,(g,x)\mapsto gx.$$

$$H_G^{\ast }(X_s;{\mathbb{F}}_2)\to H_G^{\ast }(G{\times }_{C_G(E)}{X}^E;{\mathbb{F}}_2)\cong H_{C_G(E)}^{\ast }(X^E;{\mathbb{F}}_2)\cong H^{\ast }(C_G(E);{\mathbb{F}}_2)$$

$$H^{\ast }(G;{\mathbb{F}}_p)=H_G^{\ast }(X;{\mathbb{F}}_p)\to H_G^{\ast }(X_s;{\mathbb{F}}_2)\to H^{\ast }(C_G(E);{\mathbb{F}}_2)$$

$$0\to {\lim }_{{\mathcal{A}}_{\ast }(G)}^1{H}^t(C_G(E);{\mathbb{F}}_2)\to H_G^{t+1}(X_s;{\mathbb{F}}_2)\to {\lim }_{{\mathcal{A}}_{\ast }(G)}{H}^{t+1}(C_G(E);{\mathbb{F}}_2)\to 0.$$

$$C_G(E_1)\cong G{L}_2({\mathcal{O}}_K[\frac{1}{2}]),C_G(E_2)\cong D_2({\mathcal{O}}_K[\frac{1}{2}])\cong {\mathcal{O}}_K[\frac{1}{2}{]}^{\times }\times {\mathcal{O}}_K[\frac{1}{2}{]}^{\times }.$$

$$\mathbb{Z}/4\times \mathbb{Z}\cong \mathbb{Z}[\frac{1}{2},i{]}^{\times },(n,m)\mapsto i^n(1+i{)}^m$$

$$H^{\ast }(C_{\Gamma }(E_2);{\mathbb{F}}_2)\cong H^{\ast }(\mathbb{Z}[\frac{1}{2},i{]}^{\times }\times \mathbb{Z}[\frac{1}{2},i{]}^{\times };{\mathbb{F}}_2)\cong {\mathbb{F}}_2[y_1,y_2]\otimes E(x_1,x_1^{\prime },x_2,x_2^{\prime })$$

$$H_{1}(\mathbb{Z}[\frac{1}{2},i]^{\times};\mathbb{F}_{2})\cong\mathbb{Z}[\frac{1}{2},i])^{\times}/\big{(}\mathbb{Z}[\frac{1}{2},i])^{\times}\big{)}^{2}\cong\mathbb{Z}/2\times\mathbb{Z}/2$$

$$H^{\ast }(C_{\Gamma }(E_1);{\mathbb{F}}_2)\cong H^{\ast }(G{L}_2(\mathbb{Z}[\frac{1}{2},i]);{\mathbb{F}}_2)\cong {\mathbb{F}}_2[c_1,c_2]\otimes E(e_1,e_1^{\prime },e_3,e_3^{\prime }).$$

$$H^{\ast }(G{L}_2({\mathbb{F}}_5);{\mathbb{F}}_2)\cong {\mathbb{F}}_2[c_1,c_2]\otimes E(q_1,q_3)$$

$$\pi :\mathbb{Z}[\frac{1}{2},i]\to {\mathbb{F}}_5,{\pi }^{\prime }:\mathbb{Z}[\frac{1}{2},i]\to {\mathbb{F}}_5.$$

$$\begin{array}{ccc}{D}_2(\mathbb{Z}[\frac{1}{2}])& \to & G{L}_2(\mathbb{Z}[\frac{1}{2}])\\ & & \\ \downarrow & & \downarrow \\ & & \\ D_2({\mathbb{F}}_5)& \to & G{L}_2({\mathbb{F}}_5).\end{array}$$

$$H^{\ast }(D_2({\mathbb{F}}_5);{\mathbb{F}}_2)\cong H^{\ast }({\mathbb{F}}_5^{\times }\times {\mathbb{F}}_5^{\times };{\mathbb{F}}_2)\cong {\mathbb{F}}_2[y_1,y_2]\otimes E(x_1,x_2)$$

$${\pi }^{\ast },{\pi }^{\prime \ast }:H^{\ast }(D_2({\mathbb{F}}_5);{\mathbb{F}}_2))\to H^{\ast }(D_2(\mathbb{Z}[\frac{1}{2}]);{\mathbb{F}}_2)$$

$${\pi }^{\ast }(y_i)=y_i={\pi }^{\prime \ast }(y_i),{\pi }^{\ast }(x_i)=x_i,{\pi }^{\prime \ast }(x_i)=x_i+x_i^{\prime }\text{for}i=1,2.$$

$$\begin{array}[]{ll}c_{1}\mapsto y_{1}+y_{2},\ \ c_{2}\mapsto y_{1}y_{2},\ \ q_{1}\mapsto x_{1}+x_{2}&q_{3}\mapsto y_{1}x_{2}+y_{2}x_{1}\ .\\ \end{array}$$

$$e_1={\pi }^{\ast }(q_1),e_3={\pi }^{\ast }(q_3),e_1^{\prime }={\pi }^{\prime \ast }(q_1),e_3^{\prime }={\pi }^{\prime \ast }(q_3).$$

$$\begin{array}[]{lll}c_{1}\mapsto\ y_{1}+y_{2}&c_{2}\mapsto y_{1}y_{2}\\ e_{1}\mapsto\ x_{1}+x_{2}&e_{3}\mapsto y_{1}x_{2}+y_{2}x_{1}\\ e_{1}^{\prime}\mapsto\ x_{1}+x_{1}^{\prime}+x_{2}+x_{2}^{\prime}&e_{3}^{\prime}\mapsto y_{1}(x_{2}+x_{2}^{\prime})+y_{2}(x_{1}+x_{1}^{\prime})\ .\\ \end{array}$$

$${\alpha }_{\ast }:H^{\ast }(C_{\Gamma }(E_1);{\mathbb{F}}_2)\to H^{\ast }(C_{\Gamma }(E_2);{\mathbb{F}}_2)$$

$$\begin{array}[]{lll}\tau_{*}(y_{1})=y_{2}&\tau_{*}(x_{1})=x_{2}&\tau_{*}(x_{1}^{\prime})=x_{2}^{\prime}\\ \tau_{*}(y_{2})=y_{1}&\tau_{*}(x_{2})=x_{1}&\tau_{*}(x_{2}^{\prime})=x_{1}^{\prime}\\ \end{array}$$

$$\begin{array}[]{llll}\sigma_{*}(y_{1})=y_{2}&\sigma_{*}(x_{1})=x_{2}&\sigma_{*}(x_{1}^{\prime})=x_{2}^{\prime}\\ \sigma_{*}(y_{2})=y_{1}+y_{2}&\sigma_{*}(x_{2})=x_{1}+x_{2}&\sigma_{*}(x_{2}^{\prime})=x_{1}^{\prime}+x_{2}^{\prime}\ .\\ \end{array}$$

$$\frac{0\to {\lim }_{\mathcal{A}}F\to F(E_1)\text{buildrel}\phi }{⟶{\mathrm{Hom}}_{\mathbb{Z}[{\mathfrak{S}}_3]}(S{t}_{\mathbb{Z}},F(E_2))\to {\lim }_{\mathcal{A}}^{1}F\to 0}$$

Full text

1. Introduction

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

We will show in Theorem 5.1 that for $\Lambda=\mathbb{Z}[\frac{1}{2},i]$ conjecture $C(n,\Lambda,2)$ is equivalent to the existence of an isomorphism

[TABLE]

where the classes $c_{i}$ are the Chern classes of the tautological $n$-dimensional complex representation of $GL_{n}({{\mathbb{Z}[{1\over 2},i]}})$, $E$ denotes an exterior algebra and the classes $e_{2i-1},e_{2i-1}^{\prime}$ are of cohomological degree $2i-1$ for $i=1,\ldots,n$.

Conjecture $C(n,{{\mathbb{Z}[{1\over 2},i]}},2)$ 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 $BGL_{n}(\mathbb{Z}[\frac{1}{2}])$ and comparing the set of homotopy classes of $[BP,X_{n}]$ with that of $[BP,BGL_{n}(\mathbb{Z}[\frac{1}{2}])]$ for suitable cyclic groups of order $2^{n}$ can be adapted to disprove $C(16,{{\mathbb{Z}[{1\over 2},i]}},2)$. We will not dwell on this in this paper. However, we note that étale approximations can also be used to show that if $C(n,\mathbb{Z}[\frac{1}{2},i],2)$ fails then $C(2n,\mathbb{Z}[{1\over 2}],2)$ fails as well [HL]. We also note that $C(n,\mathbb{Z}[{1\over 2}],2)$ 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 $n\geq 14$ [HL].

In this paper we give a partial calculation of $H^{*}(SL_{3}({{\mathbb{Z}[{1\over 2},i]}});\mathbb{F}_{2})$ and make a first step in an attempt to study conjecture $C(3,{{\mathbb{Z}[{1\over 2},i]}},2)$. We propose the same strategy as the one which was used in the case of $SL_{3}(\mathbb{Z}[\frac{1}{2}])$. In a first step one uses a centralizer spectral sequence introduced in [H1] in order to calculate the mod-$2$ Borel cohomology $H^{*}_{G}(X_{s};\mathbb{F}_{2})$ 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 $G=SL_{3}(\mathbb{Z}[\frac{1}{2}])$ this step was carried out in [H1] and for $G=SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ 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 $H^{*}_{G}(X,X_{s};\mathbb{F}_{2})$ and of the connecting homomorphism for the Borel cohomology of the pair $(X,X_{s})$. In the case of $G=SL_{3}(\mathbb{Z}[\frac{1}{2}])$ this was carried out by hand in [H2]. A by hand calculation looks forbidding in the case of $G=SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ 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 $SL_{3}(\mathbb{Z}[\frac{1}{2},i])$. The indices of the other elements give their cohomological degrees. These elements come from Quillen’s exterior cohomology classes in the cohomology of $GL_{3}(\mathbb{F}_{p})$ for suitable primes $p$, for example for $p=5$ (cf. section 3.2 for more details). Furthermore $\Sigma^{n}$ denotes $n$-fold suspension so that $\Sigma^{4}\mathbb{F}_{2}$ is a one dimensional $\mathbb{F}_{2}$-vector space concentrated in degree $4$.

Theorem 1.1**.**

Let $\Gamma=SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ and let $X$ be any mod-$2$ acyclic $\Gamma$-CW complex such that the isotropy group of each cell is finite. Then the centralizer spectral sequence of [H1]

[TABLE]

collapses at $E_{2}$ and gives a short exact sequence

[TABLE]

in which the second map is a map of graded algebras.

Next let

[TABLE]

be the composition of the map induced by the inclusion $X_{s}\subset X$ and the epimorphism of Theorem 1.1.

Theorem 1.2**.**

Let $\Gamma=SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ and $X$ be as in the previous theorem.

a) If $SD_{3}(\mathbb{Z}[\frac{1}{2},i])$ denotes the subgroup of diagonal matrices of $\Gamma$ then the target of $\psi$ can be identified with a subalgebra of $H^{*}(SD_{3}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})$ and $\psi$ is induced by the restriction homomorphism $H^{*}(B\Gamma;\mathbb{F}_{2})\to H^{*}(SD_{3}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})$.

b) There exists a map of graded $\mathbb{F}_{2}$-algebras

[TABLE]

with $e_{i}$ and $e_{i}^{\prime}$ of degree $2i-1$ such that the composition of $\varphi$ with $\psi$ is the isomorphism which sends $c_{i}$ to $b_{i}$, $i=2,3$, $e_{i}$ to $d_{i}$ and $e_{i}^{\prime}$ to $d_{i}^{\prime}$, $i=3,5$.

c) The homomorphism $\psi$ is surjective in all degrees, an isomorphism in degrees $*>8$ and its kernel is finite dimensional in degrees $*\leq 8$.

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 $H^{*}(GL(n,\Lambda);\mathbb{F}_{2})$ for rings of $S$-integers $\Lambda$ in a number field satisfying suitable assumptions (cf. 14.7 of [Q1]). This conjecture would hold in the case of $n=3$ and $\Lambda=\mathbb{Z}[\frac{1}{2},i]$ if the maps $\psi$ and $\psi$ 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.

Corollary 1.4**.**

Let $\Gamma=SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ and $X$ be as in Theorem 1.1. Then the following conditions are equivalent.

a) The restriction homomorphism $H^{*}(B\Gamma;\mathbb{F}_{2})\to H^{*}(SD_{3}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})$ is injective and $H^{*}(B\Gamma;\mathbb{F}_{2})$ is isomorphic as a graded $\mathbb{F}_{2}$-algebra to $\mathbb{F}_{2}[b_{2},b_{3}]\otimes E(d_{3},d_{3}^{\prime},d_{5},d_{5}^{\prime})$.

b) There is an isomorphism

[TABLE]

and the connecting homomorphism $H^{*}_{\Gamma}(X_{s};\mathbb{F}_{2})\to H^{*+1}_{\Gamma}(X,X_{s};\mathbb{F}_{2})$ 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 $H^{*}(\Gamma;\mathbb{F}_{2})$. 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.

2. 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 $\mathcal{A}(G)$ be the category whose objects are the elementary abelian $p$-subgroups $E$ of $G$, i.e. subgroups which are isomorphic to $(\mathbb{Z}/p)^{k}$ 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 $\mathcal{A}(G)$ consists precisely of those group homomorphisms $\alpha:E_{1}\to E_{2}$ for which there exists an element $g\in G$ with $\alpha(e)=geg^{-1}$ for all $e\in E_{1}$. Let $\mathcal{A}_{*}(G)$ be the full subcategory of $\mathcal{A}(G)$ 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 $E\mapsto H^{*}(C_{G}(E);\mathbb{F}_{p})$ determines a functor from $\mathcal{A}_{*}(G)$ to the category $\mathcal{E}$ of graded $\mathbb{F}_{p}$-vector spaces. The inverse limit functor is a left exact functor from the functor category $\mathcal{E}^{\mathcal{A}_{*}(G)}$ to $\mathcal{E}$. Its right derived functors are denoted by $\mathrm{lim}^{s}$. 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 $(\mathbb{Z}/p)^{k}$.

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 $EG\times_{G}X$ of a $G$ space $X$ will be denoted $H^{*}_{G}(X;\mathbb{F}_{p})$. 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 mod-$p$ acyclic $G$-CW complex $X$ such that the isotropy group of each cell is finite. Then there exists a cohomological second quadrant spectral sequence

[TABLE]

with $E_{2}^{s,t}=0$ if $s\geq r_{p}(G)$ and $t\geq 0$.

Remark 2.2**.**

The edge homomorphism in this spectral sequence is a map of algebras

[TABLE]

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

[TABLE]

for $E\in\mathcal{A}_{*}(G)$ induce compatible maps in Borel cohomology

[TABLE]

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 $H^{*}_{C_{G}(E)}(X^{E};\mathbb{F}_{2})\cong H^{*}_{C_{G}(E)}(*;\mathbb{F}_{2})\cong H^{*}(C_{G}(E);\mathbb{F}_{2})$.

Furthermore the composition

[TABLE]

is induced by the inclusions $C_{G}(E)\to G$ as $E$ varies through $\mathcal{A}_{*}(G)$.

In [H1] we have used this spectral sequence in the case $p=2$ and $G=SL_{3}(\mathbb{Z})$. Here we will use it in the case $p=2$ and $G=SL(3,\mathbb{Z}[{1\over 2},i])$. 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

[TABLE]

3. The centralizer spectral sequence for $SL_{3}(\mathbb{Z}[\frac{1}{2},i])$

3.1. The Quillen category

Let $K$ be any number field, let $\mathcal{O}_{K}$ be its ring of integers and consider the ring of $S$-integers $\mathcal{O}_{K}[\frac{1}{2}]$. Then, up to equivalence, the Quillen category of $G:=SL_{3}(\mathcal{O}_{K}[{1\over 2}])$ 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 $\mathcal{A}_{*}(G)$ has a skeleton, say $\mathcal{A}$, 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 $\mathrm{Aut}_{\mathcal{A}}(E_{2})$ is isomorphic to the group of all abstract automorphisms of $E_{2}$ which we can identify with ${\mathfrak{S}}_{3}$, the symmetric group on three elements. There are three morphisms from $E_{1}$ to $E_{2}$ and $\mathrm{Aut}_{\mathcal{A}}(E_{2})$ acts transitively on them.

3.2. The centralizers and their cohomology

For the centralizers in $H:=GL_{3}(\mathcal{O}_{K}[\frac{1}{2}])$ we find $C_{H}(E_{1})\cong GL_{2}(\mathcal{O}_{K}[\frac{1}{2}])\times GL_{1}(\mathcal{O}_{K}[\frac{1}{2}])$ resp. $C_{H}(E_{2})\cong D_{3}(\mathcal{O}_{K}[\frac{1}{2}])$ if $D_{n}(\mathcal{O}_{K}[\frac{1}{2}])$ denotes the subgroup of diagonal matrices in $GL_{n}(\mathcal{O}_{K}[\frac{1}{2}])$. This implies

[TABLE]

From now on we specialize to the case $K=\mathbb{Q}_{2}[i]$ where we have $\mathcal{O}_{K}[\frac{1}{2}]=\mathbb{Z}[\frac{1}{2},i]$. In this case the cohomology of the centralizers is explicitly known. In the sequel we abbreviate $SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ by $\Gamma$.

3.2.1. The cohomology of $C_{\Gamma}(E_{2})$

There is an isomorphism of groups

[TABLE]

and therefore we get an isomorphism

[TABLE]

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 $x_{1}^{\prime}$ come from the first factor with $x_{1}$ and $x_{1}^{\prime}$ being the dual basis to the basis of

[TABLE]

given by the image of $i$ and $(1+i)$ in the mod-$2$ reduction of the abelian group $GL_{1}(\mathbb{Z}[\frac{1}{2},i])$ and $y_{1}$ coming from $H^{2}(\mathbb{Z}/4;\mathbb{F}_{2})$; likewise with $y_{2}$, $x_{2}$ and $x_{2}^{\prime}$ coming from the second factor.

3.2.2. The cohomology of $C_{\Gamma}(E_{1})$

This cohomology has been calculated in [W]. In fact, from Theorem 1 of [W] we know

[TABLE]

In the sequel we give a short summary of this calculation. The classes $e_{1}$, $e_{1}^{\prime}$, $e_{3}$ and $e_{3}^{\prime}$ are pulled back from Quillen’s exterior classes $q_{1}$ and $q_{3}$ [Q2] in

[TABLE]

via two ring homomorphisms

[TABLE]

We choose $\pi$ such that $i$ is sent to $3$ and $\pi^{\prime}$ such that $i$ is sent to $2$.

Then consider the two commutative diagrams (with horizontal arrows induced by inclusion and vertical arrows induced by $\pi$ resp. $\pi^{\prime}$)

[TABLE]

By abuse of notation we can write

[TABLE]

with $y_{1}\in H^{2}(\mathbb{F}_{5}^{\times};\mathbb{F}_{2})$ and $x_{1}\in H^{2}(\mathbb{F}_{5}^{\times};\mathbb{F}_{2})$ 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

[TABLE]

which in term of the isomorphisms (3.6) and (3.1) are explicitly given by

[TABLE]

The cohomology of $GL_{2}(\mathbb{F}_{5})$ is detected by restriction to the cohomology of diagonal matrices and restriction is given explicitly as follows:

[TABLE]

Then $e_{1},e_{1}^{\prime},e_{3},e_{3}^{\prime}$ are defined via

[TABLE]

If $c_{1}$ and $c_{2}$ are the Chern classes of the tautological $2$-dimensional complex representation of $GL_{2}(\mathbb{Z}[\frac{1}{2}],i)$, then the restriction homomorphism from $H^{*}(GL_{2}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})$ 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

[TABLE]

3.2.3. Functoriality

We note that together with the isomorphisms (3.1) and (3.2) the restriction (3.10) also describes the map

[TABLE]

induced from the standard inclusion of $E_{1}$ into $E_{2}$.

To finish the description of $H^{*}(C_{\Gamma}(-);\mathbb{F}_{2})$ as a functor on $\mathcal{A}$ it remains to describe the action of the symetric group $Aut_{\mathcal{A}}(E_{2})\cong{\mathfrak{S}}_{3}$ of rank $3$ on $H^{*}(C_{\Gamma}(E_{2});\mathbb{F}_{2})\cong\mathbb{F}_{2}[y_{1},y_{2}]\otimes\Lambda(x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime})$ and because of the multiplicative structure we need it only on the generators.

If $\tau\in\mathrm{Aut}_{\mathcal{A}}(E_{2})$ corresponds to permuting the factors in $C_{\Gamma}(E_{2})\cong GL_{1}(\mathbb{Z}[\frac{1}{2},i])\times GL_{1}(\mathbb{Z}[\frac{1}{2},i])$ then

[TABLE]

and if $\sigma\in\mathrm{Aut}_{\mathcal{A}}(E_{2})$ corresponds to the cyclic permutation of the diagonal entries (in suitable order) then

[TABLE]

3.3. Calculating the limit and its derived functors

In Proposition 4.3 of [H1] we showed that for any functor $F$ from $\mathcal{A}$ to $\mathbb{Z}_{(2)}$-modules there is an exact sequence

[TABLE]

where $St_{\mathbb{Z}}$ is the $\mathbb{Z}[{\mathfrak{S}}_{3}]$ module given by the kernel of the augmentation $\mathbb{Z}[{\mathfrak{S}}_{3}/{\mathfrak{S}}_{2}]\to\mathbb{Z}$, and if $a$ and $b$ are chosen to give an integral basis of $St_{\mathbb{Z}}$ on which $\tau$ and $\sigma$ act via

[TABLE]

then $\varphi(x)(a)=\alpha_{*}(x)-(\sigma_{*})^{2}\alpha_{*}(x)$ and $\varphi(x)(b)=\alpha_{*}(x)-\sigma_{*}\alpha_{*}(x)$ if $x\in F(E_{1})$.

Because in our case the functor takes values in $\mathbb{F}_{2}$-vector spaces we can replace $\mathrm{Hom}_{\mathbb{Z}[{\mathfrak{S}}_{3}]}$ by $\mathrm{Hom}_{\mathbb{F}_{2}[{\mathfrak{S}}_{3}]}$ and $St_{\mathbb{Z}}$ by its mod-$2$ reduction. The following elementary lemma is needed in the analysis of the third term in the exact sequence (3.13).

Lemma 3.1**.**

a) Let $St$ be the $\mathbb{F}_{2}[{\mathfrak{S}_{3}}]$-module given as the kernel of the augmentation $\mathbb{F}_{2}[{\mathfrak{S}}_{3}/{\mathfrak{S}}_{2}]\to\mathbb{F}_{2}$. The tensor product $St\otimes St$ decomposes as $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-module canonically as

[TABLE]

where $A_{3}$ denotes the alternating group on three letters. In fact, the decomposition is given by

[TABLE]

and the first summand is isomorphic to $\mathbb{F}_{2}[{\mathfrak{S}}_{3}/A_{3}]$ while the second summand is isomorphic to $St$.

b) The tensor product $\mathbb{F}_{2}[{\mathfrak{S}}_{3}/A_{3}]\otimes St$ is isomorphic to $St\oplus St$.

Proof.

a) It is well known that $St$ is a projective $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-module, hence $St\otimes St$ is also projective. It is also well known that every projective indecomposable $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-module is isomorphic to either $St$ or $\mathbb{F}_{2}[{\mathfrak{S}}_{3}/A_{3}]$. Both modules can be distinguished by the fact that $e:=id+\sigma_{*}+\sigma_{*}$ acts trivially on $St$ and as the identity on $\mathbb{F}_{2}[{\mathfrak{S}}_{3}/A_{3}]$.

Furthemore $e$ is a central idempotent in $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$ and hence each $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-module $M$ decomposes as direct sum of $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-modules

[TABLE]

An easy calculation shows that in the case of $St\otimes St$ 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 $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-module, hence the tensor product is a projective $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-module. Because $\sigma$ acts as the identity on $\mathbb{F}_{2}[{\mathfrak{S}}_{3}/A_{3}]$ we see that the idempotent $e$ acts trivially on the tensor product and this forces the tensor product to be isomorphic to $St\oplus St$. ∎

Lemma 3.2**.**

The Poincaré series $\chi_{2}$ of $\mathrm{Hom}_{\mathbb{F}_{2}[{\mathfrak{S}}_{3}}](St,\mathbb{F}_{2}[y_{1},y_{2}]\otimes E(x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime}))$ is given by

[TABLE]

Proof.

The isomorphism of (3.1) is an isomorphism of $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-modules where the action of ${\mathfrak{S}}_{3}$ is given by (3.11) and (3.12). In particular we see that $H^{1}(GL_{1}(\mathbb{Z}[\frac{1}{2},i])\times GL_{1}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})$ is isomorphic to $St\oplus St$ generated by $x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime}$. The exterior powers of $H^{1}$ are given as

[TABLE]

and, because $E^{k}(St)$ is isomorphic to $\Sigma^{k}\mathbb{F}_{2}$ if $k=0,2$, isomorphic to $\Sigma St$ if $k=1$, and trivially otherwise, we obtain

[TABLE]

where $\mathbb{F}_{2}$ denotes the trivial $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-module whose additive structure is that of $\mathbb{F}_{2}$.

Therefore the Poincaré series $\chi_{2}$ of $\mathrm{Hom}_{\mathbb{F}_{2}[{\mathfrak{S}}_{3}]}(St,H^{*}(C_{G}(E_{2});\mathbb{F}_{2}))$ decomposes according to the decomposition of $\Lambda(x_{1},x_{2}^{\prime},x_{1}^{\prime},x_{2}^{\prime})$ as sum

[TABLE]

where $\chi_{2,0}$ is the Poincaré series of $\mathrm{Hom}_{\mathbb{F}_{2}[{\mathfrak{S}}_{3}]}(St,\mathbb{F}_{2}[y_{1},y_{2}])$, $\chi_{2,1}$ is the Poincaré series of $\mathrm{Hom}_{\mathbb{F}_{2}[{\mathfrak{S}}_{3}]}(St,St\otimes St\otimes\mathbb{F}_{2}[y_{1},y_{2}])$ and $\chi_{2,2}$ is that of $\mathrm{Hom}_{\mathbb{F}_{2}[{\mathfrak{S}}_{3}]}(St,St\otimes\mathbb{F}_{2}[y_{1},y_{2}])$.

Furthermore it is well known (and elementary to verify) that there is an isomorphism of $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-modules $St\oplus St\oplus\mathbb{F}_{2}[{\mathfrak{S}}_{3}/A_{3}]\cong\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$ and therefore an isomorphism

[TABLE]

Together with the elementary fact that the $A_{3}$-invariants $\mathbb{F}_{2}[y_{1},y_{2}]^{A_{3}}$ form a free module over $\mathbb{F}_{2}[y_{1},y_{2}]^{\mathfrak{S}}_{3}\cong\mathbb{F}_{2}[c_{2},c_{3}]$ on two generators $1$ and $y_{1}^{3}+y_{1}y_{2}^{2}+y_{2}^{3}$ of degree [math] resp. $6$ this implies

[TABLE]

and hence

[TABLE]

It is elementary to check that $St$ and $\mathbb{F}_{2}[{\mathfrak{S}}_{3}/A_{3}]$ are both self-dual $\mathbb{F}_{2}[{\mathfrak{S}}_{3}]$-modules and hence Lemma 3.1 gives

[TABLE]

and

[TABLE]

Therefore, if $\chi_{\mathbb{F}_{2}[y_{1},y_{2}]^{A_{3}}}$ denotes the Poincaré series of the $A_{3}$-invariants then

[TABLE]

[TABLE]

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

[TABLE]

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 $\Gamma=SL_{3}(\mathbb{Z}[\frac{1}{2},i])$.

a) There is an isomorphism of graded $\mathbb{F}_{2}$-algebras

[TABLE]

Furthermore, if we identify this limit with a subalgebra of $H^{*}(C_{\Gamma}(E_{1});\mathbb{F}_{2})\cong\mathbb{F}_{2}[c_{1},c_{2}]\otimes E(e_{1},e_{1}^{\prime},e_{3},e_{3}^{\prime})$ then

[TABLE]

b) There is an isomorphism of graded $\mathbb{F}_{2}$-vector spaces

[TABLE]

c) For any $s>1$

[TABLE]

Proof.

a) It is straightforward to check that the subalgebra of $\mathbb{F}_{2}[c_{1},c_{2}]\otimes E(e_{1},e_{1}^{\prime},e_{3},e_{3}^{\prime})$ generated by the elements $c_{1}^{2}+c_{2}$, $c_{1}c_{2}$, $e_{3},e_{3}^{\prime},c_{1}e_{3}+c_{2}e_{1},c_{1}e_{3}^{\prime}+c_{2}e_{1}^{\prime}$ 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 $d_{3}$, $d_{3}^{\prime}$, $d_{5}$ and $d_{5}^{\prime}$ of degree $3$, $3$, $5$ and $5$. In fact, it is clear that $c_{1}^{2}+c_{2}$ and $c_{1}c_{2}$ are algebraically independant and the elements $e_{3},e_{3}^{\prime},c_{1}e_{3}+c_{2}e_{1},c_{1}e_{3}^{\prime}+c_{2}e_{1}^{\prime}$ are exterior classes; their product is given as $c_{2}^{2}e_{3}e_{3}^{\prime}e_{1}e_{1}^{\prime}\neq 0$, and this implies easily that the exterior monomials in these elements are linearly independant over the polynomial algebra generated by $c_{1}^{2}+c_{2}$, $c_{1}c_{2}$. From now on we identify $b_{2}$, $b_{3}$, $d_{3}$, $d_{3}^{\prime}$, $d_{5}$ and $d_{5}^{\prime}$ with $c_{1}^{2}+c_{2}$, $c_{1}c_{2}$, $e_{3}$, $e_{3}^{\prime}$, $c_{1}e_{3}+c_{2}e_{1}$ and $c_{1}e_{3}^{\prime}+c_{2}e_{1}^{\prime}$.

Now we use the exact sequence (3.13) and the description of $\varphi$ to determine the inverse limit. Because $\alpha_{*}$ is injective, we see that if we identify $H^{*}(C_{\Gamma}(E_{1});\mathbb{F}_{2})$ with its image in $H^{*}(C_{\Gamma}(E_{2});\mathbb{F}_{2})$ then the inverse limit can be identified with the intersection of the image of $\alpha_{*}$ with the invariants in $\mathbb{F}_{2}[y_{1},y_{2}]\otimes E(x_{1},x_{1}^{\prime},x_{2},x_{2}^{\prime})$ with respect to the action of the cyclic group of order $3$ of $\mathrm{Aut}_{\mathcal{A}}(E_{2})\cong{\mathfrak{S}}_{3}$ generated by $\sigma$. This action has been described in (3.12) and with these formulas it is straightfoward to check that the elements

[TABLE]

all belong to the inverse limit.

Now consider the following Poincaré series

[TABLE]

Then we have the following identity

[TABLE]

with

[TABLE]

and therefore

[TABLE]

This together with the fact that $\mathrm{lim}_{\mathcal{A}_{*}(\Gamma)}H^{*}(C_{\Gamma}(E);\mathbb{F}_{2})$ contains a subalgebra which is isomorphic to $\mathbb{F}_{2}[b_{2},b_{3}]\otimes\Lambda(d_{3},d_{3}^{\prime},d_{5},d_{5}^{\prime})$ already implies that the sequence

[TABLE]

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 $\mathbb{F}_{2}[b_{2},b_{3}]\otimes E(d_{3},d_{3}^{\prime},d_{5},d_{5}^{\prime})$. 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 $\mathbb{F}_{2}[b_{2},b_{3}]\otimes E(d_{3},d_{3}^{\prime},d_{5},d_{5}^{\prime})$ of $H^{*}(C_{\Gamma}(E_{1});\mathbb{F}_{2})$ and by recalling that $C_{\Gamma}(E_{1})$ is equal to $SD_{3}(\mathbb{Z}[\frac{1}{2},i])$.

b) The two ring homomorphisms $\pi,\pi^{\prime}:\mathbb{Z}[\frac{1}{2},i]\to\mathbb{F}_{5}$ of (3.4) determine homomorphisms $SL_{3}(\mathbb{Z}[\frac{1}{2},i])\subset GL_{3}(\mathbb{Z}[\frac{1}{2},i])\to GL_{3}(\mathbb{F}_{5})$. By [Q2] we have

[TABLE]

We get a well defined homomorphism of $\mathbb{F}_{2}$-graded algebras

[TABLE]

by sending $c_{i}$ to the $i$-th Chern class of the tautological $3$-dimensional representation of $\Gamma$ and by declaring $\varphi(e_{i})=\pi^{*}(q_{i})$ and $\varphi(e_{i}^{\prime})=\pi^{\prime*}(q_{i}^{\prime})$ 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 ${\mathfrak{S}}_{3}$ on $H^{*}(GL_{3}(\mathbb{F}_{5});\mathbb{F}_{2})\cong\mathbb{F}_{2}[y_{1},y_{2},y_{3}]\otimes E(x_{1},x_{2},x_{3})$ (cf. (5.1) below).

Next we determine the composition $\psi\phi$. The universal Chern classes $c_{i}$ are the elementary symmetric polynomials in variables, say $y_{i}$, and the inclusion $GL_{2}(\mathbb{C})\subset SL_{3}(\mathbb{C})\subset GL_{3}(\mathbb{C})$ imposes the relation $y_{1}+y_{2}+y_{3}=0$. This implies that the behaviour of $\psi$ on Chern classes is given by

[TABLE]

In these equations we have identified $H^{*}(GL_{2}(\mathbb{Z}[\frac{1}{2},i];\mathbb{F}_{2})$, as in the proof of Proposition 3.3, via restriction with a subalgebra of $\mathbb{F}_{2}[y_{1},y_{2}]\otimes E(x_{1},x_{1}^{\prime},x_{3},x_{3}^{\prime})$.

In order to determine the composition $\psi\varphi$ on the classes $e_{3}$, $e_{3}^{\prime}$, $e_{5}$ and $e_{5}^{\prime}$ we calculate at the level of $\mathbb{F}_{5}$ and use naturality with respect to the homomorphisms induced by $\pi$ and $\pi^{\prime}$. In fact, the inclusion

[TABLE]

induce in cohomology a map

[TABLE]

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 to the natural action of ${\mathfrak{S}}_{3}$ on the cohomology of diagonal matrices $H^{*}(D_{3}(\mathbb{F}_{5});\mathbb{F}_{2})\cong\mathbb{F}_{2}[y_{1},y_{2},y_{3}]\otimes E(x_{1},x_{2},x_{3})$. Explicitly we get

[TABLE]

and if $i$ denotes the inclusion

[TABLE]

then (3.7) and (3.19) imply

[TABLE]

where we have identified the target of $\psi$ with a subalgebra of $H^{*}(GL_{2}(\mathbb{Z}[\frac{1}{2},i];\mathbb{F}_{2})$ and the latter via restriction with a subalgebra of $\mathbb{F}_{2}[y_{1},y_{2}]\otimes E(x_{1}x_{1}^{\prime},x_{3},x_{3}^{\prime})$.

c) The space $X$ can be taken to be the product of symmetric space $X_{\infty}:=SL_{3}(\mathbb{C})/SU(2)$ and the Bruhat-Tits building $X_{2}$ for $SL_{3}(\mathbb{Q}_{2}[i])$. Now $SL_{3}(\mathbb{Q}_{2}[i])\backslash X_{2}$ is a $2$-simplex (cf. [B]) and the projection map $X\to X_{2}$ induces a map

[TABLE]

whose fibres have the homotopy type of a $6$-dimensional $SL_{3}(\mathbb{Z}[\frac{1}{2},i])$-invariant deformation retract (cf. section 4). Therefore we get $H^{n}_{G}(X,X_{s};\mathbb{F}_{2})=0$ if $n>8$ and the inclusion $X_{s}\subset X$ induces an isomorphism $H^{n}_{G}(X;\mathbb{F}_{2})\cong H^{n}_{G}(X_{s};\mathbb{F}_{2})$ 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. ∎

4. Comments on step 2

The situation for $p=2$ and $G=SL_{3}(\mathbb{Z}[{1\over 2},i])$ is analogous to the situation for $p=2$ and $G=SL_{3}(\mathbb{Z}[{1\over 2}])$ for which step 2 was carried out in [H2] via a detailed study of the relative cohomology $H^{*}_{G}(X,X_{s};\mathbb{F}_{2})$ for $X$ equal to the product of the symmetric space $X_{\infty}:=SL_{3}(\mathbb{R})/SO(3)$ with the Bruhat-Tits building $X_{2}$ for $SL_{3}(\mathbb{Q}_{2})$; the spaces involved had a few hundred cells and the calculation was painful. In the case of $SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ with $X$ the product of $SL_{3}(\mathbb{C})/SU(3)$ with the Bruhat-Tits building for $SL_{3}(\mathbb{Q}_{2}[i])$ 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 $H^{*}_{\Gamma}(X,X_{s};\mathbb{F}_{2})$ can be studied via the spectral sequence of the projection map

[TABLE]

This gives a spectral sequence with

[TABLE]

Here $\Lambda_{s}$ indexes the $s$-dimensional cells in the orbit space of $X_{2}$ with respect to the action of $\Gamma$. The orbit space is a $2$-simplex, i.e. $\Lambda_{0}$ and $\Lambda_{1}$ contain $3$ elements and $\Lambda_{2}$ is a singleton. Furthermore $\Gamma_{\sigma}$ is the isotropy group of a chosen representative in $X_{2}$ of the cell $\sigma$ in the quotient space. For fixed $s$ all $s$-dimensional cells have isomorphic isotropy groups because the $\Gamma$-action on the Bruhat-Tits building is the restriction of a natural action of $GL_{3}(\mathbb{Z}[\frac{1}{2},i])$ 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 $SL_{3}(\mathbb{Z}[i])$ which itself appears as isotropy group of a [math]-dimensional cell in $X_{2}$. The isotropy groups of $1$-dimensional and $2$-dimensional cells are isomorphic to well-known congruence subgroups of $SL_{3}(\mathbb{Z}[i])$. By the Soulé-Lannes method the fibre $X_{\infty}$ of the projection map $p$ admits a $6$-dimensional $SL_{3}(\mathbb{Z}[i])$-equivariant deformation retract (the space of “well-rounded hermitean forms” modulo arithmetic equivalence) with compact quotient (cf. [Ash]) and therefore we have

[TABLE]

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 $SL(3,\mathbb{Z}[\frac{1}{2}])$ (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.

5. 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 $e_{2k-1},e_{2k-1}^{\prime}$ figuring in part c) will be introduced in (5.1) below.

Theorem 5.1**.**

Suppose $n\geq 2$. Then the following statements are equivalent.

a) $C(n,\mathbb{Z}[\frac{1}{2},i],2)$ holds, i.e. $H^{*}(GL_{n}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})$ is a free module over $\mathbb{Z}/2[c_{1},\ldots,c_{n}]$ where the $c_{i}$ are the mod-$2$ Chern classes of the tautological $n$-dimensional complex representation of $GL_{n}(\mathbb{Z}[\frac{1}{2},i])$.

b) The restriction homomorphism $H^{*}(GL_{n}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})\to H^{*}(D_{n}(\mathbb{Z}[\frac{1}{2},i]);\mathbb{F}_{2})$ is injective where $D_{n}(\mathbb{Z}[\frac{1}{2},i])$ denotes the subgroup of diagonal matrices in $GL_{n}(\mathbb{Z}[\frac{1}{2}])$.

c) There are isomorphisms

[TABLE]

where the classes $c_{k}$ are the Chern classes of the tautological $n$-dimensional complex representation of $GL_{n}(\mathbb{Z}[\frac{1}{2},i])$ and the classes $e_{2k-1},e_{2k-1}^{\prime}$ are of cohomological degree $2k-1$ for $k=1,\ldots,n$.

Proof.

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

In order to show that (a) implies (b) we observe that $D_{n}(\mathbb{Z}[\frac{1}{2},i])$ is the centralizer of the unique, up to conjugacy, maximal elementary abelian $2$-subgroup $E_{n}$ of $GL_{n}(\mathbb{Z}[\frac{1}{2},i])$ given by the subgroup of diagonal matrices of order $2$. Now consider the top Dickson invariant $\omega$ in $H^{*}(BGL_{n}(\mathbb{C});\mathbb{F}_{2})$, i.e. the class whose restriction to $H^{*}B(\prod_{i=1}^{n}GL_{1}(\mathbb{C}));\mathbb{F}_{2})$ is the product of all non-trivial classes of degree $2$. The image of $\omega$ in $H^{*}(GL_{n}(\mathbb{Z}[{1\over 2},i]);\mathbb{F}_{2})$ restricts trivially to the cohomology of all elementary abelian $2$-subgroups $E$ of $GL_{n}(\mathbb{Z}[{1\over 2},i])$ of rank less than $n$. If (a) holds then the image of $\omega$ is not a zero divisor in $H^{*}(GL_{n}(\mathbb{Z}[{1\over 2},i]);\mathbb{F}_{2})$ and hence Corollary I.5.8 of [HLS] implies that the restriction to the centralizer of $E_{n}$ is injective.

The implication $(b)\Rightarrow(c)$ follows from Proposition 5.3 below. ∎

Before we go on we introduce the classes $e_{2k-1}$ and $e_{2k-1}^{\prime}$. As in the case of $GL_{2}$ they are obtained from Quillen’s classes $q_{2k-1}\in H^{2k-1}(GL_{n}(\mathbb{F}_{5});\mathbb{F}_{2})$ [Q2] which restrict in the cohomology of diagonal matrices in $\mathbb{F}_{5}$ to the symmetrization of the class $y_{1}\ldots y_{k-1}x_{k}$ where $y_{k}$ is of cohomological degree $2$ corresponding to the $k$-th factor in the product $\prod_{k=1}^{n}\mathbb{F}_{5}^{\times}$ and $x_{k}$ is of cohomological degree $1$ of the same factor. Then we define

[TABLE]

where $\pi,\pi^{\prime}$ are the two ring homomorphisms $\mathbb{Z}[\frac{1}{2},i]\to\mathbb{F}_{5}$ with $\pi$ sending $i$ to $3$ and $\pi^{\prime}$ sending $i$ to $2$ which we considered earlier in section 3. If we identify the mod-$2$ cohomology $H^{*}(D_{n}(\mathbb{Z}[{1\over 2},i]);\mathbb{F}_{2})$ with $\mathbb{F}_{2}[y_{1},\ldots y_{n}]\otimes E(x_{1},x_{1}^{\prime}\ldots,x_{n},x_{n}^{\prime})$ with $y_{k}$, $k=1,\ldots,n$ of degree $2$ and $x_{k},x_{k}^{\prime}$, $k=1,\ldots,n$ of degree $1$ where as before we choose $x_{k}$ and $x_{k}^{\prime}$ to be the basis which is dual to the basis of the $k$-th factor in

[TABLE]

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 $H^{*}(D_{n}(\mathbb{Z}[\frac{1}{2},i];\mathbb{F}_{2}))$ to the symmetrization of $y_{1}\ldots y_{k-1}x_{k}$ and the class $e_{2k-1}^{\prime}$ restricts to the symmetrization of $y_{1}\ldots y_{k-1}(x_{k}+x_{k}^{\prime})$. ∎

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 $GL_{n}(\mathbb{Z}[{1\over 2}])$ for $p=2$ and of Anton [An1] for $GL_{n}(\mathbb{Z}[{1\over 3},\zeta_{3}])$ for $p=3$.

Proposition 5.3**.**

Let $n\geq 1$ be an integer. The image of the restriction map

[TABLE]

is isomorphic to

[TABLE]

Here we have identified the Chern classes $c_{i}$ and the classes $e_{2i-1}$ and $e_{2i-1}^{\prime}$ 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 $e_{2i-1}^{\prime}$ have been determined in Lemma 5.2. We remark that even though $i^{*}$ need not be injective, it is injective on the subalgebra of $H^{*}(GL_{n}(\mathbb{Z}[{1\over 2},i]);\mathbb{F}_{2})$ generated by the classes $c_{i}$, $e_{2i-1}$ and $e_{2i-1}^{\prime}$, $1\leq i\leq n$.

This proposition is an analogue of Proposition 3.6 of [An2]. Its proof uses crucially condition (5.3) below, which also plays a central role in [An2].

Proof.

In this proof we denote the subalgebra

[TABLE]

of $H^{*}(D_{n}(\mathbb{Z}[{1\over 2},i]);\mathbb{F}_{2})$ by $C_{n}$ and the image of the restriction map by $B_{n}$. We need to show that $B_{n}=C_{n}$. This is trivial if $n=1$ and for $n=2$ this follows from Theorem 1 of [W] (cf. (3.2) and (3.8) and Lemma 5.2).

The classes $c_{1},\ldots,c_{n}$ are in $B_{n}$ as images of the Chern classes with the same name and the classes $e_{1},\ldots e_{2n-1}$, $e_{1}^{\prime},\ldots e_{2n-1}^{\prime}$ are in $B_{n}$ by Lemma 5.2. Therefore we have $C_{n}\subset B_{n}$. We will show $B_{n}\subset C_{n}$ for $n\geq 2$ by induction on $n$. This will be done in three steps.

1. From the inclusions

[TABLE]

given by matrix block sum and the identifications of $D_{n-2}({{\mathbb{Z}[{1\over 2},i]}})\times D_{2}({{\mathbb{Z}[{1\over 2},i]}})$ with $D_{n}({{\mathbb{Z}[{1\over 2},i]}})$ and of $D_{n-1}({{\mathbb{Z}[{1\over 2},i]}})\times D_{1}({{\mathbb{Z}[{1\over 2},i]}})$ with $D_{n}({{\mathbb{Z}[{1\over 2},i]}})$ we see that

[TABLE]

and by induction hypothesis the latter subalgebra is equal to

[TABLE]

in particular we have

[TABLE]

2. The monomial basis in

[TABLE]

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

[TABLE]

where the $a_{i}$ are integers $\geq 0$ and $\varepsilon_{i,j}\in\{0,1\}$ for $i=1,2$ and $1\leq j\leq n$. More precisely to $I$ we associate the monomial

[TABLE]

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

In the sequel we replace the symmetrizations of the elements $y_{1}\ldots y_{i-1}(x_{i}+x_{i}^{\prime})$, $i=1,\ldots,n$, by the symmetrization of $y_{1}\ldots y_{i-1}x_{i}^{\prime}$ and by abuse of notation we continue to denote them by $e_{2i-1}^{\prime}$. This does not change the subalgebra $C_{n}$. This subalgebra

[TABLE]

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

[TABLE]

where the $k_{i}$ are integers $\geq 0$ and $\phi_{i,j}\in\{0,1\}$ for $i=1,2$ and $1\leq j\leq n$. More precisely to $K$ we associate the monomial

[TABLE]

We define a map

[TABLE]

by associating to $K\in T(n)$ the largest monomial in $S(n)$ which occurs in the decomposition of $c^{K}$ as linear combination of elements $x^{I}$ with $I\in S(n)$. The proof of the following result is elementary and is left to the reader.

Lemma 5.4**.**

The map $\alpha$ is explicitly given by

[TABLE]

with

[TABLE]

From this lemma it is obvious that $\alpha$ is injective and a sequence

[TABLE]

is in the image of $\alpha$ if and only if we have

[TABLE]

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 $I\in S(n)$ satisfies (5.3) for all $1\leq j<n$. Likewise, if $x$ is in $C_{i}\otimes C_{n-i}$ 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 $I\in S(k)\times S(n-k)$ and hence it satisfies (5.3) for all $1\leq j<i$ and $i+1\leq j<n$.

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 $I\in S(n)$. By (5.2) we have $x\in C_{n-1}\otimes C_{1}$ and $x\in C_{n-2}\otimes C_{2}$, and $I_{0}$ remains the maximal sequence in $S(n-1)\times S(1)$ resp. $S(n-2)\times S(2)$ appearing in the decomposition of $x$ as a linear combination of the monomials $x^{I}$ with $I\in S(n-1)\times S(1)$ resp. $I\in S(n-2)\times S(2)$. Hence $I_{0}$ satisfies conditions (5.3) for $1\leq j<n-1$ resp. $1\leq j<n-2$ and $j=n-1$. In particular condition condition (5.3) holds for all $1\leq j<n$ and therefore there exists $K_{0}\in T(n)$ such that $\alpha(K_{0})=I_{0}$. Then $x-c^{K_{0}}$ is still in $B_{n}$ and the maximal sequence appearing in the decomposition of $x-c^{K_{0}}$ is smaller than that of $x$. By iterating this procedure we see that $x$ belongs to $C_{n}$. ∎

Finally we relate $C(3,\mathbb{Z}[\frac{1}{2},i],2)$ to the behaviour of the restriction homomorphism

[TABLE]

For this we observe that the subgroups $\Gamma=SL_{3}(\mathbb{Z}[\frac{1}{2},i])$ and the center $Z\cong\mathbb{Z}[\frac{1}{2},i]^{\times}$ of $GL_{3}(\mathbb{Z}[\frac{1}{2},i])$ have trivial intersection and their product is the kernel of the homomorphism

[TABLE]

given as the composition of the determinant with the natural quotient map. Therefore the spectral sequence of the extension

[TABLE]

gives an isomorphism

[TABLE]

Proposition 5.5**.**

The conjecture $C(3,\mathbb{Z}[{1\over 2},i],2)$ holds if and and only if either

a) $H^{*}(SL_{3}(\mathbb{Z}[{1\over 2},i]);\mathbb{F}_{2})\cong\mathbb{F}_{2}[b_{2},b_{3}]\otimes E(d_{3},d_{3}^{\prime},d_{5},d_{5}^{\prime})$ or

b) the kernel of the map $\psi$ of Theorem 1.2 is a finite dimensional vector space for which the action of $\mathbb{Z}/3\cong(\mathbb{Z}[\frac{1}{2},i])^{\times}/(\mathbb{Z}[\frac{1}{2},i])^{\times})^{3}$ has trivial invariants.

Proof.

The quotient $\mathbb{Z}/3\cong(\mathbb{Z}[\frac{1}{2},i])^{\times}/(\mathbb{Z}[\frac{1}{2},i])^{\times})^{3}$ acts clearly trivially on $H^{*}(Z;\mathbb{F}_{2})$ and on the image of the homomorphism $\varphi$ of Theorem 1.2. Hence, the corollary follows immediately from (5.4) and Theorem 1.2. ∎