classifying space and cohomology of integer general linear group

Question

I have obtained that the classifying space $$ BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty) $$ is the Grassmannian.

I have also obtained that the mod 2 cohomology is the polynomial algebra $$ H^*(BGL(\mathbb{R}^n);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_n]. $$

Let $GL(\mathbb{Z}^n)$ be the group consisting of all $n\times n$ matrices with integer entries such that the determinants are either $1$ or $-1$. What isthe classifying space $$BGL(\mathbb{Z}^n)?$$

What is the cohomology algebra $$ H^*(BGL(\mathbb{Z}^n);\mathbb{Z}_2)=? $$ Could you give any references? Thanks.

Answer 1

Let me give a more detailed answer, to give more context to the list of references. Unfortunately, these answers grow to encyclopedic size so easily...

Stable results: as mentioned in all the answers, the question becomes easier by stabilizing $GL_n(\mathbb{Z})$ to $GL_\infty(\mathbb{Z})$. In this case, one can use K-theory (or even motivic) methods. This is essentially what is done in the following papers:

• S.A. Mitchell. On the plus construction for $BGL(\mathbb{Z}[1/2])$ at the prime 2. Math. Z. 209 (1992), no. 2, 205–222.

• D. Arlettaz, M. Mimura, K. Nakahata, N. Yagita. The mod 2 cohomology of the linear groups over the ring of integers. Proc. Amer. Math. Soc. 127 (1999), no. 8, 2199–2212.

In particular, the last paper explicitly answers the stable question: there is a ring isomorphism $$ H^{\bullet}(BGL(\mathbb{Z})^+,\mathbb{Z}/2)\cong H^{\bullet}(BO,\mathbb{Z}/2)\otimes H^{\bullet}(SU,\mathbb{Z}/2) $$ (plus some more statements about Hopf algebra structure and Steenrod operations). This is based on explicit models for the 2-completed classifying space, eventually using Voevodsky's solution of the Milnor conjecture (resp. the Quillen-Lichtenbaum conjecture relating étale and algebraic K-theory).

I would expect that the result could actually be generalized to some extent, say to get some statements with odd torsion coefficients (using the Rost-Voevodsky solution of the Milnor-Bloch-Kato conjecture) or other rings of $S$-integers - but I am not aware of papers doing this.

Unstable results: The unstable case, the actual computation of $H^\bullet(GL_n(\mathbb{Z}),\mathbb{Z}/2)$ is more complicated, the K-theoretic or motivic methods do no longer work. Descriptions of an étale version of the classifying space can still be obtained (see the Topological models for arithmetic of Dwyer-Friedlander), but it is usually not possible to compare this to the actual classifying space.

The classifying space: the classifying space itself is not actually easy to describe or helpful. A better approximation to the classifying space is the locally symmetric space $GL_n(\mathbb{Z})\backslash GL_n(\mathbb{R})/O(n)$. There is more geometry to study this space, but $GL_n(\mathbb{Z})$ acts non-freely, with finite isotropy. Analysis of L^2-cohomology of this space is the basis of Borel's computation of rational cohomology of $GL_n(\mathbb{Z})$. With finite coefficients, the space is difficult to understand, cohomology is influenced by the finite subgroups and their normalizers as well as compactly supported cohomology of the locally symmetric space (which I would like to view as cusp forms with finite coefficients, but this is not mathematically precise).

Explicit computations: there are not so many. The computation of $H^\bullet(SL_2\mathbb{Z},\mathbb{Z}/2)$ is an exercise. Rank two, i.e. $SL_3$ resp. $GL_3$ is already a substantial work:

• C. Soulé. The cohomology of $SL_3(\mathbb{Z})$. Topology 17 (1978), no. 1, 1–22.

I am not aware of (complete all-degree) computations with higher rank - although Voronoi methods have been used to partially compute homology of $SL_n(\mathbb{Z})$ for $n\leq 8$, the small primes are usually excluded. I would expect that for $SL_4(\mathbb{Z})$ everything is controlled by the finite subgroups, but I could not make this sufficiently precise. See the paper of Dutour-Sikiric, Ellis,Schürmann or the paper of Elbaz-Vincent, Gangl, Soulé.

Quillen conjecture, exotic cohomology: to illustrate how much more complicated the unstable case is, compared to the stable case (although this is slightly different from the situation in the question). There is a conjecture of Quillen which in particular predicts that the restriction map $H^\bullet(GL_n\mathbb{Z}[1/2],\mathbb{Z}/2)\to H^\bullet(D_n(\mathbb{Z}[1/2]),\mathbb{Z}/2)$ is injective, where $D_n$ is the subgroup of diagonal matrices. This is true in low degree by work of Mitchell ($n=2$) and Henn ($n=3$), and stably (see the paper of Arlettaz et al. mentioned above). It is however false for $n\geq 32$ by

• W. Dwyer. Exotic cohomology for $GL_n(\mathbb{Z}[1/2])$. Proc. Amer. Math. Soc. 126 (1998), 2159--2167.

Further information can be found in K.P. Knudson: Homology of linear groups. Birkhäuser 2001.

Answer 2

As far as I know the answer is not known. If $R$ satisfies Bass' stable rank condition, e.g. if it is a field or Dedekind domain, the group homology $H_*(GL_n(R);A)$ with trivial coefficients $A$ stabilizes and the stable homology is equal to that of $K(R)$, the algebraic K-theory spectrum of $R$. These spectra are hard to understand, though some things are known in the case $R = \mathbb{Z}$ after a lot of work by great mathematicians. For example, the homotopy groups of $K(\mathbb{Z})$ are known except in degrees divisible by $4$. If you only care about the prime 2, the rational homotopy groups can be deduced from Borel's work and the 2-primary subgroups of the homotopy groups of $K(\mathbb{Z})$ are completely known, see Corollary VI.9.8 of Weibel's K-book.

In terms of (co)homology, here is one remark: I think it should be comparably hard to study $H^*(GL(\mathbb{Z});\mathbb{Z}/2\mathbb{Z})$ and $H^*(GL(\mathbb{Q});\mathbb{Z}/2\mathbb{Z})$. This is because we have Quillen's localization fiber sequence $\bigvee_p K(\mathbb{F}_p) \to K(\mathbb{Z}) \to K(\mathbb{Q})$ and his stable computation of the (co)homology of general linear groups over finite fields. He computes explicitly $H^*(GL_n(\mathbb{F}_q);\mathbb{F}_l)$ with $q$, $l$ coprime in theorem 3 of On the cohomology and K-theory of the general linear groups over a finite field (and stably he also computes it if $q$ and $l$ are not coprime). Alternatively, theorem 5 of that paper gives a computation of the stable homology as the homology of the homotopy fiber of a certain self-map of $BU$.

Answer 3

I'm pretty sure that the stable cohomology with $\mathbb{Z}/2$ coefficients is not known, but Borel calculated $H^{\ast}(\text{GL}(\mathbb{Z});\mathbb{Q})$ in

A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 235–272 (1975).