Stable homology of arithmetic groups

Question

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Suppose that $F/Q$ is a number field.

Using automorphic forms, Borel computed the ($R$-) stable cohomology of $\SL_n(O_F)$, and as a result, computed $K_i(O_F)\otimes Q$. Nowadays, using work of Suslin, Voevodksy, Rost, etc, one has an almost "complete" understanding of the actual integral groups $K_i(Z)$, say, modulo Vandiver's conjecture. This does not directly give the stable cohomology of $\SL_n$, because one set of groups is computing homotopy and the other is computing cohomology, but let us not worry about that distinction for the moment.

Borel also computed the ($R$-) stable cohomology of $\Sp_{2n}(O_F)$. My question, loosely speaking, is whether one can describe the stable integral cohomology of $\Sp_{2n}(Z)$ in as detailed away as the algebraic K-groups of $Z$ "describe" the integral cohomology of $\SL_n(Z)$.

---

Let me summarize some of what I have found out (following up some of the answers below), mostly though emails from experts.

For affine objects, which certainly includes $Z$, K-theory is about the monoidal category $P(A)$ of projective finitely generated A-modules, and Hermitian K-theory is about the monoidal category $P(A)_h$ of objects in $P(A)$ equipped with a non-degenerate symmetric (or skew-symmetric) form. One of the issues with computing or working with such a theory over $Z$ is that irritating issues arise in characteristic $2$, as one might expect with quadratic forms present. It seems that one might be in good shape to understand the groups $K^h_i(Z[1/2])$. For usual K-theory, there is an excision formula relating $K_i(Z)$ to $K_i(Z[1/2])$ and $K_i(F_2)$. The latter group is "easy" (or at least was computed by Quillen).

Of interest to me in $K_i(Z)$ are the Soulé classes. The next thing I will be thinking about is whether Soule classes can give rise to elements in the stable cohomology of $\Sp_2n(Z)$.

Andy said some very interesting things, but I will probably be awarding the bounty to Oscar, since the paper he linked to was more directly relevant to what I was trying to find out. (Sorry Andy...but it looks like you have lots of reputation anyway!)

Answer 1

The algebraic K-groups of Z are to the homology of SL_n(Z) as the Hermitian K-groups of Z are to the homology of Sp_2g(Z). There is a paper by Berrick and Karoubi here in which they discuss, and make some calculations of, the Hermitian K-theory of Z and Z[1/2].

Answer 2

It is known that $H_2$ of $Sp_{2g}(Z)$ is $Z$ for $g$ at least 4 and $Z+(Z/2Z)$ for $g=3$. The latter calculation plus references for the first calculation can be found in the following paper.

• MR0372056 (51 #8273) Stein, Michael R. The Schur multipliers of ${\rm Sp}_{6}(Z)$, ${\rm Spin}_{8}(Z)$, ${\rm Spin}_{7}(Z)$, and $F_{4}(Z)$. Math. Ann. 215 (1975), 165–172.

Also, it is easy to see that $Sp_{2g}(Z)$ is perfect for $g$ at least 2, so $H_1(Sp_{2g}(Z);Z)=0$.

One can then calculate $H^2$ and $H^3$ integrally using the universal coefficients theorem together with their rational values.

Answer 3

I don't know what $H_3(Sp_{2g}(Z))$ is, but I want to give a reference for a related fact. Let $Mod_g$ be the mapping class group. The action on $H_1$ of the surface preserves the algebraic intersection form and gives a representation $Mod_g\to Sp_{2g}(Z)$ which is well-known to be surjective. Rationally, the stable cohomology of $Sp_{2g}(Z)$ injects into the stable cohomology of $Mod_{g}$ and gives "one half" of the stable cohomology of $Mod_{g}$.

Anyway, I do know a reference for some stable integral homology calculations for the mapping class group. Namely, in The low-dimensional homotopy of the stable mapping class group, Ebert proves that in a stable range we have $H_3$ of the mapping class group equal to $Z/12Z$ and $H_4$ equal to $Z^2$. He does this by using the fact that Madsen and Weiss's work identifies the infinite loop space $(Mod_{\infty})^+$ with a space whose first few homotopy groups are known. Ebert then builds with his bare hands the first few stages of the Postnikov tower for this space.

Presumably this either has been done or could be done for $Sp_{2g}(Z)$ by a sufficiently motivated homotopy theorist.