TITLE: Symplectic K-theory
QUESTION [14 upvotes]: For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary matrices, let $\operatorname{BSp}(R)$ be a classifying space, let $\operatorname{BSp}(R)^+$ be the result of applying Quillen's $+$-construction (with respect to $\operatorname{ESp}(R)$) and define $\operatorname{KSp}_*(R) := \pi_*(\operatorname{BSp}(R)^+)$. [Is this the right definition?]

Is the following statement true (and is there a reference)?
Let $\tilde{\operatorname{Sp}}_{2n}(R)$ be the Steinberg group of $\operatorname{Sp}_{2n}(R)$. There is an exact sequence 
$$
1 \to  \operatorname{KSp}_2(R) \to \tilde{\operatorname{Sp}}_{2n}(R) \to \operatorname{Sp}_{2n}(R) \to 1
$$
for $n$ large enough. [Is the kernel always $\pi_2(\operatorname{BSp}_{2n}(R)^+)$?]
What is the right analogue of the ``fundamental theorem of K-theory'', i.e. how can $\operatorname{KSp}_*(R[t])$ and $\operatorname{KSp}_*(R[t,t^{-1}])$ be expressed in terms of $\operatorname{KSp}_*(R)$?

So essentially I am wondering what changes if one replaces $\operatorname{GL}(R)$ by $\operatorname{Sp}(R)$ in the definition of algebraic K-theory. I am mostly interested in the case where $R$ is a field. In that case $\operatorname{KSp}_1(R)$, $\operatorname{KSp}_1(R[t])$, and $\operatorname{KSp}_1(R[t,t^{-1}])$ all vanish at least if the characteristic is not $2$.

REPLY [9 votes]: Matthias has already pointed out that the map $\widetilde{\mathrm{ESp}}_{2n}(R)\rightarrow Sp_{2n}(R)$ need not be surjective. In fact, the same phenomenon arises already for $\mathrm{SL}_n$ (the image of $\widetilde{\mathrm{E}}(n, R) = \mathrm{St}(n, R)$ is exactly the elementary subgroup $\mathrm{E}(n, R)$ which is typically strictly smaller than $\mathrm{SL}(n, R)$).
I would also like to add that from Lavrenov's result it follows that an unstable version of Gersten's formula holds (i.e. $\mathrm{H}_3(\widetilde{\mathrm{ESp}}_{2n}(R), \mathbb{Z}) = \pi_3(B\mathrm{Sp}_{2n}(R)^+)$ for $n\geq 4$). The proof is the same as in the stable case (it is only relevant that $\widetilde{\mathrm{ESp}_{2n}}$ is a universal central extension of $\mathrm{ESp}_{2n}$).
There is a theorem by Grunewald, Mennicke and Vaserstein which states the following:

If $R$ is a "locally principal" ring (i.e. localization $R_\mathfrak{M}$ at every maximal ideal $\mathfrak{M}\triangleleft R$ is a principal ideal domain) then $\mathrm{K_1Sp}_{2n}(R[x_1,\ldots x_n]) = \mathrm{K_1Sp_{2n}}(R)$, $n\geq 2$.
If $R$ is a p.i.d. then $\mathrm{K_1Sp}_{2n}(R[x, x^{-1}])=\mathrm{K_1Sp}_{2n}(R)$, $n\geq 2$.

Both statements apply to a rather narrow class of rings (either assumption implies that Krull dimension of $R$ is $\leq 1$). There exist also theorems asserting triviality of $\mathrm{K_1Sp}_{2n}(R[t_1,\ldots, t_n])$ for certain classes of rings (e.g in this paper of Kopeiko such theorem is proved for rings of geometric type).