Different definitions of unstable orthogonal K_2

Abstract: Many approaches to higher algebraic $\mathrm K$-theory and
Hermitian $\mathrm K$-theory are known. For example, stable Quillen's
groups $\mathrm K_n(R)$ (defined e.\,g. via the $+$-construction) and
stable Karoubi--Villamayor groups $\mathrm{KV}_n(R)$ (defined via standard
simplicial scheme) coincide for $n\geq 1$ if $R$ happens to be a regular
ring. These theories use infinite-dimensional algebraic groups such as
$\mathrm{GL}_\infty(R)$ in their definition. In this talk we will discuss
an {\it unstable} analogue of such result for the functor $\mathrm K_2$.

The interest for the unstable Quillen's $\mathrm K_2$-groups, in
particular, comes from the fact that they appear in Steinberg's
presentation of the groups of points of algebraic groups by means of
generators and relations. On the other hand,Karoubi--Villamayor $\mathrm
K_2$-groups can be interpreted as fundamental groups in the unstable
$\mathbb{A}^1$-homotopy category $\mathscr{H}_\bullet(k)$ of F.~Morel and
V.~Voevodsky (using results of A.\,Asok, M.\,Hoyois and M.\,Wendt).
Conjecturally, for any split simple group $G=\mathrm G(\Phi,R)$ with
$\mathrm{rk}\,\Phi\geq5$ and regular ring $R$ holds an equality
\begin{align}
\label{conj} \pi_1^{\mathbb A^1}(G)(R)=\pi_2(\mathrm BG^+).
\end{align}

We remark that the Nisnevich localization
$\mathrm{a_{Nis}}\,\pi_1^{\mathbb
A^1}(G)(R)$ of $\mathbb A^1$-fundamental groups was recently computed by
F.\,Morel and A.\,Sawant, and coincides with the unramified Milnor
$\mathrm{\underline{K}_2^M}$ or Milnor-Witt
$\mathrm{\underline{K}_2^{MW}}$ sheaf depending on $\Phi$.

Conjecture~(\ref{conj}) is parallel to the Serre's problem and
Bass--Quillen
conjecture, and we adopt Quillen--Suslin and Lindel--Popesque results for
this case. In particular, for $\Phi=\mathsf A_l,\,\mathsf D_l$ this
conjecture is already proven for a regular ring $R$ containing a field $k$
of characteristic $\neq 2$, $l\geq7$.

As a corollary, one can obtain the following results.
\begin{itemize}
\item The group $\mathrm{Spin}_{2l}(k[t_1,\ldots, t_n])$ admits an
explicit presentation by means of generators and relation (generalizing
Steinberg's presentation in the case $n=0$).
\item $\mathrm H_2(\mathrm{Spin}_{2l}(k[t_1,\ldots,t_n]),\,\mathbb Z\big)
= \mathrm K^\mathrm{M}_2(k).$
\item $\mathrm H_2 (\mathrm{O}_{2l}(R[t]), \mathbb Z) = \mathrm H_2
(\mathrm{O}_{2l}(R), \mathbb Z).$
\end{itemize}


The talk is based on my joint work with Sergey Sinchuk and Egor
Voronetsky.