algebraic k theory - Symplectic Steinberg group

Let me do a bit of necromancy here and address the third question.

A Note on Milnor–Witt K-Theory and a Theorem of Suslin by K. Hutchinson and L. Tao provides a description for $H_2left(Sp(F),mathbb{Z}right)=H_2left(SL(2,F),mathbb{Z}right)$ for an infinite field $F$ as Milnor—Witt K-theory $K_2^{MW}(F)$, introduced by F. Morel in 2003 in his study of $mathbb{A}^1$-homotopy theory.

$K_*^{MW}(F)$ is a graded associative ring generated by the symbols $[u]$, $uin F^*$ of degree $+1$ and one symbol $eta$ of degree $-1$ modulo the following relations:

• For $ain Fsetminus{0,1}$, $[a]cdot[1-a]=0$;


• For $a,b,in F^*$, $[ab]=[a]+[b]+eta[a][b]$;


• For $uin F^*$, $[u]eta=eta[u]$;


• $eta^2[-1]+2eta=0$.

The proof is based on Matsumoto—Moore presentation for $H_2left(Sp(F),mathbb{Z}right)$ and the coincidence of $K_2^{MW}(F)$ with $K_2^{MM}(F)$.

PS. The equality $H_2left(Sp(F),mathbb{Z}right)=H_2left(SL(2,F),mathbb{Z}right)$ has something to do with $mathsf{A}_1=mathsf{C}_1$ (see this MO question).