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 H2left(Sp(F),mathbbZright)=H2left(SL(2,F),mathbbZright)H2left(Sp(F),mathbbZright)=H2left(SL(2,F),mathbbZright) for an infinite field FF as Milnor—Witt K-theory KMW2(F)KMW2(F), introduced by F. Morel in 2003 in his study of mathbbA1mathbbA1-homotopy theory.
KMW∗(F)KMW∗(F) is a graded associative ring generated by the symbols [u][u], uinF∗uinF∗ of degree +1+1 and one symbol etaeta of degree −1−1 modulo the following relations:
- For ainFsetminus0,1ainFsetminus0,1, [a]cdot[1−a]=0[a]cdot[1−a]=0;
- For a,b,inF∗a,b,inF∗, [ab]=[a]+[b]+eta[a][b][ab]=[a]+[b]+eta[a][b];
- For uinF∗uinF∗, [u]eta=eta[u][u]eta=eta[u];
- eta2[−1]+2eta=0eta2[−1]+2eta=0.
The proof is based on Matsumoto—Moore presentation for H2left(Sp(F),mathbbZright)H2left(Sp(F),mathbbZright) and the coincidence of KMW2(F)KMW2(F) with KMM2(F)KMM2(F).
PS. The equality H2left(Sp(F),mathbbZright)=H2left(SL(2,F),mathbbZright)H2left(Sp(F),mathbbZright)=H2left(SL(2,F),mathbbZright) has something to do with mathsfA1=mathsfC1mathsfA1=mathsfC1 (see this MO question).
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