Triviality of the product map μ2(A) ⊗ H2(SL2(A), Z) → H3(SL2(A), Z) over local domains with |A/mA| ≠ 2

Determine whether, for every local domain A whose residue field satisfies |A/mA| ≠ 2, the product map P*: p2(A) ⊗ H2(SL2(A), Z) → H3(SL2(A), Z) induced by the group multiplication p: p2(A) × SL2(A) → SL2(A), (a, X) ↦ aX, is trivial, where p2(A) denotes the subgroup of units of A consisting of 2-roots of unity and H2(·, Z), H3(·, Z) denote integral group homology groups.

Background

The paper studies the low-dimensional homology groups of SL2(A) and PSL2(A) via the central extension 1 → p2(A) → SL2(A) → PSL2(A) → 1 and associated spectral sequences. A key ingredient is the product map P*: p2(A) ⊗ Hi(SL2(A), Z) → Hi+1(SL2(A), Z) induced by the group multiplication p: p2(A) × SL2(A) → SL2(A). For local domains A with residue field size greater than two, the authors show this product map is trivial for i = 0 and i = 1 and ask whether the same holds for i = 2.

The authors note that they do not know the answer even for general infinite fields, but they establish triviality of P* for several important classes of fields (finite fields, quadratically closed fields, and real closed fields). Resolving the triviality in full generality impacts the exactness of the nine-term sequence connecting H3(SL2(A), Z) and H3(PSL2(A), Z) over local domains, and clarifies the role of the p2(A) ⊗ H2(SL2(A), Z) term in these exact sequences.