A refined scissors congruence group and the third homology of SL2.

There is a natural connection between the third homology of SL 2 (A) and the refined Bloch group RB (A) of a commutative ring A. In this article we investigate this connection and as the main result we show that if A is a universal GE 2 -domain such that − 1 ∈ A × 2 , then we have the exact sequence H 3 (SM 2 (A) , Z) → H 3 (SL 2 (A) , Z) → RB (A) → 0 , where SM 2 (A) is the group of monomial matrices in SL 2 (A). Moreover, we show that RP 1 (A) , the refined scissors congruence group of A , is naturally isomorphic to the relative homology group H 3 (SL 2 (A) , SM 2 (A) ; Z). [ABSTRACT FROM AUTHOR]