The low dimensional homology groups of the elementary group of rank two

Abstract: In this article we study the first, the second and the third homology groups of the elementary group $\textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that $-1\in (A^{\times})^2$ or $|A^{\times}/(A^{\times})^2|\leq 4$.