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

In this article, we study the first, second, and third homology groups of the elementary group E2(A), where A is a commutative ring. In particular, we establish a refined Bloch–Wigner type exact sequence over a semilocal ring (subject to mild restrictions on its residue fields) in which either -1∈(A×)2 or |A×/(A×)2|≤4.