Third homology of general linear groups over rings with many units (Q420737)

A. Suslin proved that for an infinite field \(F\) one has the homological stability: NEWLINE\[NEWLINE H_{n}(GL_{n}(F),{\mathbb Z})\cong H_{n}(GL_{n+1}(F),{\mathbb Z})\cong H_{n}(GL_{n+2}(F),{\mathbb Z})\cdots NEWLINE\]NEWLINE and conjectured that the kernel of NEWLINE\[NEWLINE H_{n}(GL_{n-1}(F),{\mathbb Z})\rightarrow H_{n}(GL_{n}(F),{\mathbb Z})NEWLINE\]NEWLINE is a torsion group. This conjecture has a positive answer for \(n\leq 4.\) It is also known that the kernel of the homomorphism \( H_{3}(GL_{2}(F),{\mathbb Z})\rightarrow H_{3}(GL_{3}(F),{\mathbb Z})\) is a \(2\)-power torsion group. In the paper the author generalizes this result to the ring \(R\) with many units and applies it to study the indecomposable part of \(K_{3}(R).\)