Third homology of general linear groups over rings with many units

TL;DR

This paper investigates the third homology groups of general linear groups over rings with many units, identifying the kernel of a specific homology map and its elements' order, with applications to algebraic K-theory.

Contribution

It describes the kernel of the homology map between GL_2 and GL_3 over rings with many units and analyzes the order of its elements, advancing understanding of K_3(R).

Findings

Kernel of H_3(GL_2(R)) to H_3(GL_3(R)) map identified

Elements in the kernel have order at most two

Application to the indecomposable part of K_3(R)

Abstract

For a commutative ring R with many units, we describe the kernel of H_3(inc): H_3(GL_2(R), Z) --> H_3(GL_3(R), Z). Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable part of K_3(R).