Third homology of general linear groups

The third homology group of GL_n® is studied, where R is a `ring with many units’ with center Z®. The main theorem states that if K_1(Z®)_Q \simeq K_1®_Q, (e.g. R a commutative ring or a central simple algebra), then H_3(GL_2®, Q) --> H_3(GL_3®, Q) is injective. If R is commutative, Q can be replaced by a field k such that 1/2 is in k. For an infinite field R (resp. an infinite field R such that R*=R*^2), we get a better result that H_3(GL_2®, Z[1/2] --> H_3(GL_3®, Z[1/2]) (resp. H_3(GL_2®, Z) --> H_3(GL_3®, Z)) is injective. As an application we study the third homology group of SL_2® and the indecomposable part of K_3®.