Third homology of general linear groups

The third homology group of GLn(R) is studied, where R is a ‘ring with many units’ with center Z(R). The main theorem states that if K1(Z(R))⊗Q≃K1(R)⊗Q (e.g. R a commutative ring or a central simple algebra), then H3(GL2(R),Q)→H3(GL3(R),Q) is injective. If R is commutative, Q can be replaced by a field k such that 1/2∈k. For an infinite field R (resp. an infinite field R such that R*=R*2), we get the better result that (resp. H3(GL2(R),Z)→H3(GL3(R),Z)) is injective. As an application we study the third homology group of SL2(R) and the indecomposable part of K3(R).