Third homology of general linear groups

Abstract. 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 a better result that H3(GL2(R),Z [ 12]) → H3(GL3(R),Z [ 12]) (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). 1