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 is in 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[1/2] --> H3(GL3(R), Z[1/2]) (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).
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