Third homology of general linear groups

Details Third homology of general linear groups Third homology of general linear groups

2003-12-22

arxiv.org/abs/math/0312409v2

Mathematics

K-Theory and Homology

24 pages, Latex, new title, some results are generalized, added references

Scientific paper

The third homology group of GL_n(R) is studied, where R is a `ring with many units' with center Z(R). The main theorem states that if K_1(Z(R))_Q \simeq K_1(R)_Q, (e.g. R a commutative ring or a central simple algebra), then H_3(GL_2(R), Q) --> H_3(GL_3(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 H_3(GL_2(R), Z[1/2] --> H_3(GL_3(R), Z[1/2]) (resp. H_3(GL_2(R), Z) --> H_3(GL_3(R), Z)) is injective. As an application we study the third homology group of SL_2(R) and the indecomposable part of K_3(R).

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