Third homology of SL_2 and the indecomposable K_3

Details Third homology of SL_2 and the indecomposable K_3 Third homology of SL_2 and the indecomposable K_3

2011-08-30

arxiv.org/abs/1108.5900v1

Mathematics

K-Theory and Homology

10 pages, Latex

Scientific paper

It is known that, for an infinite field F, the indecomposable part of K_3(F) and the third homology of SL_2(F) are closely related. In fact, there is a canonical map \alpha: H_3(SL_2(F),Z)_F* --> K_3(F)^ind. Suslin has raised the question that, is \alpha an isomorphism? Recently Hutchinson and Tao have shown that this map is surjective. They also gave some arguments about its injectivity. In this article, we improve their arguments and show that \alpha is bijective if and only if the natural maps H_3(GL_2(F), Z)--> H_3(GL_3(F), Z) and H_3(SL_2(F), Z)_F* --> H_3(GL_2(F), Z) are injective.

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