In this article we study the homology of nilpotent groups. In particular a certain vanishing result for the homology and cohomology of nilpotent groups is proved.
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.
For a commutative ring R with many units, we describe the kernel of H_3(inc): H_3(GL_2(R), Z) --> H_3(GL_3(R), Z). Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable part of K_3(R).
In this article we prove a generalization of the Bloch-Wigner exact sequence over commutative rings with many units. When the ring is a domain, we get a generalization of Suslins Bloch-Wigner exact sequence over infinite fields. Our proof is different and is easier, even in its general form. But nevertheless we use some of Suslins results which relates the Bloch group of the ring to the third homology group of the general linear group of the ring. From there we take an easier path.
