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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 central perfect extension of groups $A rightarrowtail G twoheadrightarrow Q$, we study the maps $H_3(A,mathbb{Z}) to H_3(G, mathbb{Z})$ and $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$ provided that $Asubseteq G$. First we show that the image of $H
In this paper the third homology group of the linear group GL_2(R) with integral coefficients is investigated, where R is a commutative ring with many units.
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).
We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham diffe
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.