For an infinite field $F$, we study the cokernel of the map of homology groups $H_{n+1}(mathrm{GL}_{n-1}(F),mathbb{k}) to H_{n+1}(mathrm{GL}_{n}(F),mathbb{k})$, where $mathbb{k}$ is a field such that $(n-2)!in mathbb{k}^times$, and the kernel of the
natural map $H_{n}big(mathrm{GL}_{n-1}(F),mathbb{Z}big[frac{1}{(n-2)!} big] big) to H_{n}big(mathrm{GL}_{n}(F),mathbb{Z}big[frac{1}{(n-2)!} big]big)$. We give conjectural estimates of these cokernel and kernel and prove our conjectures for $nleq 4$.
In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N$ is $2(c(n-1)-1)$-tame as a
$G/N$-module, $c$ the nilpotency class of $N$, then $mathrm{vb}_j(G):=sup_{Minmathcal{A}_G}dim_mathbb{Q} H_j(M,mathbb{Q})$ is finite for all $0leq jleq n$, where $mathcal{A}_G$ is the set of all finite index subgroups of $G$.
For an abelian group $A$, we give a precise homological description of the kernel of the natural map $Gamma(A) to Aotimes_mathbb{Z} A$, $gamma(a)mapsto aotimes a$, where $Gamma$ is whiteheads quadratic functor from the category of abelian groups to itself.
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 $H
_3(A, mathbb{Z})to H_3(G, mathbb{Z})/rho_ast(Aotimes_mathbb{Z} H_2(G, mathbb{Z}))$ is $2$-torsion where $rho: A times G to G$ is the usual product map. When $BQ^+$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$.
Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3(mathrm{SL}_2(A),mathbb{Z}[frac{1}{2}]) to H_3(mathrm{SL}_2(F),mathbb{Z}[frac{1}{2}])to m
athcal{RP}_1(k)[frac{1}{2}]to 0$, where $mathcal{RP}_1(k)$ is the refined scissors congruence group of $k$. Let $Gamma_0(mathfrak{m}_A)$ denote the congruence subgroup consisting of matrices in $mathrm{SL}_2(A)$ whose lower off-diagonal entry lies in the maximal ideal $mathfrak{m}_A$. We also prove that there is an exact sequence $0to overline{mathcal{P}}(k)[frac{1}{2}]to H_2(Gamma_0(mathfrak{m}_A),mathbb{Z}[frac{1}{2}])to H_2(mathrm{SL}_2(A),mathbb{Z}[frac{1}{2}])to I^2(k)[frac{1}{2}]to 0$, where $I^2(k)$ is the second power of the fundamental ideal of the Grothendieck-Witt ring $mathrm{GW}(k)$ and $overline{mathcal{P}}(k)$ is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) $mathcal{P}(k)$ of $k$.
In this article we extend the Bloch-Wigner exact sequence over local rings, where their residue fields have more than nine elements. Moreover, we prove Van der Kallens theorem on the presentation of the second $K$-group of local rings such that their
residue fields have more than four elements. Note that Van der Kallen proved this result when the residue fields have more than five elements. Although we prove our results over local rings, all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings.
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 alp
ha 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 differen
t 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.
