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Register a new userA note on third homology of GL_2
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Authors
B. Mirzaii
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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.
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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 $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$.
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).
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.
Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by extended cyclic homology, which we define and compute here, based on the extended noncommutative de Rham complex introduced previously by the authors.
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