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Behrooz Mirzaii

Publications and source records attributed to Behrooz Mirzaii.

26 records · Page 2Linked to original sources
arXiv2011

Third homology of SL_2 and the indecomposable K_3

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 α: H_3(SL_2(F),Z)_F* --> K_3(F)^ind. Suslin has raised the question that, is αan 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 αis 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.

math.KT↗
arXiv2007

Third homology of general linear groups

The third homology group of GL_n(R) is studied, where R is a `ring with many units' with center Z(R). The main theorem states that if K_1(Z(R))_Q \simeq K_1(R)_Q, (e.g. R a commutative ring or a central simple algebra), then H_3(GL_2(R), Q) --> H_3(GL_3(R), Q) is injective. If R is commutative, Q can be replaced by a field k such that 1/2 is in k. For an infinite field R (resp. an infinite field R such that R*=R*^2), we get a better result that H_3(GL_2(R), Z[1/2] --> H_3(GL_3(R), Z[1/2]) (resp. H_3(GL_2(R), Z) --> H_3(GL_3(R), Z)) is injective. As an application we study the third homology group of SL_2(R) and the indecomposable part of K_3(R).

math.KT↗
arXiv2007

Homology of GL_n over algebraically closed fields

In this paper we define higher pre-Bloch groups p_n(F) of a field F. When our base field is algebraically closed we study its connection to the homology of the general linear groups with finite coefficient Z/l where l is a positive integer. As a result of our investigation we give a necessary and sufficient condition for the map H_n(GL_{n-1}(F), Z/l) --> H_n(GL_{n}(F), Z/l)$ to be bijective. We prove that this map is bijective for n < 5. We also demonstrate that the divisibility of p_n(C) is equivalent to the validity of the Friedlander-Milnor Isomorphism Conjecture for (n+1)-th homology of GL_n(C).

math.KT↗
arXiv2006

Homology of SL_n and GL_n over an infinite field

The homology of GL_n(F) and SL_n(F) is studied, where F is an infinite field. Our main theorem states that the natural map H_4(GL_3(F), k) --> H_4(GL_4(F), k) is injective where k is a field with char(k) \neq 2, 3. For algebraically closed field F, we prove a better result, namely, H_4(GL_3(F), Z) --> H_4(GL_4(F), Z) is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K_4(F).

math.KT↗
arXiv2001

Homology stability for unitary groups

In this paper homology stability for unitary groups over a ring with finite unitary stable rank is established. Homology stability of symplectic groups and orthogonal groups appears as a special case of our results.

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