Homology of GL _n over algebraically closed fields

Abstract

Abstract. In this paper we define higher pre-Bloch groups pn(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/lZ where l is a positive integer. As a result of our investigation we give a necessary and sufficient condition for the map Hn(GLn−1(F ), Z/lZ) → Hn(GLn(F ), Z/lZ) to be bijective. We prove that this map is bijective for n ≤ 4. We also demonstrate that the divisibility of pn(C) is equivalent to the validity of the Friedlander-Milnor Isomorphism Conjecture for (n + 1)-th homology of GLn(C).