Homology of GL n over algebraically closed fields

Abstract

In this paper we define higher pre-Bloch groups 𝔭n(F) of a field F. When the base field is algebraically closed, we study its connection to the homology of the general linear groups with coefficients in ℤ/l ℤ, where l is a positive integer. As a result of our investigation we give a necessary and sufficient condition for the natural map Hn(GLn−1(F), ℤ/l ℤ) → Hn(GLn(F), ℤ/l ℤ) to be bijective. We prove that this map is bijective for n⩽4. We also demonstrate that a certain property of 𝔭n(ℂ) is equivalent to the validity of the Friedlander–Milnor isomorphism conjecture for (n+1)th homology of GLn(ℂ).