The extended Bloch group and algebraic K-theory

Abstract

We define an extended Bloch group for an arbitrary field F, and show that this group is canonically isomorphic to K_3^ind(F) if F is a number field. This gives an explicit description of K_3^ind(F) in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3-manifold with finite volume and invariant trace field k has a fundamental class in K_3^ind(k) tensor Z[1/2].