The extended Bloch group and algebraic K-theory

Abstract

Abstract We define an extended Bloch group for an arbitrary field F, and show that this group is naturally isomorphic to K 3 ind ( F ) $K_3^{\operatorname{ind}}(F)$ if F is a number field. This gives an explicit description of K 3 ind ( F ) $K_3^{\operatorname{ind}}(F)$ 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 [ M ] $[M]$ in K 3 ind ( k ) ⊗ ℤ [ 1 2 ] $K_3^{\operatorname{ind}}(k)\otimes \mathbb {Z}[\frac{1}{2}]$ . If M is closed, 2 [ M ] $2[M]$ is well defined in K 3 ind ( k ) $K_3^{\operatorname{ind}}(k)$ , and if M has cusps, 8 [ M ] $8[M]$ is well defined in K 3 ind ( k ) $K_3^{\operatorname{ind}}(k)$ .