Pre-Bloch invariants of 3-manifolds with boundary

Let M be an oriented hyperbolic 3-manifold with finite volume. In [W.D. Neumann, J. Yang, Bloch invariants of hyperbolic 3-manifolds, Duke Math. J. 96 (1999) 29–59. [9]], Neumann and Yang defined an element β ( M ) of Bloch group B ( C ) for M. For this β ( M ) , volume and Chern–Simons invariant of M is represented by a transcendental function. In this paper, we define β ( M , ρ , C , o ) ∈ P ( C ) for an oriented 3-manifold M with boundary, a representation of its fundamental group ρ : π 1 ( M ) → PSL ( 2 , C ) , a pants decomposition C of ∂M and an orientation o on simple closed curves of C. Unlike in the case of finite volume, we construct an element of pre-Bloch group P ( C ) , and we need essentially the pants decomposition on the boundary. The volume makes sense for β ( M , ρ , C , o ) and we can describe the variation of volume on the deformation space.