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 , 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.