A combinatorial invariant for Spherical CR structures

We study a cross-ratio of four generic points of S which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in S to the pre-Bloch group P(C). If M is a 3-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a P(C)-valued invariant for M . We show that when applying to it the Bloch-Wigner function, it is zero. Under some conditions on M , we show the invariant lies in the Bloch group B(k), where k is the field generated by the cross-ratio. For a CR triangulation of Whitehead link complement, we show its invariant is a torsion in B(k) and for a triangulation of the complement of the 52-knot we show that the invariant is not trivial and not a torsion element.