A combinatorial invariant for Spherical CR structures

We study a cross-ratio of four generic points of $S^3$ which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in $S^3$ to the pre-Bloch group $\mathcal {P}(\C)$. If $M$ is a $3$-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a $\mathcal {P}(\C)$-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 $\mathcal B(k)$, where $k$ is the field generated by the cross-ratio. For a CR triangulation of Whitehead link complement, we show its invariant is a non-trivial torsion in $\mathcal B(k)$.