A combinatorial invariant for Spherical CR structures

TL;DR

This paper introduces a new invariant for spherical CR 3-manifolds derived from cross-ratios of points in $S^3$, linking geometric configurations to algebraic structures like the pre-Bloch group and Bloch group.

Contribution

It constructs a homomorphism from configurations of four points in $S^3$ to the pre-Bloch group and defines a new invariant for spherical CR manifolds, revealing its torsion properties in specific cases.

Findings

The invariant is zero when applying the Bloch-Wigner function.

Under certain conditions, the invariant lies in the Bloch group $ ext{B}(k)$.

For the Whitehead link complement, the invariant is a non-trivial torsion.

Abstract

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 $\mathcal B(k)$.