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September 2013 A combinatorial invariant for spherical CR structures
Elisha Falbel, Qingxue Wang
Asian J. Math. 17(3): 391-422 (September 2013).

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 $\mathcal{P}(\mathbb{C})$. If $M$ is a 3-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a $\mathcal{P}(\mathbb{C})$-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 the Whitehead link complement, we show its invariant is a torsion in $\mathcal{B}(k)$ and for a triangulation of the complement of the 52-knot we show that the invariant is not trivial and not a torsion element.

Citation

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Elisha Falbel. Qingxue Wang. "A combinatorial invariant for spherical CR structures." Asian J. Math. 17 (3) 391 - 422, September 2013.

Information

Published: September 2013
First available in Project Euclid: 8 November 2013

zbMATH: 1296.57013
MathSciNet: MR3119793

Subjects:
Primary: 19M05, 57M27, 57M50

Rights: Copyright © 2013 International Press of Boston

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Vol.17 • No. 3 • September 2013
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