The volume and Chern-Simons invariant of a representation
type: Item
instance of: scholarly article
title: The volume and Chern-Simons invariant of a representation
zbMATH Open document ID: 1246.58019
DOI: 10.1215/00127094-2009-058
author: Christian K. Zickert
published in: Duke Mathematical Journal
publication date: 2010-03-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/0710.2049
review text: The Cheeger-Chern-Simons invariant is a characteristic class of flat \(\mathrm{PSL}(2,\mathbb{C})\)-bundles. In the case of the canonical flat \(\mathrm{PSL}(2,\mathbb{C})\)-bundle of a closed hyperbolic \(3\)-manifold \(M\), its evaluation on the fundamental class is the complex volume, \(\text{Vol}(M)+i\,\text{CS}(M)\in\mathbb{C}/i\pi^2\mathbb{Z}\), where \(\text{CS}(M)=2\pi^2\text{cs}(M)\) for the Chern-Simons invariant \(\text{cs}(M)\). To study this invariant, by using the bundle map of the canonical flat \(\mathrm{PSL}(2,\mathbb{C})\)-bundle over \(M\) to the universal flat \(\mathrm{PSL}(2,\mathbb{C})\)-bundle over \(B(\mathrm{PSL}(2,\mathbb{C}))\) (endowing \(\mathrm{PSL}(2,\mathbb{C})\) with the discrete topology), it is enough to study the characteristic class of this universal bundle. This class lies in \(H^3(B(\mathrm{PSL}(2,\mathbb{C})),\mathbb{C}/\pi^2\mathbb{Z})\), and therefore can be considered as a homomorphism \(\hat c_2:H_3(\text{PSL}(2,\mathbb{C}))\to\mathbb{C}/\pi^2\mathbb{Z}\). Moreover \(M\) determines a fundamental class \([M]\in H_3(\mathrm{PSL}(2,\mathbb{C}))\) so that \(\hat c_2([M])\) is the complex volume multiplied by \(i\). In [Geom. Topol. 8, 413--474 (2004; Zbl 1053.57010)], \textit{W. D. Neumann} gave an explicit expression for \(\hat c_2\). For the determination of \([M]\), Neumann also constructed a group, \(\widehat{\mathcal{B}}(\mathbb{C})\), called extended Bloch group, which is isomorphic to \(H_3(\mathrm{PSL}(2,\mathbb{C}))\), and defined a map \(R:\widehat{\mathcal{B}}(\mathbb{C})\to\mathbb{C}/\pi^2\mathbb{Z}\) that corresponds to \(\hat c_2\), and an element in \(\widehat{\mathcal{B}}(\mathbb{C})\) that corresponds to \([M]\), obtaining a formula for the complex volume. This formula also applies to cusped hyperbolic \(3\)-manifolds. The goal of the present paper is to introduce a new way of obtaining the complex volume, which turns out to be more efficient for the numerical computations. It uses the subgroup \(P\) of upper…
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5688240
zbMATH Keywords: Cheeger-Chern-Simons invariant
zbMATH Keywords: Chern-Simons invariant
zbMATH Keywords: complex volume
zbMATH Keywords: hyperbolic \(3\)-manifold
zbMATH Keywords: tame \(3\)-manifold
zbMATH Keywords: \(\mathrm{PSL}(2,\mathbb{C})\)-representation
zbMATH Keywords: extended Bloch group
reviewed by: Jesús A. Álvarez López
describes a project that uses: Snap
MaRDI profile type: Publication
OpenAlex ID: W3105109878
arXiv ID: 0710.2049
cites work: On Culler-Shalen seminorms and Dehn filling
cites work: Real places and torus bundles
cites work: Plane curves associated to character varieties of 3-manifolds
cites work: A dilogarithmic formula for the Cheeger-Chern-Simons class
cites work: The extended Bloch group and the Cheeger-Chern-Simons class
cites work: TRACE FIELDS OF TWIST KNOTS
cites work: Q4789131
cites work: Q3759556
cites work: Q4028063
cites work: Extended Bloch group and the Cheeger-Chern-Simons class
cites work: Bloch invariants of hyperbolic \(3\)-manifolds
cites work: Volumes of hyperbolic three-manifolds
description: scientific article