The volume and Chern-Simons invariant of a representation
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 triangular matrices in \(\mathrm{PSL}(2,\mathbb{C})\) with \(1\) on the diagonal. First, the author shows that the relative homology group \(H_3(\mathrm{PSL}(2,\mathbb{C}),P)\) can be computed using a complex generated by ideal hyperbolic simplices endowed with a decoration consisting of a horosphere at each ideal vertex and an identification of the horosphere with \(\mathbb C\). This decoration endowes each ideal simplex with a flattening, which is used to define a map \(\Psi:H_3(\mathrm{PSL}(2,\mathbb{C}),P)\to\widehat{\mathcal{B}}(\mathbb{C})\). Next the author considers the more general setting of a tame \(3\)-manifold \(M\) with a boundary-parabolic representation \(\rho:\pi_1(M)\to\mathrm{PSL}(2,\mathbb{C})\) (it maps each peripheral subgroup to a parabolic subgroup); now, \(M\) may not be hyperbolic and the boundary components of \(\overline M\) may not be tori. Such a \(\rho\) defines a fundamental class \([\rho]\in H_3(\mathrm{PSL}(2,\mathbb{C}),P)\) by using a decoration consisting of a conjugation of the image of each peripheral subgroup into \(P\). A topological triangulation of \(M\) is used to construct a representative of \([\rho]\) in the complex of decorated ideal simplices by using a developing map. The image of \([\rho]\) in \(\widehat{\mathcal{B}}(\mathbb{C})\) is independent of the decoration, and the complex volume of \(\rho\) is defined by the formula \(i(\text{Vol}(\rho)+i\,\text{CS}(\rho))=R\circ\Psi([\rho])\). This equals the complex volume of \(M\) when \(\rho\) is the geometric representation of a hyperbolic \(3\)-manifold.
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