# zbMATH — the first resource for mathematics

The volume and Chern-Simons invariant of a representation. (English) Zbl 1246.58019
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)], 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.

##### MSC:
 58J28 Eta-invariants, Chern-Simons invariants 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
Snap
Full Text:
##### References:
 [1] S. Boyer and X. Zhang, On Culler-Shalen seminorms and Dehn filling , Ann. of Math. (2) 148 (1998), 737–801. JSTOR: · Zbl 1007.57016 [2] D. Calegari, Real places and torus bundles , Geom. Dedicata 118 (2006), 209–227. · Zbl 1420.57047 [3] D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of $$3$$ -manifolds, Invent. Math. 118 (1994), 47–84. · Zbl 0842.57013 [4] J. L. Dupont and C. K. Zickert, A dilogarithmic formula for the Cheeger-Chern-Simons class , Geom. Topol. 10 (2006), 1347–1372. · Zbl 1130.57013 [5] S. Goette and C. K. Zickert, The extended Bloch group and the Cheeger-Chern-Simons class , Geom. Topol. 11 (2007), 1623–1635. · Zbl 1201.57019 [6] O. Goodman, Snap, http://www.ms.unimelb.edu.au/$$\sim$$snap J. Hoste and P. D. Shanahan, Trace fields of twist knots , J. Knot Theory Ramifications 10 (2001), 625–639. · Zbl 1003.57014 [7] C. Maclachlan and A. W. Reid, The arithmetic of hyperbolic $$3$$-manifolds , Grad. Texts in Math. 219 , Springer, New York, 2003. · Zbl 1025.57001 [8] S. Mac Lane, Homology , reprint of the 1975 edition, Classics in Mathematics, Springer, Berlin, 1995. [9] R. Meyerhoff, “Density of the Chern-Simons invariant for hyperbolic $$3$$-manifolds” in Low-Dimensional Topology and Kleinian Groups (Coventry/Durham, England, 1984) , London Math. Soc. Lecture Note Ser. 112 , Cambridge Univ. Press, Cambridge, 1986, 217–239. · Zbl 0622.57008 [10] W. D. Neumann, “Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic $$3$$-manifolds” in Topology ’90 (Columbus, Ohio, 1990) , Ohio State Univ. Math. Res. Inst. Publ. 1 , de Gruyter, Berlin, 1992, 243–271. · Zbl 0768.57006 [11] -, Extended Bloch group and the Cheeger-Chern-Simons class , Geom. Topol. 8 (2004), 413–474. · Zbl 1053.57010 [12] W. D. Neumann and J. Yang, Bloch invariants of hyperbolic $$3$$ -manifolds, Duke Math. J. 96 (1999), 29–59. · Zbl 0943.57008 [13] W. D. Neumann and D. Zagier, Volumes of hyperbolic three-manifolds , Topology 24 (1985), 307–332. · Zbl 0589.57015 [14] E. H. Spanier, Algebraic Topology , corrected reprint, Springer, New York, 1981.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.