On Volumes and Chern-Simons Invariants of Geometric 3-Manifolds

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J. Math. Sci. Univ. Tokyo
Vol. 3 (1996), No. 3, Page 723--744.

Hilden, Hugh M. ; Lozano, MarÃa Teresa ; Montesinos-Amilibia, JosÃ© MarÃa
On Volumes and Chern-Simons Invariants of Geometric 3-Manifolds
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

Abstract:
Let

$M_n(K)$ be the hyperbolic 3-manifold obtained by

$n$ -cyclic covering of

$S^3$ branched over a hyperbolic knot

$K$ . A method to compute the volume and the Chern-Simons invariant of

$M_n(K)$ is given. The value of the volume of

$M_n$ is

$n$ times the value of the volume of the corresponding hyperbolic orbifold. This volume can be obtained by appying the SchlÃ¤ffli Formula for the volume to the cone-manifold family,

$(K,Î± )$ , with singularity

$K$ . The same approch is followed for the Chern-Simons invariant, after proving a "SchlÃ¤ffli Formula" for a generalized Chern-Simons function on the family of cone-manifold structures

$(K,Î± )$ .

Mathematics Subject Classification (1991): Primary 57M50, 51M10, 51M25

Mathematical Reviews Number: MR1432115

Received: 1996-04-08

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