The Chern-Simons Invariants of Hyperbolic Manifolds Via Covering Spaces

One important invariant of a closed Riemannian 3-manifold isthe Chern–Simons invariant 1]. The concept was generalizedto hyperbolic 3-manifolds with cusps in 11], and to geometric(spherical, euclidean or hyperbolic) 3-orbifolds, as particularcases of geometric cone-manifolds, in 7]. In this paper, westudy the behaviour of this generalized invariant under changeof orientation, and we give a method to compute it for hyperbolic3-manifolds using virtually regular coverings 10]. We confineourselves to virtually regular coverings because a coveringof a geometric orbifold is a geometric manifold if and onlyif the covering is a virtually regular covering of the underlyingspace of the orbifold, branched over the singular locus. Thereforeour work is the most general for the applications in mind; namely,computing volumes and Chern–Simons invariants of hyperbolicmanifolds, using the computations for cone-manifolds for whicha convenient Schläfli formula holds (see 7]). Among otherresults, we prove that every hyperbolic manifold obtained asa virtually regular covering of a figure-eight knot hyperbolicorbifold has rational Chern–Simons invariant. We giveexplicit examples with computations of volumes and Chern–Simonsinvariants for some hyperbolic 3-manifolds, to show the efficiencyof our method. We also give examples of different hyperbolicmanifolds with the same volume, whose Chern–Simons invariants(mod ) differ by a rational number, as well as pairs of differenthyperbolic manifolds with the same volume and the same Chern–Simonsinvariant (mod ). (Examples of this type were also obtainedin 12] and 9], but using mutation and surgery techniques,respectively, instead of coverings as we do here.) 1991 MathematicsSubject Classification 57M50, 51M10, 51M25.