The Chern-Simons invariants of hyperbolic manifolds via covering spaces

Abstract

The Chern-Simons invariant was extended to 3-dimensional geometric cone manifolds in [H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia, J. Math. Sci. Univ. Tokyo 3 (1996), no. 3, 723–744; MR1432115 (98h:57056)]. The present paper is about the behavior of this generalized invariant under change of orientation and with respect to virtually regular coverings. (A virtually regular cover is a cover with the property that the branching index is constant along the fiber over each point of the branching set.) As one might suspect, CS(−M)=−CS(M). However, unlike the volume, the Chern-Simons invariant is not multiplicative with respect to branched coverings. There is a correction term depending on the intersection number of longitudes of inverse images of the singular set with the inverse image of the longitude of the singular set. The paper concludes with applications of the main formula to specific examples.