Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds

In this paper we prove some results on combinatorics of triangulations of 3-dimensional pseudo-manifolds, improving on results of [NZ], and apply them to obtain a simplicial formula for the Chern-Simons invariant of an ideally triangulated hyperbolic 3-manifold. Combining this with [MN] gives a simplicial formula for the η invariant also. In effect, the main ingredient in the formula is the sum of the “Rogers dilogarithm” of the complex parameters of the ideal tetrahedra of the triangulation, but the choice of the appropriate branch of the Rogers dilogarithm for each simplex involves unexpected combinatorics (cf. Remark 4 below for this interpretation of the formula). The combinatorial part of this paper (Sects. 4–6) is self-contained and of independent interest. For instance, T. Yoshida [Y2] has used these combinatorics (in the version of [NZ]) to study character varieties and boundary slopes in the spirit of Culler-Shalen [CS]. In the remainder of this Introduction we summarize the application to the ChernSimons invariant. All manifolds in this paper are assumed to be oriented. If M is a complete hyperbolic 3-manifold which is compact, then its Chern-Simons invariant CS(M) is well-defined modulo 2π. If M is non-compact then Bob Meyerhoff has shown in [M] that there is still a natural definition of CS(M) which is well-defined modulo π . Let V(M) = Vol(M)+ i CS(M), which is well-defined modulo i2πZ or iπZ. A formula for V(M ) mod iπZ, as M ′ varies over the hyperbolic Dehn surgeries on M , was conjectured in [NZ] and proved by T. Yoshida in [Y1]. This formula is of theoretical interest but is not practical for actually computing V(M ). It is not hard to reverse the derivation in [NZ] to obtain a computable formula in terms of an ideal triangulation of M . However, the resulting formula involves an unknown constant which depends on the combinatorics of the triangulation of M and which seems hard to determine in general. Using a result of Dupont [D] we can find a version of the formula in which this constant is at least a rational multiple of iπ (Theorem 1 below). Using a more careful analysis of the relevant combinatorics we are able to give a version (Theorem 2) in which the constant is conjecturally in (iπ/6)Z and is thus determined up to a six-fold ambiguity (since it lives in C mod iπZ ). Suppose M has an ideal triangulation which subdivides it into n ideal tetrahedra