Problems for Chern - Simons Invariants

This paper makes certain observations regarding some conjectures of Milnor and Ramakrishnan in hyperbolic geometry and algebraic K-theory. As a consequence of our observations, we obtain new results and conjectures regarding the rationality and irrationality of Chern-Simons invariants of hyperbolic 3-manifolds. In this paper, by a hyperbolic 3-manifold, we shall mean a complete, oriented hyperbolic 3-manifold with finite volume. So a hyperbolic 3-manifold is compact or has finitely many cusps as its ends (see e.g. [20]). By a cusped manifold we shall mean a non-compact hyperbolic 3-manifold. A hyperbolic 3-manifold M is a quotient of the hyperbolic 3-space H3 by a discrete subgroup Γ of PSL2(C) with finite covolume. The isometry class of M determines the discrete subgroup up to conjugation. To each subgroup Γ of PSL2(C), we can associate the trace field of Γ, that is, the subfield of C generated by traces of all elements in Γ. The trace field clearly depends only on the conjugacy class of Γ, so one can define it to be the trace field of the hyperbolic 3-manifold. However, the trace field is not an invariant of commensurability class of Γ, although it is not far removed. There is a notion of invariant trace field due to Alan Reid (see [17]) which is a subfield of trace field and does give an invariant of commensurability classes. Let Γ be the subgroup of Γ generated by squares of elements of Γ. The invariant trace field k(M) of M is defined to be the trace field of Γ. We will use invariant trace fields throughout this paper. We want to emphasize here that each invariant trace field is a number field together with a specific embedding in C. Recall that a number field is totally real if all its embeddings into C have image in R and totally imaginary if none of its embeddings has image in R. A CM-field is a number field with complex multiplication, i.e., it is a totally imaginary quadratic extension of a totally real number field. A familiar class of CM-fields is the class of cyclotomic fields. We will also use a slightly more general notion. We say that an embedding σ : F →֒ C of a number field F is a CM-embedding if σ(F ), as a subfield of C, is an imaginary quadratic extension of a totally real field. So F is a CM-field if and only if all its embeddings are CM-embeddings. The Chern-Simons invariant of a compact (4n−1)-dimensional Riemannian manifold is an obstruction to conformal immersion of the Riemannian manifold in Euclidean space [4]. For hyperbolic 3-manifolds Meyerhoff [9] extended the definition to allow manifolds with cusps. The Chern-Simons invariant CS(M) of a hyperbolic 3-manifold M is an element in R/πZ. It is rational (also called torsion) if it lies in πQ/πZ. The main application of our paper to Chern Simons invariants is the following theorem.