O ct 2 00 6 K - invariants of hyperbolic 3 - manifolds Igor Nikolaev

An invariant of the 3-dimensional manifolds appearing in the Ktheory of certain operator algebras is introduced. The operator algebra in question is a crossed product C-algebra attached to the monodromy of a 3-dimensional manifold M which fibers over the circle. The invariant is a triple (Λ, i, [I]) consisting of an order Λ in an algebraic number field K, an embedding i : K → R and an equivalence class of ideals [I] in Λ. As a consequence, one gets a rational number d(Λ) which a homotopy invariant of M . Our approach is a blend of topology and K-theory of operator algebras.