Cut numbers of 3-manifolds

We investigate the relations between the cut number, c(M), and the first Betti number, b ₁(M), of 3-manifolds M. We prove that the cut number of a "generic" 3-manifold M is at most 2. This is a rather unexpected result since specific examples of 3-manifolds with large 61 (M) and c(M) ≤ 2 are hard to construct. We also prove that for any complex semisimple Lie algebra g there exists a 3-manifold M with b ₁ (M) = dim g and c(M) ≤ rank g. Such manifolds can be explicitly constructed.