Boundary Curves of Incompressible Surfaces

Let M be an compact orientable 3 manifold whose boundary ∂M consists of a single torus. If a meridian and longitude in this torus are chosen, then isotopy classes of smoothly embedded circles in ∂M that do not bound disks correspond bijectively with elements of QP = Q ∪ {1/0} , regarded as slopes of these curves. We show in this paper that the set of slopes coming from boundary curves of incompressible, ∂ incompressible surfaces in M is finite. There is also a generalization to the case that ∂M consists of n tori T1, ··· , Tn . Given a curve system in ∂M consisting of finitely many disjoint smoothly embedded circles that do not bound disks, then by choosing parallel orientations for the circles in each component of ∂M , we get an element of H1(∂M) . Ignoring orientations amounts to factoring out multiplication by ±1 in each component, yielding a quotient of H1(∂M) which can be identified with the set CS(∂M) of isotopy classes of curve systems in ∂M . Each factor H1(Ti) of H1(∂M) is the integer lattice in H1(Ti;R) ≈ R , and H1(Ti;R)/±1 is a cone, so CS(∂M) can be viewed as the integer lattice in a product of cones, the space H1(∂M ;R)/Z n 2 where Z n 2 acts by inversions in the factors H1(Ti;R) .