Boundary Curves of Incompressible Surfaces

This is a Tex version, made in 2004, of a paper that appeared in Pac. J. Math. 99 (1982), 373-377, with some revisions in the exposition. 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 1 E Q[f 1= 0g, 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 H 1 N@MO. Ignoring orientations amounts to factoring out multiplication by1 in each component, yielding a quotient of H 1 N@MO which can be identified with the set CSN@MO of isotopy classes of curve systems in@M. Each factorH1NTiO ofH1N@MO is the integer lattice inH1NTi;RO R 2 , and H 1 NT i ;RO= 1 is a cone, so CSN@MO can be viewed as the integer lattice in a product of cones, the space H1N@M;RO=Z n where Z n acts by inversions in the factors H 1 NT i ;RO.