Essential disks and semi-essential surfaces in 3-manifolds

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily ∂-incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory and is analogous to the analysis of incompressible surfaces using branched surfaces. The surfaces in question are carried by a finite set of “carriers.” We hope our theory of carriers will be used in other papers to describe objects representing limits of essential disks in a handlebody or in a 3-manifold having compressible boundary.