Incompressible maps of surfaces
by
Ulrich Oertel
Rutgers University, Newark

Suppose [`M] is a manifold obtained from a manifold M by Dehn filling, and suppose for simplicity that [`M] is closed and M subset [`M] has one boundary torus. Suppose [`M] is Haken, containing an incompressible surface S. When intersections of S with the Dehn filling solid torus are suitably minimized, it is well-known that the surface [^S]=S \cap M is incompressible and \partial-incompressible. We prove a similar result for incompressible maps of surfaces, where incompressible maps of surfaces are defined in such a way as to avoid the issue of the simple loop conjecture for maps from surfaces to 3-manifolds. Thus, for a closed surface S, a map f:S --> [`M] is incompressible if every simple loop in S which is mapped to a null-homotopic curve in [`M] bounds a disc in S. A corollary of the most general technical result we prove says that if K subset S³ is a knot in the 3-sphere and X is the knot exterior, then there is an essential map [^f]:(P, \partialP) --> (X, \partialX) of a planar surface, with [^f] restricted to one component of \partialP mapping to an integer slope embedded simple closed curve in \partialX and with all other components of \partialP mapping to meridian circles in \partialX.

Date received: February 6, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagh-38.