This paper is about maps
of compact 3-manifolds which map the boundary of the domain (possibly
nonhomeomorphically) into the boundary of the range. F. Waldhausen has
shown that such a map between compact, orientable, irreducible S-manifolds
with nonempty, incompressible boundary is homotopic to a homeomorphism
if and only if the map induces an isomorphism at the fundamental group
level. The main theorem of this paper states that the above theorem remains
valid if the assumption of incompressible boundary is dropped. A study of
disk sums of bounded 3-manifolds will be required in order to prove the
above-mentioned theorem. This investigation involves theorems about disk sums of
bounded 3-manifolds analogous to the classical Kneser theorem for closed
3-manifolds.
