  A Heegaard splitting of an open 3-manifold is the partition of the manifold
into two non-compact handlebodies which intersect on their common boundary.
This paper proves several non-compact analogues of theorems about compact
Heegaard splittings. The main theorem is: if N is a compact, connected,
orientable 3-manifold with non-empty boundary, containing no S^2 components,
and if M is obtained from N by removing the boundary then any two Heegaard
splittings of M are properly ambient isotopic. This is a non-compact analogue
of the classifications of splittings of (closed surface) x I and (closed
surface) x S^1 by Scharlemann-Thompson and Schultens. Work of Frohman-Meeks and
a non-compact analogue of the Casson-Gordon theorem on weakly reducible
Heegaard splittings are key tools.
