Math Forum / Mathematics / Research / April 2009

I seem to have found the answer to my question. Swarup, GA, 1975: An
unknotting criterion. J. Pure Appl. Algebra, 6 (1975) and Dicks, W.
and Dunwoody, M.J. Groups acting on Graphs, Cambridge studies in
advanced mathematics 17, Cambridge University Press, Cambridge-New
York-Melbourne (1989), are cited by Hillman, Four-manifolds,
Geometries, and Knots as providing the result that if a codimension-2
sphere bounds an immersed codimension-1 disk which has no double-
points on its boundary, then the complement of the sphere is homotopy
equivalent to the circle. The results of Levine et al. then show that
in dimensions five and up, there is an embedded disk, whereas in
dimension four this is true in the topological category by Freedman,
but unknown (to my knowledge) in the PL/smooth categories.

This does raise a related question. In dimension 3 and 4, the
existence of such an immersed disk is guaranteed if the fundamental
group vanishes. But in higher dimensions, is it possible to have a non-
trivially embedded sphere where the fundamental group of the
complement vanishes? Clearly by the result of Swarup it would have to
have some nontrivial homotopy group.