Exceptional manifolds for generalized Schoenflies theorem (Q1067661)

It is shown that if \(1<q<p-1\) then every inessential embedded \(S^{p+q}\) in a \((p+q+1)\)-manifold M bounds an embedded disc if and only if every inessential embedded \(S^ p\times S^ q\) in M bounds an embedded \(D^{p+1}\times S^ q\) or \(S^ p\times D^{q+1}\). (The result and the idea behind it are reminiscent of a theorem of Alexander on tori in \(S^ 3.)\) It is remarked that if the ambient manifold is 2-connected then Irwin's embedding theorem may be used to extend the result to the cases \(1<q\leq q\). However a counterexample is given to show that it does not hold in general when \(q=1\).